Wigner Active and Passive Rotation Matrices Applied to NMR Tensor

Home , Rotation of axes, Wigner rotation

Received: 17 November 2015 | Revised: 9 November 2016 | Accepted: 31 December 2016 DOI: 10.1002/cmr.a.21385

RESEARCH ARTICLE

Wigner active and passive rotation matrices applied to NMR tensor

Pascal P. Man

Sorbonne Universités, UPMC Univ Paris 06, CNRS, FR2482, Institut des Abstract matériaux de Paris-Centre, F-75005 Paris, NMR Hamiltonians are double contraction of two spherical rank-2 tensors, their France space parts are represented by a spherical tensor and their spin parts are composed Correspondence of spherical tensor operators. The comprehension of modern NMR experiments is Sorbonne Universités, UPMC Univ Paris very often based on the rotation of these tensors. We present the active and pas- 06, CNRS, FR2482, Institut des matériaux sive rotations in a progressive way from position vector to spherical tensor opera- de Paris-Centre, F-75005 Paris, France Email: [email protected] tor via space function, spherical harmonic, and vector operator. The passive rotation of a physical quantity is described by the rotation of coordinate system. Both the left- and right-handed rotation conventions are applied whereas the right-handed rotation convention is mainly used in the literature. Throughout the article, we explore the equivalence between the active rotation of a physical quantity in one direction and the rotation of the coordinate system in the opposite direction. The article presents redundant mathematical demonstrations between active and passive rotations, but they clarify the meanings of some important expressions not well developed in the literature.

KEYWORDS active rotation, canonical transformation, Euler angles, passive rotation, Wigner rotation matrix

1 | INTRODUCTION In this article, we present the active and passive rotations in a progressive way from position vector to spherical NMR Hamiltonians may be defined with Cartesian or tensor operator via space function, spherical harmonic, and spherical tensors.1 In the latter case, they are double con- vector operator. The passive rotation of a physical quantity tractions of two spherical rank-2 tensors, one representing is described by the rotation of coordinate system. the space part and the other the spin part. Very often, the Section 2 describes two equivalent points of view for space part has to be expressed in several coordinate sys- the rotation of axes by an angle h in a plane of 3-D Eucli- tems (principal-axis system, MAS rotor system, laboratory- dean space using left- and right-handed conventions. axis system), whereas the spin part is mainly described in Throughout the article, right-handed coordinate systems are the laboratory-axis system in simple pulse experiments. For used for left- and right-handed rotations. example, the electric-field gradient tensor of quadrupole Active, passive, and canonical transformations are spins in powder is well studied in static, MAS, DOR, extensively discussed in the literature. As we mainly deal DAS, and MQMAS experiments. Multi-pulse experiments with spin operators and the density matrix operator in 1 fi on spin-2 systems reduce the homonuclear dipole-dipole NMR, we de ne not only the active and passive rotations Hamiltonian. These methods aim to narrow the line widths of operators but also the canonical transformation of opera- using rotations. Descriptions of rotation in classical and tors in Section 3. quantum mechanics are numerous in literature. We explore Section 4 details the active and passive rotations of the active and passive points of view, which are sources of space function Ψ(r) about a single axis using active RA(a, confusion. z) and passive RP(a, z) rotation operators. We also use

Dirac notations,2 which have the advantages to represent not only the space function by ket state jiW but also the coordinate system by bra position basis hjr . This is not the case in standard notation about the rotation of space function such as RA(a, z) Ψ(r). We show that right- and left- handed rotations of coordinate system about z-axis by angle a are expressed mathematically by the application of the operators RP(a, z) and RA(a, z) to bra position basis, respectively. In contrast, right- and left-handed active rotations of space function about z-axis by angle a are expressed by the application of RA(a, z) and RP(a, z) to ket state, respectively. In Section 5, active and passive rotation operators for space function are expressed in terms of Euler angles about fixed or rotated axes. These rotation operators are extensively discussed in the literature. Some examples are provided for clarifying the notations used in these rotation operators. We recall in Section 6 the reasons for which rotations of spin components or those of spherical tensor operator components should be performed about fixed axes. As a result, we only present active and passive rotations about fixed axes in the remaining of the article. Section 7 presents the Wigner active and passive rota- FIGURE 1 (A) Right-handed counter-clockwise rotations about 0 tion matrices because they are involved in the active and the axis n; (B) left-handed clockwise rotations about the axis n passive rotations of spherical harmonics, spherical tensors, and spherical tensor operators. with spherical components. In each case we provide a sim- Section 8 presents the spherical harmonics, which are ple example. space functions. Properties of spherical tensors are Section 11 provides the definitions of spherical tensor deduced from those of spherical harmonics. Furthermore, operator and an application example about the excitation of spherical harmonics are related to Wigner rotation matri- a spin by an off-resonance RF pulse using Wigner rotation ces via simple relations. These relations allow us to matrices. determine the type (active or passive) of Wigner rotation matrix. Section 9 details the active rotations of spherical har- 2 | ROTATION IN 3-D EUCLIDEAN monics about fixed axes and the rotations of coordinate SPACE system about fixed axes, which allows us to define the rotations of spherical tensors. For simplicity, we replace A right-handed rotation in 3-D Euclidean space is defined the passive rotations of spherical harmonics by rotations of by an axis of rotation (n) through the origin (O) and an coordinate system. We explore the two points of view (ac- angle (h) about that axis (Figure 1A). This rotation is tive and passive) of the two rotation operators denoted as R(n, h). The positive sense of rotation3 is coun- fixed a; b; c fixed c; b; a RA ð Þ and RP ð Þ in Dirac notations. In each ter-clockwise when looking down the axis n toward the ori- case we provide a simple example. gin O. This description of rotation only involves the right Section 10 details the rotations of vector operator and hand. A second description of rotation involves the two those of coordinate system. A vector operator is the sim- hands: right-handed rotation (Figure 1A) and left-handed plest spherical tensor operator, whose rank is 1. In NMR, rotation (Figure 1B). In left-handed rotation, the positive the spin I is a vector operator. The study of rotation of sense of rotation is clockwise when looking down the axis vector operator is more complex than that of space func- n0 toward the origin O. tion because vector operator is sandwiched by the rotation Figure 2 adopted from Jackson4 shows a right-handed operator and its adjoint operator. In contrast, the rotation orthonormal Cartesian coordinate system (O, x, y, z). The operator is sandwiched by a bra position basis and a ket point B in the fgx; y plane is specified by the position vec- state in the case of space function. We discuss the rota- tor V with components (x1, y1, 0) along the axes as shown tions of vector operator with Cartesian components and in Figure 2A. The latter shows that a left-handed rotation MAN | 3of64

FIGURE 3 Two equivalent points of view for a rotation of axes by an angle a in a plane: (A) the vector V is fixed in the plane while a FIGURE 2 Two equivalent points of view for a rotation of axes the axes are right-hand rotated about z-axis by a positive angle in fi by an angle a in a plane: (A) the vector V is fixed in the plane, passive point of view; (B) the axes are xed while the vector V is 0 whereas the axes are left-hand rotated about z-axis by a positive angle left-hand rotated to a new position V about z-axis by a positive angle a a in passive point of view; (B) the axes are fixed while the vector V in active point of view is right-hand rotated to a new position V0 about z-axis by a positive angle a in active point of view Heine10 presents a similar example (Figure 3). Fig- ure 3A shows that a right-handed rotation about z-axis by about the z-axis by angle a applied to the coordinate sys- angle a applied to the coordinate system transforms (O, x, tem transforms (O, x, y, z) into (O, x0, y0, z0), keeping the y, z) into (O, x0, y0, z0), keeping the point B fixed. Fig- point B fixed. Figure 2B shows that a right-handed rotation ure 3B shows that a left-handed rotation about z-axis by about the z-axis by angle a applied to the position vector angle a applied to the position vector transforms V into V0, transforms V into V0, keeping the coordinate system (O, x, keeping the coordinate system (O, x, y, z) fixed. In either fi 0 ; 0 ; 0 ; 0 ; y, z) xed. In either case, the primed coordinates ðx1 y1 0Þ case, the primed coordinates ðx1 y1 0Þ are given by are given by 0 1 0 10 1 x0 cos h sin h 0 x 0 1 0 10 1 @ 1 A @ A@ 1 A 0 h h y0 ¼ sin h cos h 0 y : (2) x1 cos sin 0 x1 1 1 @ 0 A @ h h A@ A: 0 0010 y1 ¼ sin cos 0 y1 (1) 0 0010 Equation (2) is deduced from Eq. (1) by changing the We replace h in Eq. (1) with the positive angle a used sign of h as the senses of rotation described by angles a in in Figure 2. This means that left-handed rotation of the Figure 3 are the opposite of those in Figure 2. This means coordinate system (O, x, y, z) keeping the position vector that right-handed rotation of the coordinate system (O, x, y, V fixed is equivalent to right-handed rotation of the posi- z) keeping the position vector V fixed is equivalent to left- tion vector V keeping the coordinate system (O, x, y, z) handed rotation of the position vector V keeping the coor- fixed. dinate system (O, x, y, z) fixed. A single transformation matrix (Eq. (1)) can represent In practice, Figure 2 is suitable for the analysis of rotation either left-handed rotation of coordinate system or right- of vector because Figure 2B describes the right-handed rota- handed rotation of position vector. An important point to tion of vector.9,11 In contrast, Figure 3 is suitable for the note is that we use the same symbol a for two opposite analysis of rotation of coordinate system because Figure 3A senses of rotation.5–9 These opposite senses are shown by describes the right-handed rotation of coordinate system.12,13 arrows. This point is not emphasized enough in the litera- Figure 4 shows the typical rotation angle notations suitable ture. for rotation of vector. In this case, all rotations are described 4of64 | MAN

x, y, z) keeping the position vector V fixed. Another way to express the same thing is the following. Right-handed counter-clockwise rotation of the position vector V keeping the coordinate system (O, x, y, z) fixed is equivalent to right-handed clockwise rotation of the coordinate system (O, x, y, z) keeping the position vector V fixed.

3 | ACTIVE, PASSIVE, AND CANONICAL TRANSFORMATIONS OF OPERATOR

Thaller17 suggests a test of the isotropy of space in performing an experiment with a physical system S in a fixed coordinate system C and then repeating the experiment in a rotated coordinate system C0. The response of S to an experiment should not depend on its orientation in space. This can be done in three ways. 1. Rotate the system S but not the observer (Figure 5A). The experiment on the rotated physical system S0 is described by an observer sitting in the fixed coordinate system C. FIGURE 4 Two equivalent points of view for a rotation of axes 18 by an angle a in a plane: (A) the vector V is fixed in the plane while the The observer represents the measuring apparatus. axes are right-hand rotated about z-axis by a negative angle a in 2. Rotate the observer but not the system S (Figure 5B). passive point of view; (B) the axes are fixed while the vector V is right- The experiment on the system S in the fixed coordinate hand rotated to a new position V0 about z-axis by a positive angle a in system C is described by an observer sitting in the active point of view rotating coordinate system C0. 3. Rotate the system S and the observer (Figure 5C). The procedure consists in rotating the whole experimental 1,3,11,14–16 by right-handed rotations in right-handed coordinate setup (the physical system and the observer). The exper- systems. The sense of rotation is either negative as shown in iment on the rotated system S0 is performed and Figure 4A by angle a or positive as shown in Figure 4B by described in the rotated coordinate system C0. If the angle a. As a result, two rotation matrices are involved. That space is isotropic, the rotated system S0 in the rotated corresponding to Figure 4A describes a right-handed rotation coordinate system C0 behaves exactly as the system S 3 of coordinate system about z-axis by angle h: did in the fixed coordinate system C.

0 1 0 10 1 In classical mechanics, “rotate the system S but not the x0 cos h sin h 0 x 1 1 observer” is called an active transformation, whereas “ro- @ y0 A @ sin h cos h 0 A@ y A: (3) 1 ¼ 1 ” 0 0010 tate the observer but not the system S is called a passive transformation.17 This angle h is negative (a) in Figure 4A. That corre- In quantum mechanics, the physical system is repre- sponding to Figure 4B describes a right-handed rotation of sented by a state vector in Hilbert space of the physical vector about z-axis by angle h:3 system and observables are provided by measuring apparatus (or an observer in classical mechanics) sitting in a coor- 0 1 0 10 1 dinate system. As a result, moving a measuring apparatus 0 x cos h sin h 0 x 19 1 1 means moving the coordinate system. Observables (or @ y0 A ¼ @ sin h cos h 0 A@ y A: (4) 1 1 more generally operators) T defined before moving the 0 0010 measuring apparatus becomes observables T 0 after moving This angle h is positive (a) in Figure 4B. the measuring apparatus. In other words, observables are Replacing h by a in Eq. (3) and by a in Eq. (4) yield defined for a coordinate system. 0 ; 0 ; 20 21 5 the same primed coordinates ðx1 y1 0Þ. Therefore, right- Shankar, Rembold, and Auletta and Wang provide handed positive rotation of the position vector V keeping the definition of active and passive transformations. The the coordinate system (O, x, y, z) fixed is equivalent to state vectors in Hilbert space are affected in an active trans- right-handed negative rotation of the coordinate system (O, formation and left alone in the passive case. The operators MAN | 5of64

FIGURE 5 Transformations of physical system: (A) In an active transformation, the physical system S is transformed with respect to a fixed coordinate system C. (B) In a passive transformation, the coordinate system C is transformed while the physical system S is left unchanged. (C) In a canonical transformation, both the physical system S and the coordinate system C are simultaneously transformed in the same way. C and C0 are called body-attached coordinate system or the measurement apparatus are left alone in an active system that remains fixed in an active transforma- transformation and are affected in the passive case. This tion. But the matrix elements of operators are usually presentation of active and passive transformations opposes changed:25 transformed states against transformed operators. The coor- RA W T W ! R W TR W : (7) dinate system is implicitly related to measuring apparatus 1 2 A 1 A 2 or operators. In calculating matrix elements, the transformation opera- 22 23 Thompson and Lipkin also distinguish three types of tor RA can always be shifted from the state vectors to the transformations as does Thaller:17 active, passive, and operators T. Equation (7) becomes9 canonical transformations. State vectors are represented by R kets. Dynamical operators are denoted by T and transfor- W W A W y W : hj1 Tji2 ! hjð1 RAÞ TRA 2 (8) mation operators by Ri. 26 9 In NMR, we essentially manipulate spin operators and Sakurai and Napolitano and Brink and Satchler also distinguish two approaches to describing RA. density matrix operators, scarcely the state vector of the R W A W spin system. Therefore, we have to define the active and Approach 1: ji! jiRA is a transformation of state passive rotations of an operator. We focus on operators and vector with operators T unchanged (Eq. (7)). RA y coordinate systems, whereas state vector is hidden. The Approach 2: T !ðRAÞ TRA is a transformation of active transformation may be involved in one or two coor- operators T with state vectors unchanged (Eq. (8)). 1 In other words, we can either apply the active rotation dinate systems. Two coordinate systems are involved in R W A W passive transformation. operator to state vector (ji! jiRA ) and leave operators unchanged, or we can leave state vectors unchanged R and rotate operators [T A R yTR ]. Both yield the 3.1 Active transformation of operator !ð AÞ A | same matrix elements: Three entities are involved in our approach: state vector, operator, and right-handed coordinate system. An active RA y hjW1 TjiW2 ! hjRAW1 TRjiAW2 ¼ hjðW1 RAÞ TRA W ; transformation operator R changes the physical system but 2 A (9) leaves the coordinate system unchanged.24 That is,22 so the same physics.

If RA is a right-handed counter-clockwise (Figure 1A) R W A W W W0 ; ji! RAji¼ jiRA ¼ ji (5) active rotation operator, the transformation defined in Eq. (7) describes a right-handed counter-clockwise rota- jiW0 is the transformed state vector and tion of the physical system, keeping the operators T R T !A T: (6) fixed. It is equivalent to saying that the transformation defined in Eq. (8) describes a right-handed clockwise The operators T are unchanged by an active transfor- rotation of the operators T, keeping the physical system mation, because operators are defined for a coordinate fixed. The same matrix element describes either the 6of64 | MAN

RP 0 y right-handed counter-clockwise rotation of the state T ! T ¼ðRPÞ TRP as a left-handed rotation of the coor- vectors or the right-handed clockwise rotation of the dinate system, keeping the operators T fixed. It is also operators. called left-handed passive rotation of operators T. To distinguish these two ways to describe a matrix ele- RA ment, we call the first way (jiW ! jiRAW ) a right-handed (Figure 1A) active rotation of state vector and the second 3.3 | Canonical transformation of operator RA y way [T !ðRAÞ TRA] a left-handed (Figure 1B) active A canonical transformation by the operator R changes both rotation of operators T. In both ways, the coordinate system A the physical system and the coordinate system. We have,22 remains fixed.

RA 0 jiW ! RAjiW ¼ jiRAW ¼ jiW ; (16) 3.2 | Passive transformation of operator and

A passive transformation operator R leaves the physical R P T !A T0: (17) system unchanged, but changes the coordinate system. The change in coordinate system is also called the change in Recall in quantum mechanics, observables are provided description.27 This means two coordinate systems are by a measuring apparatus. Moving a measuring apparatus involved in passive transformation. We have22 means moving the coordinate system. The transformation deﬁned by Eqs. (16) and (17) is R jiW !P jiW (10) called canonical because it transforms state vector and operators but leaves the matrix elements invariant:28 and

R T !P T0; (11) W W RA W0 0W0 W W : hj1 Tji2 ! 1 T 2 ¼ hj1 Tji2 (18) T and T 0 being the same operators expressed in terms of the old and new coordinate systems, respectively. Matrix The important point is that matrix elements do not elements are usually changed under a passive transforma- change by a unitary transformation of state vector and tion: a unitary transformation of observables. An important class of canonical transformations are those which R 23 hjW TjiW !P hjR W TRji¼W hjðW R ÞyTR jiW correspond to symmetries of the physical system. 1 2 P 1 P 2 1 P P 2 They are translations, rotations, space inversion, and ¼ W T0jiW : 1 2 time-reversal. (12) Equation (16) allows us to rewrite the matrix element in Eq. (18) as As it is well-known (see Section 4) that y y W0 T0 W0 R W T0 R W W R T0R W ðRAÞ ¼ RP; (13) 1 2 ¼ A 1 jiA 2 ¼ hj1 ðÞA Aji2 ¼ hjW TjiW : the operators T 0 in Eqs. (11) and (12) are defined by19,23 1 2 (19) As a result, Eq. (17) becomes RP 0 y y T ! T ¼ðRPÞ TRP ¼ RATðRAÞ : (14) RA 0 y: Therefore, Eq. (12) becomes T ! T ¼ RATðRAÞ (20) R W W P W 0W W y W The transformation in Eq. (20) differs with hj1 Tji2 ! T ¼ ðRPÞ TRP R 1 2 1 2 T !ðA R ÞyTR discussed in active transformation of oper- W yW : A A ¼ 1 RATðRAÞ 2 (15) ator. As the latter is a left-handed active rotation of operators T in a fixed coordinate system, the transformation in If RA is a right-handed counter-clockwise active rotation operator, it follows that R is also a right-handed counter- Eq. (20) is a right-handed active rotation of operators T in P fi clockwise, but passive rotation operator. We have just pro- a xed coordinate system. R posed that the transformation T A R yTR deduced In quantum mechanics textbooks on symmetry operations !ð AÞ A 29 from Eq. (8) describes a left-handed active rotation of oper- (Chapter 3 in Ballentine, Section 14 of Chapter 3 in 30 ators T, keeping the coordinate system fixed. We deduce Biedenharn and Louck, Complement BVI in Cohen-Tan- 31 32 that the transformation T RP T0 R T R y in Eq. (14) noudji et al., Pages 99-101 in Davydov, Chapter 1 in ! ¼ A ð AÞ 27 28 describes a right-handed active rotation of operators T, Haywood, Chapter 29 in Hecht, Section 7.4 in Konishi 33 keeping the coordinate system fixed. Thanks to Eq. and Paffuti, Chapter 2 and Section 6.3 in Mathur and 14 34 35 (14), it is equivalent to describing the transformation Singh, Chapter 13 in Messiah, Section 4.10 in Miller, MAN | 7of64

Section 2 in Rose,36 Page 3 in Tung,37 Chapter 3 in Zare,38 We first provide the definitions of terms such as Hilbert Pages 102 and 116 in Zettili39), the active transformation is space, scalar product, position representation, and wave func- defined as the following: In conjunction with the transformation used in quantum mechanics that are important for the tion of the state vectors induced by a symmetry operation comprehension of some common notations. Especially, scalar (unitary transformation) given by Eq. (16), each operator T product and wave function have two possible interpretations. undergoes the transformation defined in Eq. (20). In the canonical transformation by the operator RA, both 4.1 | Hilbert space operators and coordinate system rotate together. The right- handed active rotation of operators T in a fixed coordinate A Hilbert space is an abstract vector space with a scalar system is followed by a right-handed rotation of the fixed product and a positive definite norm.28,39,43 In quantum coordinate system. The rotated coordinate system is called mechanics, physical properties of a system are represented body-attached coordinate system. by linear operators R, L, ... and a state vector Φ or state In fact, the active transformation corresponds to our for short by ket state jiU or wave function Φ(r), and by active rotation in one coordinate system, the fixed coordi- bra state hjU or complex conjugate UðrÞ of wave function. nate system associated with the unprimed operator. The While ket state belongs to a Hilbert space, its bra state canonical transformation corresponds to our active rotation belongs to the corresponding dual Hilbert space. Wave in two coordinate systems:1 the fixed coordinate system is functions, complex-valued functions of space, are square- associated with the unprimed operator and the body- integrable functions. Their space possesses the properties attached coordinate system is attached to the operator and of an infinite-dimensional Hilbert space.39 rotates with it. Equation (18) means that the primed opera- The quantum state is an abstract concept, but it contains tor in the body-attached coordinate system has the same all the information about the system. Observables are Hermi- matrix elements as those of the unprimed operator in the tian operators, which take real eigenvalues only. A wave fixed coordinate system before the active rotation. function is associated with a state, whereas a space function is not necessary a state. An operator acting on a ket produces another ket. However, the result of an operator on a state is 4 | ROTATION OF SPACE not necessary a state, it may be a space function.35,44 In our FUNCTION ABOUT A SINGLE AXIS case, a rotation operator converts an old state into a new state because rotation operators are unitary operators. The study of active rotation of space function about a single One of the key benefits of Dirac notation2 is that operators, axis involves one or two coordinate systems. When one coor- kets, and bras are independent of any representation basis. dinate system is used,40 it corresponds to a space-fixed coor- Very often it is convenient to use some less abstract represen- dinate system (O, x, y, z).38 When two coordinate systems tations of states and operators to get numerical results. are used,36 one corresponds to the space-fixed coordinate In finite-dimensional Hilbert space, we expand any ket system (O, x, y, z) and the other is attached to the space func- as a linear combination of the eigen kets of an observ- tion, rotates with it, and is called body-attached coordinate able.45 If Q is such an observable in a 2-D vector space system (O, x0, y0, z0) or molecule-fixed frame.38,41 Viewed in with eigen kets {jii , jij }, then an arbitrary state jiU can be its body-attached coordinate system, the space function expressed as a linear combination of {jii , jij }. These two remains motionless. We previously shown1 that Rose36 used eigen kets are basis vectors in an orthonormal coordinate this approach for deducing the active rotation operator system. That is, involving Euler angles. As Rose did not use figures to sup- 40 42 U : port his presentation, Wolf and Bouten understood that ji¼aijiþi ajjij (21) Rose dealt with passive rotation of space function. The components are given by hii j U ¼ a and hij j U ¼ a . In this section, we formulate the rotation of a space i j Therefore, the state Φ may be written as an abstract ket function in one direction and the equivalent rotation of state jiU , a superposition of ket basis vectors (Eq. (21)), or coordinate system in the opposite direction using right and a set of numbers (expansion coefficients hii j U ¼ a and left hands. Two approaches are used: the standard one i hij j U ¼ a in Eq. (21)). The three kets jiU , jii , and jij involving rotation operator and space function [R (a, j A belong to the same Hilbert space. The set of components z)Ψ(r)] and that involving Dirac notations [hjr RAða; zÞjiW ]. {a , a } of jiU along the eigen kets of Q completely The second approach has the advantage to exhibit both the i j describe jiU . They are called the Q representation of coordinate system represented by bra position basis and the jiU .29,45 Changing representations does not change the space function represented by ket space function. In the state ket. The ket state jiU itself is representation-free. Sim- standard approach [R (a, z)Ψ(r)], the coordinate system is A ilarly, the scalar product is also representation-free.35,46 not explicitly defined. 8of64 | MAN

When we take a scalar product in quantum mechanics, we Ket position basis: jir ; 8r 2 R3: (25) use a common basis in both states. In Eq. (21), the ket state is expressed in a countable A scalar product or a bra-ket pair can be interpreted in basis, identified by an integer. On the other hand, the posi- two ways.39 First, a bra-ket pair corresponds to a vector pro- tion basis is uncountable, identified by real numbers. jection. The projection of jiU onto the eigen ket jii is hii j U , The ket position bases verify26,28,39 the closure relation that on jij is hij j U . Therefore, Eq. (21) can be written as Z X 3 jiU ¼ hii j U jii þ hij j U jij ¼ jia hia j U ; (22) jir hjr d r ¼ 1; (26) a¼i;j P and the orthonormality condition because the closure relation jia hj¼a 1 is available. a¼i;j Second, the quantity hiW j U represents the probability hr0jir ¼ dðr r0Þ; (27) amplitude that the system’s ket state jiU before a measure- the latter being the highly localized Dirac delta func- ment will be found to be in another ket state jiW after the tion5,28,33,44,48 defined by measurement. Some Hilbert spaces are of infinite number of dimensions, Z where the observables can take on an infinite number of pos- f ðrÞ¼ dðr r0Þf ðr0Þd3r0 (28) sible values. The typical examples are function spaces where vectors are functions. In quantum mechanics, wave functions for any function f that is continuous at r. Suppose the parti- are often space functions.32,43 For states with continuous cle is at position r . The corresponding ket is r . The eigenvalues, such as two wave functions W ðrÞ and W ðrÞ, 0 0 1 2 position representation of r 35,49 or its wave function is the scalar product is defined by integral33,47 0 d Z ðr r0Þ. fi 3 The norm of the continuous ket position bases is in - W W ¼ WðrÞW ðrÞd r: (23) 1 2 1 2 nite.29,39 Therefore, the kets jir are not square inte- 28,39 fi grable. As Hilbert space admits only vectors of nite W W 29 ’ In other words, 1 2 corresponds to the integral of norm, the kets jir are not elements of the particle s Hil- W W 46 34,39,49 1ðrÞ 2ðrÞ. As we mentioned above, the scalar product bert space but remain a valid basis that spans the W W W W ’ 28,39 1 2 is the projection of 2 onto 1 . However, particle s Hilbert space. W W U the more general interpretation of 1 2 , independent of The scalar product hir j of jir with an arbitrary state U U 49 representation, is that it represents the probability amplitude ket jiis a square-integrable function of r, ðrÞ. The W W 26 U U for the ket state 2 to be found in the ket state 1 ; result of an operator L on jiis Lji. Its position repre- operators are not explicitly involved in this case. sentation is hjr LjiU . If an observable N is involved, we are interested in calculating the mean value hjW NjiW , which can be checked by performed the same experiment on the same state many 4.3 | Wave function times. The state Ψ appears as both a bra and a ket. Space function Ψ(r) of a particle in 3-D space is defined hjW NjiW is also called expectation value. by24,25 WðrÞ¼hir j W ; (29) 4.2 | Position representation which is not an equation in the usual sense. Ψ(r) is also known as normalized Schrodinger€ wave function for ket We focus on the position representation corresponding to state jiW 14,28,39,47 with measuring position operator R for a spinless particle in 3-D Z Z space. The eigen kets of R are jir with corresponding 0 0 3 0 0 0 3 0 eigenvalues r,26 the position vectors of the particle. The jiW ¼ jir hjr Wid r ¼ jir Wðr Þd r : all space all space eigen kets are labeled by the eigenvalues. That is, (30) A particle0s wave function defines not only its position Rjir ¼ rrji: (24) but also all the spatial properties of the particle. The pro- These eigenvalues are continuous as every value of duct of the wave function with its complex conjugate is the position is measurable. The position representation may be density probability25,44 for finding the particle at position r. defined by the continuous set of ket position basis jir .So Equation (29) relates the conventional notation of the we have abstract wave function Ψ in position representation with MAN | 9of64

Dirac notation of the wave function.46 It shows that the Before the rotation, the physical system is described by wave function is the projection of the ket state jiW onto the ket state jiW , whose space function Ψ(r) is the r-posi- the bra position base hjr .39,50 tion representation of jiW in (O, x, y, z). Ψ(r) is a mathe- The use of the position eigenstates as basis states to rep- matical function of r, the first meaning (1) of a space resent operators and states is referred to as the position rep- function. After the rotation, the physical system is 0 resentation. Equation (30) is the expansion of an arbitrary described by the new ket state jiW A ¼ RAða; zÞjiW , whose W 0 Ψ0 ket state jiin terms of the ket position bases jir , where space function A(r) is the r-position representation of fi W0 the expansion coef cient is the wave function in position A in the same coordinate system (O, x, y, z). As whole representation. The summation over the index of a com- functions, the mathematical expression of Ψ(r) differs with 0 0 plete set of eigenvalues in Eq. (22) becomes an integral that of Ψ A(r), that is, Ψ(r)6¼Ψ A(r). over continuous eigenvalues in Eq. (30). Ψ(r) is also the As the rotation operator RA(a, z) is a unitary operator, 13,33,44,51 0 probability amplitude for finding a particle at posi- the value Ψ A(r) of the space function at the position r is tion r in an orthonormal basis of 3-D space. equal to the value Ψ(r1) of the space function before rota- Therefore, the symbol Ψ(r) has two different meanings tion at the position r1 (Figure 6A): in different contexts:44 0 W ðrÞ¼Wðr1Þ: (31) 1. The first meaning is WðrÞ¼jiW , Ψ(r) means a mathe- A 0 matical function of r, the whole function without being The second meaning (2) of space functions Ψ(r) and Ψ A(r) explicit the actual value of r. It is the position represen- is used in Eq. (31). 3 tation of the abstract ket state jiW . As the list of compo- The two positions r1 and r in (O, x, y, z) are related by nents of a vector in a basis of 3-D vector space is a the right-handed active rotation matrix Az(a): particular representation of that vector, Ψ(r) and jiW are similar to the components of a vector in a particular 0 1 cos a sin a 0 basis and the vector, respectively. Similarly, WðpÞ¼ r ¼ A ðaÞr ¼ @ sin a cos a 0 Ar : (32) jiW is the momentum representation of the abstract ket z 1 1 001 state jiW , but Ψ(p) is the Fourier transform of Ψ(r).26 It is important to note that the mathematical expression of Formally, we can introduce the right-handed passive Ψ Ψ 3 (r) differs with that of (p). rotation matrix Pz(a): 2. The second meaning of Ψ(r) is Eq. (29), that is, the A a P a : r1 ¼ zð Þr ¼ zð Þr (33) probability amplitude Ψ(r) for finding the particle in the state jiW at the some specific position r,52 or more gen- Equation (33) relates the components of position vectors erally the value Ψ(r) of the space function Ψ at some using rotation matrices. 31 specific position r, not the whole function. A scalar Equation (33) does not mean the two positions r and r1 product is a number, and hir j W is the value of Ψ at are the same position vector viewed in two different coordi- P a the position r.Ifhir j W ¼ 0, it does not mean jiW and nate systems. It suggests that the two positions r and z( )r jir are orthogonal because they do not belong to the (Figure 6B) are viewed in two different coordinate systems, P a 0 0 0 same Hilbert space. It just means the value of the wave that of r is (O, x, y, z) and that of z( )r is (O, x , y , z ). The function Ψ at the specific position r is zero. latter is generated by the left-handed rotation of (O, x, y, z) about z-axis by angle a. Notice that we use the same symbol A particle in state jir is located at position r. In con- a for two rotation angles of opposite signs. In other words, trast, a particle in state jiW can be found at different posi- the components of position r in (O, x, y, z) are identical to 0 0 0 tions with corresponding probability amplitudes. The action those of position Pz(a)r in (O, x , y , z ). of an operator L on a space function is related to its action The left-handed rotation of coordinate system is related to 29 P a on the abstract vector space by the rule LWðrÞ¼hjr LjiW the fact that the position r1= z( )r is deduced from the posi- P a with WðrÞ¼hjr Wi. tion r by a left-handed rotation. As a result, z( )maybe either a right-handed passive rotation matrix or a left-handed passive rotation matrix, depending on the circumstance. 4.4 | Active rotation of space function in one We illustrate this property of P (a) with Figure 7. Fig- coordinate system z ure 7A shows the space function Ψ(y) in the fixed coordi- Consider the right-handed active rotation9,11,22,32,34,38,40,53 nate system (O, x, y, z). The action of a right-handed active a Ψ a p about z-axis by angle of the space function in the space- rotation of (y) about z-axis by angle ¼ 2 is shown in W0 fixed coordinate system (O, x, y, z). The rotation operator Figure 7B where the space function becomes AðxÞ. The a P a p is denoted by RA( , z). action of zð ¼ 2Þ on r is 10 of 64 | MAN

FIGURE 6 (A) Right-handed active rotation of space function about z-axis by angle a shown in a single coordinate system (O, x, y, z). 0 Ψ(r1) is the initial space function before rotation and Ψ A(r) is the rotated space function. (B) Equivalent left-handed rotation of coordinate system from (O, x, y, z)to(O, x0,y0,z0). Position vector r is attached to (O, x, y, 0 0 0 z) and position vector Pz(a)r is attached to (O, x ,y,z)

0 10 1 cosða ¼ pÞ sinða ¼ pÞ 0 x B 2 2 CB C P a p @ a p a p A@ A zð ¼ 2Þr¼ sinð ¼ 2Þ cosð ¼ 2Þ 0 y 001z 0 10 1 0 1 010 x y B CB C B C ¼ @ 100A@ y A ¼ @ x A: (34) 001 z z

In fact, the position vector r represents the coordinate FIGURE 7 Equivalence between right-handed active rotation of system. That is, x- and y-axes become y- and –x-axes, space function about z-axis by angle a=p/2 transforming the space 0 respectively. In Figure 7C, the original x- and y-axes are function Ψ(y) in (A) to the space function Ψ A(x) in (B) and left- black lines and the rotated axes are green lines. The green handed rotation of coordinate system about z-axis by angle a = p/2 coordinate system is deduced from the black one by a left- transforming black (O, x, y, z) to green (O, x, y, z) in (C); the space a p function is Ψ(y) in black (O, x, y, z) and Ψ(x) in green (O, x, y, z) handed rotation about z-axis by angle ¼ 2. The space function becomes Ψ(x) in the green coordinate system like the space function in Figure 7B. 0 0 24 W r r Equation (31) becomes The space function ðÞis the -position representation of the initial ket state jiW in (O, x0, y0, z0). The same W0 W W P a : space function Ψ designates the initial space function in AðrÞ¼ ðr1Þ¼ ðÞzð Þr (35) (O, x, y, z) and that in (O, x0, y0, z0). This may confuse us, P a The position z( )r is described in the left-hand rotated but it is correct, because the ket state jiW is viewed in two 0 0 0 coordinate system (O, x , y , z ). coordinate systems. Its space function is Ψ(r)in(O, x, y, MAN | 11 of 64

0 0 0 0 2 3 z) and WðÞr in (O, x , y , z ). As whole functions, the 0 1 a; W W W P a hr W ¼hjr RAð zÞji¼ r1ji¼ ðÞzð Þr mathematical expression of WðÞr0 differs with that of A R a; z r W : (41) WðÞr . ¼ h Pð Þ ji We deduce from Eq. (35) that the value Ψ0 (r) of the A The operator RA(a, z)defined in Eq. (36) is the right- fi right-hand rotated space function at position r in the xed handed counter-clockwise active rotation operator for space coordinate system (O, x, y, z) (Figure 6A) is identical to function. In hjr RAða; zÞjiW , it behaves as a right-handed W P a the value ðÞzð Þr of the initial space function at position active rotation (Figure 2A) operator when it is applied to P (a)r in the left-hand rotated coordinate system (O, x0, y0, z the ket state RAða; zÞjiW ¼ jiRAða; zÞW or as a left-handed 0 z ) (Figure 6B). In other words, a right-handed active passive rotation (Figure 2A) operator when it is applied to rotation of space function in a fixed coordinate system y the bra position basis hjr RAða; zÞ¼hjr ½RPða; zÞ . (Figure 6A) is equivalent to a left-handed rotation of We have introduced right-handed and left-handed types fi coordinate system keeping the space function xed of rotation in our discussions. The coordinate system in (Figure 6B). Dirac notations is expressed by bra position basis hjr in Introducing the operator of right-handed active rotation hjr RAða; zÞjiW . If we only use right-handed rotations, we about z-axis by angle a denoted by1,40,53,54 will say: The operator RA(a, z)inhjr RAða; zÞjiW behaves as a counter-clockwise active rotation operator (Fig- RAða; zÞ¼expðiaLzÞ; (36) ure 4B) when it is applied to the ket state or as a clock- 11,29,32,34,40 Eq. (35) becomes wise passive rotation operator (Figure 4A) when it is 1 2 3 applied to the bra position basis. In this sentence, we W0 ðrÞ ¼ R ða; zÞWðrÞ ¼ Wðr Þ ¼ WðÞP ðaÞr : (37) A A 1 z may replace counter-clockwise by positive and clockwise 40 Two interpretations of the action of RA(a, z) are possible: by negative.

1. By definition, the action of right-handed operator RA(a, z) upon the space function Ψ(r) must give us a new 4.5 | Active rotation of space function in two 0 32 — function Ψ A(r) of same argument. The first equality coordinate systems canonical transformation 1 denoted by ¼ describes a right-handed active rotation The two coordinate systems are the space-fixed coordinate of the space function from Ψ to Ψ0 , their arguments r A system (O, x, y, z) in Figure 8 and the rotated body- being the same. As the argument r of the space function attached coordinate system (O, x0, y0, z0) in Figure 8B. The also represents the coordinate system, the right-handed space function and its body-attached coordinate system operator R (a, z) rotates the space function from Ψ to A rotate together as a whole. Ψ0 , keeping the coordinate system fixed. We may write A Figure 8A is identical to Figure 6A. It shows the space Eq. (37) as 0 functions Ψ(r) and Ψ A(r) of the initial ket state jiW and 0 the right-hand rotated ket state jiW A ¼ RAða; zÞjiW in (O, W0 1 a; W 2 W 3 W P a : AðrÞ ¼½ðRAð zÞ rÞ ¼ ðr1Þ ¼ ðÞzð Þr (38) x, y, z), respectively. Equations deduced in the previous 3 2. In contrast, the equality denoted by ¼ in Eq. (37) subsection remain valid here. Therefore, we have (see Eq. (37)) describes the action of RA(a, z) as a left-handed rotation of coordinate system from r, which is associated with 0 1 2 0 W ðrÞ ¼ RAða; zÞWðrÞ ¼ Wðr Þ with r1 ¼ PzðaÞr: (42) (O, x, y, z), to Pz(a)r, which is associated with (O, x , A 1 0 0 Ψ fi y , z ), keeping the space function xed. 0 Figure 8B shows the rotated ket state jiW A in two In quantum mechanics, an active rotation operator coordinate systems. Its r-position representation [the first Ψ0 RA(a, z) is a unitary transformation, that is, meaning (1) of space function] is the space function A(r) in (O, x, y, z). Its r0-position representation should be the y 1 ½R ða; zÞ ¼ ½R ða; zÞ ¼ R ða; zÞ¼R ða; zÞ: (39) 0 0 0 0 0 A A A P space function Ψ A(r )in(O, x , y , z ). As whole functions, 0 the mathematical expression of Ψ A(r) differs with that of The operator RP(a, z) will be introduced in Section 4.6. The Ψ0 0 Ψ0 Ψ0 0 definition of a space function in Eq. (29) allows us to write A(r ), that is, A(r)6¼ A(r ). It is important to note that in Figure 8B we have used Ψ 0 ½ðR ða;zÞW rÞ¼hjr R ða;zÞWi¼hjr R ða;zÞjiW the space function (r ) associated with the initial ket state A DEA A jiW to denote the right-hand rotated space function in its a; y W a; W ¼ ½RAð zÞ r ¼hiRAð zÞrj body-attached coordinate system. We have used the prop- 3 erty that the rotated space function as a whole function in ¼hiRPða;zÞrjW ¼WðÞPzðaÞr : (40) its body-attached coordinate system remains the same as Equation (38) written in Dirac notation becomes that before rotation Ψ(r), but its argument is r0 in (O, x0, y0, 12 of 64 | MAN

The positions r0 and r are identical in Figure 8B, but they are viewed in two coordinate systems, r0 in (O, x0, y0, z0) and r in (O, x, y, z). Therefore,3

0 r ¼ PzðaÞr: (44) Equation (44) relates the components of position vectors using rotation matrix. Substituting Eq. (44) into Eq. (43) yields

W0 W 0 W P a : AðrÞ¼ ðr Þ¼ ðÞzð Þr (45) Combining Eqs. (42) and (45) yields

W0 1 a; W 4 W 0 3 W P a : AðrÞ ¼ RAð zÞ ðrÞ ¼ ðr Þ ¼ ðÞzð Þr (46) Furthermore, the space function is motionless in its body-attached coordinate system. We have an obvious expression concerning the values of space function, which is not always provided in the literature:

W 0 2 W ; ðr Þ¼ ðr1Þ (47)

r1 being the initial position vector in (O, x, y, z) before rotation of the body-attached coordinate system and r0 the FIGURE 8 Canonical transformation (or active rotation) of space corresponding position vector in (O, x0, y0, z0) after rotation function about z-axis by angle a in two coordinate systems, the space- of the body-attached coordinate system. This corresponds fixed coordinate system (O, x, y, z) and the body-attached coordinate 0 0 0 to the canonical transformation. Equation (47) means the system (O, x ,y,z). (A) Before rotation: Ψ(r1) is the space function; 0 0 0 0 space function is described in its body-attached coordinate (O, x, y, z) and (O, x ,y,z) coincide. (B) After rotation: Ψ A(r) is the rotated space function in the space-fixed coordinate system and Ψ(r0)is system. the rotated space function in its body-attached coordinate system The active rotations of a space function in one and two coordinate systems are equivalent. This is supported by the z0). As whole functions, the mathematical expression of same series of equalities in Eqs. (37) and (46). The equal- Ψ0 (r) differs with that of Ψ(r0), that is, Ψ0 (r) Ψ(r0). 4 A A 6¼ ity ¼ in Eq. (46) is due to the implication of the body- 0 0 0 The angle a describes the rotation of (O, x , y , z )in 56 attached coordinate system. Van de Wiele and Deva- Figure 8B, whereas it describes the rotation of the space 57 nathan also use two coordinate systems. function in Figure 8A when one coordinate system is used. Equations (46) and (47) may be written in Dirac nota- W0 As the rotated ket state jiA in Figure 8B is viewed in tion: two coordinate systems, the value Ψ0 (r) of the rotated A Ψ0 1 4 0 space function A at the rotated position r in (O, x, y, z) W0 hjr R ða; zÞjiW ¼hr jiW hr A ¼ A is equal to the value Ψ(r0) of the rotated space function Ψ 3 P a W a; W ; at the position r0 in the rotated body-attached coordinate ¼h zð Þrji¼hRPð zÞrji (48) 0 0 0 36 system (O, x , y , z ): and W0 W 0 : r0 W 2 r W : (49) AðrÞ¼ ðr Þ (43) h ji¼ 1ji The second meaning (2) of space function is used in The comments for the active rotation of space function Eq. (43), where the positions r and r0 are those shown in in one coordinate system remains valid: The right-handed Figure 8B. counter-clockwise active rotation operator for space func- a a; W Equation (43) differs with Eq. (31). In Eq. (43) the tion RA( , z)inhjr RAð zÞjibehaves as a right-handed argument of the space function Ψ is the rotated position r0 active rotation (Figure 2B) operator when it is applied to 0 0 a; W a; W in the rotated body-attached coordinate system (O, x , y , the ket state RAð zÞji¼ jiRAð zÞ or as a left-handed 0 z ), whereas in Eq. (31) it is the initial position r1 in the passive rotation (Figure 2A) operator when it is applied to 55 a; a; y fixed coordinate system (O, x, y, z). the bra position basis hjr RAð zÞ¼hjr ½RPð zÞ . MAN | 13 of 64

of space function is used: these two space functions Ψ(r) 0 0 and Ψ P(r ) represent the components of the ket state jiW in (O, x, y, z) and (O, x0, y0, z0), respectively. As whole functions, the mathematical expression of Ψ(r) differs with 0 0 0 0 that of Ψ P(r ), that is, Ψ(r)6¼Ψ P(r ). As the ket state jiW remains unchanged under passive rotation, whose operator is a unitary operator, we have40

W0 0 W : Pðr Þ¼ ðrÞ (50) The second meaning (2) of space function is used in 0 0 Eq. (50): the value Ψ P(r ) of the space function after passive rotation at the position r0 in (O, x0, y0, z0)is identical to the value Ψ(r) of the space function before passive rotation at the position r in (O, x, y, z). These two identical positions r and r0 are those shown in Fig- ure 9A. The components of r0 in (O, x0, y0, z0) and those of r in (O, x, y, z) are related by the right-handed passive rotation 3 matrix Pz(a): FIGURE 9 (A) Right-handed passive rotation of space function 0 P a : (or rotation of coordinate system) about z-axis by angle a. The space r ¼ zð Þr (51) Ψ r function is viewed in two coordinate systems. ( ) is the space Formally, we can introduce the right-handed active rota- function in the space-fixed coordinate system (O, x, y, z) and Ψ0 (r0) P tion matrix A (a):3 is the space function in the rotated coordinate system (O, x0,y0,z0). z P a 1 0 P a 0 A a 0 (B) Equivalent left-handed active rotation of space function from r ¼ zð Þ r ¼ zð Þr ¼ zð Þr A a fi 0 1 position vector r to position vector z( )r in the space- xed cos a sin a 0 coordinate system (O, x, y, z) B C ¼ @ sin a cos a 0 Ar0: (52) 001 4.6 | Passive rotation of space function Equation (52) relates the components of position vectors Contrary to the active rotation of space function, which can using rotation matrices. be described in one and in two coordinate systems, two Equation (52) does not mean the two positions r and r0 coordinate systems are naturally involved in the passive in Figure 9A are related by an active rotation. It suggests rotation of space function. Figure 9A shows these two 0 0 that the two positions r and Az(a)r are related by a left- coordinate systems related by a right-handed rotation about handed active rotation about z-axis by angle a in (O, x0, y0, z-axis by angle a: the space-fixed coordinate system (O, x, z0). The left-handed active rotation of position vector is y, z) and the rotated one (O, x0, y0, z0). The operator for suggested by the fact that the position P ðaÞr0 is deduced right-handed passive rotation of space function is denoted z from the position r0 by a left-handed rotation. As r0 and r by R (a, z). P represent the same position vector viewed in two coordi- Under passive rotation, the key state jiW remains nate systems, it follows that the two positions r and Az(a)r unchanged as we mentioned in Section 3, but its space in Figure 9B are also related by a left-handed active rota- function as a whole function depends on the coordinate tion about z-axis by angle a in (O, x, y, z). Notice that we system. It is tempting to define the whole function of its use the same symbol a for two rotation angles of opposite space function in (O, x, y, z)byΨ(r), the r-position represen- signs. As a result, Az(a) may be either a right-handed tation of W , and that in (O, x0, y0, z0)byΨ(r0), the r0-posi- ji active rotation matrix or a left-handed active rotation tion representation of W . These two space functions only ji matrix, depending on the circumstance. In other words, r differ by their arguments. We already met a similar definition 0 0 and Az(a)r are equivalent to r and Az(a)r . Equation (50) of space functions in the above subsection where the rotated becomes24 coordinate system is the rotated body-attached coordinate system. This is not the case here; (O, x0, y0, z0) in Figure 9A W0 0 W 3 W A a 0 : Pðr Þ¼ ðrÞ ¼ ðÞzð Þr (53) is not the rotated body-attached coordinate system. We have to denote the space function by Ψ(r)in(O, x, Equation (53) relates the values of space functions at 0 0 0 0 0 y, z) and by Ψ P(r )in(O, x , y , z ). The first meaning (1) different position vectors. In particular, the equality 14 of 64 | MAN

3 0 3 3 WðrÞ ¼ WðÞAzðaÞr is equivalent to WðrÞ ¼ WðÞAzðaÞr . 2. In contrast, the equality denoted by ¼ in Eq. (57) This suggests that the space function is expressed in its describes the action of RP(a, z) as a left-handed active body-attached coordinate system, Ψ(r) before the left- rotation of space function in fixed coordinate system. 0 handed rotation and WðÞAzðaÞr after the left-handed rotation of the space function and its body-attached coordinate It is worth noting that Eqs. (37) and (57) have the same 3 0 system. In other words, the equality WðrÞ ¼ WðÞAzðaÞr structure. Figure 8B about the active rotation of space describes a left-handed active rotation of space function in function described in two coordinate systems and Fig- two coordinate systems, whereas the above subsection ure 9A about the passive rotation of space function also in describes a right-handed active rotation of space function two coordinate systems look similar, but the space func- 0 in two coordinate systems. As an active rotation of space tions are different. In Figure 8B, the space function Ψ A(r) function in one coordinate system and in two coordinate is expressed in the fixed coordinate system and Ψ(r0) in the 3 0 systems are equivalent, the equality WðrÞ ¼ WðÞAzðaÞr rotated body-attached coordinate system. It is the opposite 0 0 also suggests a left-handed active rotation of a space func- in Figure 9A. The space function Ψ P(r ) is expressed in tion in the fixed coordinate system (O, x, y, z). the rotated coordinate system and Ψ(r) in the fixed coordi- 0 0 We deduce from Eq. (53) that the value Ψ P(r ) of the nate system. space function at the position r0 in the rotated coordinate In quantum mechanics, a passive rotation operator 0 system (Figure 9A) is equal to the value WðÞAzðaÞr of the RP(a, z) is a unitary transformation, that is, left-hand rotated space function Ψ at the position Az(a)r in a; y a; 1 a; : the fixed coordinate system (Figure 9B). In other words, a ½RPð zÞ ¼ ½RPð zÞ ¼ RAð zÞ (58) right-handed rotation of coordinate system keeping the Equation (57) written in Dirac notation becomes space function fixed (Figure 9A) is equivalent to a left- handed active rotation of space function in a fixed coordi- 0W0 1 0 a; W 2 W hr P ¼hjr RPð zÞji¼hrji nate system (Figure 9B). 3 W A a 0 a; 0 W : The right-handed counter-clockwise passive rotation ¼ ðÞzð Þr ¼ hRAð zÞr ji (59) 0 0 operator RP(a, z ) about z -axis by angle a for a space func- 0 In hjr RPða; zÞjiW , RP(a, z) behaves as a right-handed pas- tion40,58 is sive rotation operator (Figure 3A) when it is applied to the bra 0 RPða; z Þ¼expðiaLz0 Þ: (54) The z0- and z-axes, which are parallel, are the rotation axis in (O, x0, y0, z0) and (O, x, y, z), respectively. Wolf40 established the passive rotation operator about fixed axes for space function involving Euler angles. Weissbluth59 like many authors58,60,61 chooses z instead of z0 for rotation axis:

RPða; zÞ¼expðiaLzÞ: (55) In this article, we also choose this notation. Therefore,

W0 0 a; W 0 : Pðr Þ¼RPð zÞ ðr Þ (56) As a result, the right-handed counter-clockwise active rotation operator in Eq. (36) and the right-handed counter- clockwise passive rotation operator in Eq. (55) differ only by the sign of the rotation angle. Putting Eqs. (53) and (56) together yields29,32

W0 0 1 a; W 0 2 W 3 W A a 0 : Pðr Þ ¼ RPð zÞ ðr Þ ¼ ðrÞ ¼ ðÞzð Þr (57) As with the active rotation of space function, two inter- FIGURE 10 Two equivalent points of view for a rotation of pretations of Eq. (57) are possible:40 axes by an angle a in a plane: (A) the vector V is fixed in the plane while the axes are right-hand rotated about z-axis by a positive angle 1 1. The first equality denoted by ¼ describes a right-handed a in passive point of view; (B) the axes are fixed, whereas the vector 0 passive rotation of space function from Ψ to Ψ P, their V is right-hand rotated to a new position V0 about z-axis by a arguments r0 being the same. negative angle a in active point of view MAN | 15 of 64

0 position basis ðÞhjr RPða; zÞ jiW and as a left-handed active rotation operator (Figure 3B) when it is applied to the ket state:

0 0 hjr ðÞ¼RPða; zÞjiW hir j RAða; zÞW : (60) If we only use right-handed rotations, we will say: 0 The operator RP(a, z)inhjr RPða; zÞjiW behaves as a counter-clockwise passive rotation operator (Figure 10A) when it is applied to the bra position basis and as a clockwise active rotation operator (Figure 10B) when it is applied to the ket state (Eq. (60)). In this sentence, we may replace counter-clockwise by positive and clockwise by negative. 2 Equality ¼ in Eq. (59) provides us with

0 hjr RPða; zÞ¼hjr : (61)

0 0 Although the two positions r and r are identical, the two FIGURE 11 Euler angles are traditionally deﬁned with rotated bra position bases hjr0 and hrj are different. It is interesting coordinate systems to note that the initial bra position basis hjr is deduced from the ﬁnal bra position basis hjr0 using the passive rota- a tion operator RP( , z). The components of the two positions In Section 4, we show that right- and left-handed rota- 0 r and r are related by Eq. (51). tions of coordinate system about z-axis by angle a are Equation (61) may be rewritten as expressed mathematically by the application of the opera-

tors RP(a, z) and RA(a, z) to the bra position basis, respec- 0 tively. In contrast, right- and left-handed active rotations of hjr ¼ hjr RAða; zÞ: (62) space function about z-axis by angle a are expressed by

In words, the passive rotation of a space function using the the application of RA(a, z) and RP(a, z) to the ket space passive rotation operator RP(a, z) is visualized by an active function, respectively. We extend these rules to rotation rotation of the bra position basis from hjr to hjr0 or by an operators deﬁned with Euler angles. As three rotation axes active rotation of coordinate system from (O, x, y, z)to(O, are involved, we need to distinguish rotations about rotated x0, y0, z0). axes with those about ﬁxed axes.

5 | ROTATION OF SPACE 5.1 | Active rotation operator about rotated FUNCTION IN TERMS OF EULER axes ANGLES We already provided the right-handed active rotation operator about axes of rotated body-attached coordinate system Several ways to parameterize the rotation matrix for the and about fixed axes in terms of Euler angles.1 We simply orientation of a rigid body in a coordinate system or the recall the main results written in Dirac notations. relative orientation of two coordinate systems are available. The coordinate systems shown in Figure 11 are the Some of them are: direction cosine matrix,62 Euler body-attached coordinate system of the space function. The – angles,3,24,55,63 angle-axis of rotation24,55,62 64 denoted by first rotation about the z-axis by angle a transforms the (h, n), the four Cayley-Klein parameters,63,65 and quater- coordinate system (O, x, y, z) into a new coordinate system nions.25,64 Siemens et al.66 discussed in details the benefits (O, x0, y0, z0). The second rotation about the y0-axis by of using angle-axis of rotation parametrization over that of angle b transforms (O, x0, y0, z0) into a new coordinate system Euler angles in NMR. In contrast, we focus on Euler angle (O, x00, y00, z00). The third rotation about the z″-axis by angle c parametrization, which is extensively used in the literature transforms (O, x00, y00, z00) into a new coordinate system other than that about NMR. We want to clarify its implica- (O, x000, y000, z000). This is the traditional way14,37,67–71 to tions in active and passive rotations by exploring the equiv- introduce Euler angles. The successive rotations are made alence between the active rotation of a physical quantity in about rotated axes. one direction and the rotation of the coordinate system in We denoted the operator for right-handed active rotation the opposite direction. In particular, we emphasize on of space function about rotated axes of body-attached coor- which Euler angle, a or c, is the first rotation angle. dinate system by9,14,25,28,29,34,36,38,39,47,52,55,57,61,72–74 16 of 64 | MAN

rot c; b; a c b a : The two passive rotation matrices P (c, b, a) RA ð Þ¼expði Lz00 Þ expði Ly0 Þ expði LzÞ (63) rotatedZ3Y2Z1 P c; b; a 1,3 and fixedZ1Y2Z3ð Þ are identical. When active rota- The order of Euler angles in the arguments of tions are performed with body-attached coordinate systems, rot c; b; a RA ð Þ follows that of Euler angles in the three ele- the first rotation angle is a. When active rotations are per- mentary active rotation operators. This notation is not uni- formed in fixed coordinate system, the first rotation angle fi a versal. The rst rotation angle is the right-hand side is c. This matrix3,67 is rot c; b; a argument of RA ð Þ because this operator is applied to PfixedZ1Y2Z3ðc; b; aÞ¼ProtatedZ3Y2Z1ðc; b; aÞ a ket state jiW . Therefore, we have the generalization of 0 1 CaCbCc SaSc SaCbCc þ CaSc SbCc Eq. (41) for one coordinate system: B C ¼ @ CaCbSc SaCc SaCbSc þ CaCc SbSc A: W000 1 rot c; b; a W 2 W hr A ¼hjr RA ð Þji¼hr1ji CaSb SaSb Cb 3 ¼ W½ProtatedZ3Y2Z1ðc; b; aÞr ; (64) (69) and of Eq. (48) for two coordinate systems: For simplicity, we have replaced cos and sin by C and S in Eq. (69). W000 1 rot c; b; a W 4 000 W We mention that the active rotation operator about hr A ¼hjr RA ð Þji¼hr ji fixed axes Rfixedðh; nÞ in angle-axis of rotation parametriza- 3 W P c; b; a ; A ¼ ½rotatedZ3Y2Z1ð Þr (65) tion24–26,34,36,39,47,55,56,76 is P c b a 1,3 rotatedZ3Y2Z1( , , ) being the passive rotation matrix. fixed h; h : RA ð nÞ¼expði n LÞ (70) Note that Rfixed h; n Rfixed a; b; c . 5.2 | Active rotation operator about fixed A ð Þ A ð Þ axes 1 rot c; b; a fi 5.3 | Example V.1 We manipulated the expression of RA ð Þ de ned in Eq. (63), which depends on angular momentum operators of In Section 4, we have checked that right-handed active rota- successively rotated body-attached coordinate systems, so tion of space function about z-axis by angle a is equivalent that the expression depends on angular momentum operators to left-handed rotation of coordinate system about z-axis by fi fixed a; b; c of xed coordinate system. The operator RA ð Þ for angle a. We extend this rule to rotation about rotated axes of the right-handed active rotation of space function about fixed body-attached coordinate system and to rotation about fixed 9,11,14,22,24,25,28,29,34,36,38,39,42,47,52,56, 57,60,61,67,71–77 axes is axes by Euler angles with simple examples.

Rfixed a; b; c exp iaL exp ibL exp icL : (66) A ð Þ¼ ð zÞ ð yÞ ð zÞ 5.3.1 | Active rotation about rotated axes fixed a; b; c The order of Euler angles in the arguments of RA ð Þ Right-handed active rotation of space function about follows that of elementary active rotation operators about fixed rotated axes of its body-attached coordinate system is for- axes. The notation is not universal. The Euler angle for the first mulated as rotation is c because this operator is applied to a ket state W . ji rot c; b; a W c b a W : rot c; b; a RA ð Þji¼ expði Lz00 Þ expði Ly0 Þ expði LzÞji This order is the reverse of that in RA ð Þ. Equation (64) for one coordinate system becomes29 (71) The first, second, and third right-handed active rotations of rW000 1 r Rfixed a; b; c W 2 r W h A ¼hj A ð Þji¼h 1ji the space function are about z-axis by angle a, about y0- 3 b ″ c ¼ W½PfixedZ1Y2Z3ðc; b; aÞr ; (67) axis by angle , and about z -axis by angle , respectively. The same result can be obtained with left-handed rota- P c b a 1,3 fixedZ1Y2Z3( , , ) being the passive rotation matrix. tion of coordinate system about rotated axes as Equation (65) for two coordinate systems (or canonical transformation) becomes9,14,36 rot c; b; a c b a : 1 fixed 4 hjr RA ð Þ¼hjr expði LzÞ expði Ly0 Þ expði Lz00 Þ hr W000 ¼hjr R ða; b; cÞjiW ¼hr000jiW A A (72) 3 ¼ W½PfixedZ1Y2Z3ðc; b; aÞr : (68) We have to rename the rotation axes. This is a reason why W000 4 000 W The equality hr A ¼hr jiin Eq. (68) means the rotation operators about rotated axes should not be used. W000 fi fi rotated state A appears in the xed coordinate system The rst, second, and third left-handed rotations of coordi- hjr exactly as jiW does in the rotated body-attached coordinate system are about z-axis by angle c, about y0-axis by nate system hjr000 .14 angle b, and about z″-axis by angle a, respectively. MAN | 17 of 64

FIGURE 13 Left-handed rotations of coordinate system about

axes of rotated coordinate systems, the space function Ψ(x, y, z)=px fi fi fi FIGURE 12 Right-handed active rotations of space function remains xed: (A) initial con guration; the rst (B), second (C), and third (D) rotations of the coordinate system are about z-axis by angle Ψ(x, y, z)=px (represented by an arrow for simplicity) about axes of c p 0 b p 00 a p its body-attached coordinate system: (A) initial configuration; the first = /4, about y -axis by angle = /2, and about z -axis by angle = , (B), second (C), and third (D) rotations of the space function are respectively about z-axis by angle a=p, about y0-axis by angle b=p/2, and about z00-axis by angle c=p/4, respectively. In (D), the rotated space orbital has quantum numbers n=2, ‘=1, and m=1. It is 71 function lies in the {y, z} plane related to spherical harmonics by rot 1 The active rotation operator R ðc; b; aÞ has two mean- p ¼ pffiffiffi Y ; Y ; : (73) A x 2 1 1 1 1 ings: (a) When it is applied to space function or ket state as in Eq. (71), it is the right-handed active rotation opera- Its two lobes extend along the x-axis. The space func- tor. (b) When it is applied to coordinate system via bra tion is represented by an arrow for simplicity. The first position basis hjr as in Eq. (72), it is the left-handed pas- (Figure 12B), second (Figure 12C), and third (Figure 12D) sive rotation operator. right-handed active rotations of the space function are Consider the right-handed active rotation of the space about z-axis by angle a=p, about y0-axis by angle b=p/2, W ; ; ″ c p function ðx y zÞ¼px-orbital about rotated axes of its and about z -axis by angle = /4, respectively. In its body- body-attached coordinate system (Figure 12). This atomic attached coordinate system (O, x000, y000, z000), the space 18 of 64 | MAN

W ; ; 000 fi function ðx y zÞ¼px lies on the x -axis by de nition. Figure 12D shows that the right-hand actively rotated space function lies in the fgy; z plane of the fixed coordinate system (O, x, y, z). We should obtain the same result with left-handed rotation of coordinate system about rotated axes (Figure 13), fi W ; ; fi keeping xed the space function ðx y zÞ¼px. The rst (Figure 13B), second (Figure 13C), and third (Figure 13D) left-handed rotations of coordinate system are about z-axis by angle c=p/4, about y0-axis by angle b=p/2, and about z″-axis by angle a=p, respectively. Figure 13D shows that W ; ; 000; 000 the space function ðx y zÞ¼px lies in the fgy z plane of left-hand rotated coordinate system (O, x000, y000, z000), in agreement with Figure 12D. We have checked that right-handed active rotation of space function about rotated axes of its body-attached coordinate system by Euler angles (a, b, c) is equivalent to left-handed rotation of coordinate system about rotated axes by Euler angles (c, b, a). The standard formulation is:77 right-handed active rotation of space function about rotated axes of its body-attached coordinate system by Euler angles (a, b, c) is equivalent to right-handed rotation of coordinate system about rotated axes by Euler angles (c, b, a).

5.3.2 | Active rotation about ﬁxed axes Right-handed active rotation of space function about ﬁxed axes is formulated as

fixed c; b; a W a b c W : RA ð Þji¼ expði LzÞ expði LyÞ expði LzÞji (74) The first, second, and third right-handed active rotations of the space function are about z-axis by angle c, about y-axis by angle b, and about z-axis by angle a, respectively. The same result can be obtained with left-handed rota- FIGURE 14 Right-handed active rotations of space function Ψ fi tion of coordinate system about fixed axes as (x, y, z)=px about axes of the xed coordinate system (O, x, y, z): (A) initial configuration; the first (B), second (C), and third (D) rotations of the space function are about z-axis by angle c=p/4, about fixed c; b; a a b c : b p a p hjr RA ð Þ¼hjr expði LzÞ expði LyÞ expði LzÞ y-axis by angle = /2, and about z-axis by angle = , respectively. (75) In (D), the rotated space function lies in the {y, z} plane We do not need to rename the fixed rotation axes. The Consider the right-handed active rotation of the space fi W ; ; fi rst, second, and third left-handed rotations of coordinate function ðx y zÞ¼px about xed axes (Figure 14). The system are about z-axis by angle a, about y-axis by angle space function is represented by an arrow for simplicity. b, and about z-axis by angle c, respectively. The first (Figure 14B), second (Figure 14C), and third fixed a; b; c The active rotation operator RA ð Þ has two (Figure 14D) right-handed active rotations of the space meanings: (a) When it is applied to space function or ket function are about z-axis by angle c=p/4, about y-axis by state as in Eq. (74), it is the right-handed active rotation angle b=p/2, and about z-axis by angle a=p, respectively. operator. (b) When it is applied to coordinate system via Figure 14D shows that the right-hand actively rotated space W ; ; ; bra position basis hjr as in Eq. (75), it is the left-handed function of ðx y zÞ¼px lies in the fgy z plane of the passive rotation operator.77 fixed coordinate system (O, x, y, z). MAN | 19 of 64

W ; ; 000; 000 ðx y zÞ¼px lies in the fgy z plane of its left-hand rotated coordinate system (O, x000, y000, z000), in agreement with Figure 14D. We have checked that right-handed active rotation of space function about fixed axes by Euler angles (c, b, a)is equivalent to left-handed rotation of coordinate system about fixed axes by Euler angles (a, b, c). The standard formulation is:77 right-handed active rotation of space function about fixed axes by Euler angles (c, b, a) is equivalent to right-handed rotation of coordinate system about fixed axes by Euler angles (a, b, c).

5.4 | Passive rotation operator about rotated axes The ﬁrst rotation about the z-axis by angle a transforms the coordinate system (O, x, y, z) into a new coordinate system (O, x0, y0, z0) (Figure 11). This corresponds to the passive rotation of space function about a single axis. This transformation is described by Eq. (59): 0W0 1 0 a; W 2 W : hr P ¼hjr RPð zÞji¼hrji (76) In fact, we apply the rotation operator to hjr0 . Then, the second rotation about the y0-axis by angle b transforms the coordinate system (O, x0, y0, z0) into a new coordinate system (O, x00, y00, z00). The right-handed pas- 0 sive rotation operator RP(b, y )=exp(ibLy0) transforms Eq. (76) to

00W00 1 00 a; b; 0 W 2 W : hr P ¼hjr RPð zÞRPð y Þji¼hrji (77)

00 In fact, we apply the rotation operator to hjr RPða; zÞ. Finally, the third rotation about the z″-axis by angle c transforms the coordinate system (O, x00, y00, z00) into a new FIGURE 15 Left-handed rotations of coordinate system about coordinate system (O, x000, y000, z000). The right-handed pas- axes of the fixed coordinate system (O, x, y, z), the space function 00 sive rotation operator RP(c, z )=expðicLz00 Þ transforms Ψ x, y, z =p fi fi fi ( ) x remains xed: (A) initial con guration; the rst (B), Eq. (77) to second (C), and third (D) rotations of the coordinate system are about z-axis by angle a=p, about y-axis by angle b=p/2, and about z-axis 000W000 1 000 a; b; 0 c; 00 W 2 W : by angle c=p/4, respectively hr P ¼hjr RPð zÞRPð y ÞRPð z Þji¼hrji (78) In fact, we apply the rotation operator to hjr000 R ða; zÞR ðb; y0Þ. Right-handed active rotation of space function about P P We denote the right-handed passive rotation operator rotated axes of its body-attached coordinate system (Fig- about rotated axes by12 ure 12D) agrees with that about fixed axes (Figure 14D). We should obtain the same result with left-handed rota- rot a; b; c a b c : RP ð Þ¼expði LzÞ expði Ly0 Þ expði Lz00 Þ (79) tions of coordinate system (Figure 15), keeping fixed the W ; ; fi rot a; b; c space function ðx y zÞ¼px. The rst (Figure 15B), sec- The order of Euler angles in the arguments of RP ð Þ ond (Figure 15C), and third (Figure 15D) rotations of the follows that of elementary passive rotation operators about coordinate system are about z-axis by angle a=p, about y- rotated axes. This notation is not universal. This operator is rot c; b; a axis by angle b=p/2, and about z-axis by angle c=p/4, similar to RA ð Þ, which involves angular momentum respectively. Figure 15D shows that the space function operators in successively rotated coordinate systems. 20 of 64 | MAN

58 rot Silver presented the passive rotation of a space func- a;b;c a b 0 c 00 RP ð Þ¼ÈÉexpði LzÞexpði Ly Þexpði Lz Þ tion in the traditional way without Dirac notation and b b a ÈÉexpði Ly0 Þexpði Ly0 Þ ¼ expði LzÞ deﬁned the passive rotation operator by expðibLy0 ÞexpðicLz00 ÞexpðibLy0 Þ expðibLy0 Þ: rot;Silver c; b; a c b a : (90) RP ð Þ¼expði Lz00 Þ expði Ly0 Þ expði LzÞ (80) Replacing the inner three elementary rotation operators in The Euler angles and rotation axes differ with those in Eq. (90) by Eq. (89) yields Eq. (79). As this passive rotation operator describes the transformation of a space function Ψ(r), it becomes obviously that rot a; b; c a c b : RP ð Þ¼expði LzÞ expði LzÞ expði Ly0 Þ (91) rot;Silver c; b; a W 000 c c RP ð Þ ðr Þ Inserting exp(i Lz) exp(i Lz) into Eq. (91) yields 000 ¼ expðicL 00 Þ expðibL 0 Þ expðiaL ÞWðr Þ: (81) z y z rot a; b; c c c a RP ð Þ¼fgexpði LzÞ expði LzÞ expði LzÞ The generalization of Eq. (59) to Euler angles is c b c expði LzÞ expði Ly0 Þ¼expði LzÞ 000W000 1 000 rot a; b; c W 2 W c a c b : hr P ¼hjr RP ð Þji¼hrji fgexpði LzÞ expði LzÞ expði LzÞ expði Ly0 Þ 3 000 (92) ¼ W½ArotatedZ1Y2Z3ða; b; cÞr : (82) The inner three elementary rotation operators in Eq. (92) commute, which yields 5.5 | Passive rotation operator about ﬁxed rot a; b; c c a b : axes RP ð Þ¼expði LzÞ expði LzÞ expði Ly0 Þ (93)

We determine the right-handed passive rotation operator Inserting exp (iaLz) exp(iaLz) into Eq. (93) yields about fixed axes from that about rotated axes defined in rot c; b; a c a b Eq. (79). RP ð Þ¼expði LzÞ expði LzÞ expði Ly0 Þ a a c First we express the angular momentum operator Ly0 in fgÈÉexpði LzÞ expði LzÞ ¼ expði LzÞ terms of angular momentum operator Ly (Figure 11). Ly0 is expðiaLzÞ expðibLy0 Þ expðiaLzÞ expðiaLzÞ: derived from an active rotation of Ly about z-axis by angle (94) a (see Section 10 about rotation of vector operator): Finally, substituting Eq. (86) into Eq. (94) yields the a a : Ly0 ¼ expði LzÞLy expði LzÞ (83) right-handed passive rotation operator about fixed 12,42,56,58,67,70,75,76,78,79 Applying the following transformation formula1,74 axes

y y fixed c; b; a c b a : Uf ðAÞU ¼ f ðUAU Þ (84) RP ð Þ¼expði LzÞ expði LyÞ expði LzÞ (95) rot a; b; c to Eq. (83) yields The orders of Euler angles in the two operators RP ð Þ fixed c; b; a rot c; b; a and RP ð Þ are in the reverse order as for RA ð Þ b 0 a b a : fixed a; b; c expði Ly Þ¼expði LzÞ expði LyÞ expði LzÞ (85) and RA ð Þ. We have manipulated the expression of Rrotðc; b; aÞ, so that it only depends on angular momentum Equation (85) can be rewritten as A operators of the fixed coordinate system without taking into account the space function or the coordinate system. expðiaLzÞ expðibLy0 Þ expðiaLzÞ¼expðibLyÞ (86) Although Silver58 defined the passive rotation operator Similarly, we express the angular momentum operator about rotated axes with Eq. (80), which differs with Lz00 in terms of angular momentum operator Lz: Eq. (79) used by us, he deduced for the passive rotation operator about fixed axes the same relation as Eq. (95). L 00 ¼ expðibL 0 ÞL expðibL 0 Þ: (87) z y z y The same order of Euler angles appears in the two expres- Applying the formula in Eq. (84) to Eq. (87) yields sions of Silver (Eqs. (80) and (95)). This may generate conflicting results. This is another reason why rotation

expðicLz00 Þ¼expðibLy0 Þ expðicLzÞ expðibLy0 Þ: (88) operators about rotated axes should not be used. Equation (82) becomes Equation (88) can be rewritten as 000W000 1 000 fixed c; b; a W 2 W hr P ¼hjr RP ð Þji¼hrji expðibLy0 Þ expðicLz00 Þ expðibLy0 Þ¼expðicLzÞ: (89) 3 000 ¼ W½AfixedZ3Y2Z1ða; b; cÞr ; (96) Inserting exp(ibLy0) exp(ibLy0) into the right-hand side of Rrotða; b; cÞ yields P with3,67 MAN | 21 of 64

A ða; b; cÞ¼A ða; b; cÞ passive rotation operator about rotated axes. (b) When it is 0 rotatedZ1Y2Z3 fixedZ3Y2Z1 1 CaCbCc SaSc CaCbSc SaCc CaSb applied to space function or ket state as in Eq. (100), it is B C the left-handed active rotation operator about rotated axes. ¼ @ SaCbCc þ CaSc SaCbSc þ CaCc SaSb A: b c b c b Consider the right-handed rotations of coordinate system S C S S C fi W ; ; keeping xed the space function ðx y zÞ¼px (Fig- (97) ure 16). The space function is represented by an arrow for Due to various expressions for rotation operators, it is simplicity. The first (Figure 16B), second (Figure 16C), recommended24 to add A or P and fixed or rotated in their and third (Figure 16D) right-handed rotations of the coordi- definitions to distinguish their developed forms in rotated nate system are about z-axis by angle a=p, about y0-axis by and fixed coordinate systems. angle b=p/2, and about z″-axis by angle c=p/4, respec- We mention that the passive rotation operator about fixed tively. Figure 16D shows that the space function fixed h; 12,56,59,76,80 Wðx; y; zÞ¼p lies on the z‴-axis of the rotated coordi- axes RP ð nÞ in angle-axis parametrization is x nate system (O, x000, y000, z000). Rfixedðh; nÞ¼expðihn LÞ: (98) We should obtain the same result with left-handed P W ; ; active rotations of the space function ðx y zÞ¼px (Fig- fixed h; fixed c; b; a : Note that RP ð nÞRP ð Þ ure 17) about rotated axes of its body-attached coordinate system. The first (Figure 17B), second (Figure 17C), and 5.6 | Example V.2 third (Figure 17D) left-handed rotations of the space function are about z-axis by angle c=p/4, about y0-axis by angle In Section 4, we have checked that right-handed rotation of b=p/2, and about z″-axis by angle a=p, respectively. Fig- coordinate system about z-axis by angle a keeping the ure 17D shows that the rotated space function lies on the space function fixed is equivalent to left-handed active z-axis of the fixed coordinate system, in agreement with rotation of space function about z-axis by angle a in fixed Figure 16D. coordinate system. We extend this rule to rotations about We have checked that right-handed rotation of coordi- rotated axes and fixed axes by Euler angles with simple nate system about rotated axes by Euler angles (a, b, c) examples. keeping the space function fixed is equivalent to left- handed active rotation of space function about rotated axes c 5.6.1 | Passive rotation about rotated axes of its body-attached coordinate system by Euler angles ( , b, a). The standard formulation77 is: right-handed rotation Right-handed rotation of coordinate system about rotated of coordinate system about rotated axes by Euler angles (a, axes (Eq. (79)), keeping the space function fixed, is formu- b, c) keeping the space function fixed is equivalent to lated as right-handed active rotation of space function about rotated axes of its body-attached coordinate system by Euler angles 000 rot a; b; c 000 a b c : hjr RP ð Þ¼hjr expði LzÞ expði Ly0 Þ expði Lz00 Þ (c, b, a). (99) The first, second, and third right-handed rotations of coor- fi dinate systems are about z-axis by angle a, about y0-axis 5.6.2 | Passive rotation about xed axes by angle b, and about z″-axis by angle c, respectively. Right-handed rotation of coordinate system about fixed The same result can be obtained with left-handed active axes keeping fixed the space function is formulated as rotation of space function about rotated axes of its body- attached coordinate system as 000 fixed c;b;a 000 c b a : hjr RP ð Þ¼hjr expði LzÞexpði LyÞexpði LzÞ (101) rot a; b; c W a b c W : fi RP ð Þji¼expði Lz00 Þ expði Ly0 Þ expði LzÞji (100) The rst, second, and third right-handed rotations of coordinate system are about z-axis by angle c, about y-axis by We have to rename the rotation axes. This is another rea- angle b, and about z-axis by angle a, respectively. son why rotation operators about rotated axes should not The same result can be obtained with left-handed active fi be used. The rst, second, and third left-handed active rota- rotation of the space function about fixed axes as tions of space function are about z-axis by angle c, about Rfixedðc;b;aÞjiW ¼ expðicL ÞexpðibL ÞexpðiaL ÞjiW : (102) y0-axis by angle b, and about z″-axis by angle a, respec- P z y z tively. We do not need to rename the fixed rotation axes. The rot a; b; c The passive rotation operator RP ð Þ has two first, second, and third left-handed active rotations of the meanings: (a) When it is applied to coordinate system via space function are about z-axis by angle a, about y-axis by bra position basis hjr000 as in Eq. (99), it is the right-handed angle b, and about z-axis by angle c, respectively. 22 of 64 | MAN

FIGURE 17 Left-handed active rotations of space function Ψ(x,

y, z)=px about rotated axes of its body-attached coordinate system: (A) initial configuration; the first (B), second (C), and third (D) FIGURE 16 Right-handed rotations of coordinate system about rotations of the space function are about z-axis by angle c=p/4, about 0 00 axes of rotated coordinate systems, the space function Ψ(x, y, z)=px y -axis by angle b=p/2, and about z -axis by angle a=p, respectively remains fixed: (A) initial configuration; the first (B), second (C), and third (D) rotations of the coordinate system are about z-axis by angle a=p, about y0-axis by angle b=p/2, and about z00-axis by angle c=p/4, and third (Figure 18D) right-handed rotations of the coordi- respectively nate system are about z-axis by angle c=p/4, about y-axis by angle b=p/2, and about z-axis by angle a=p, respec- fixed c; b; a The passive rotation operator RP ð Þ has two tively. Figure 18D shows that the space function W ; ; ‴ meanings: (a) When it is applied to coordinate system via ðx y zÞ¼px lies on the z -axis of the rotated coordi- bra position basis hjr000 as in Eq. (101), it is the right-handed nate system (O, x000, y000, z000). passive rotation operator about fixed axes.77 (b) When it is Right-handed rotation of coordinate system about applied to space function or ket state as in Eq. (102), it is the rotated axes (Figure 16D) agrees with that about fixed axes left-handed active rotation operator about fixed axes. (Figure 18D). Consider the right-handed rotations of coordinate system We should obtain the same result when we perform fi W ; ; keeping xed the space function ðx y zÞ¼px (Fig- left-handed active rotations of the space function W ; ; fi fi ure 18). The space function is represented by an arrow for ðx y zÞ¼px about xed axes (Figure 19). The rst (Fig- simplicity. The first (Figure 18B), second (Figure 18C), ure 19B), second (Figure 19C), and third (Figure 19D) MAN | 23 of 64

FIGURE 18 Right-handed rotations of coordinate system about axes of the fixed coordinate system (O, x, y, z), the space function FIGURE 19 Left-handed active rotations of space function Ψ(x, Ψ fi fi fi (x, y, z)=px remains xed: (A) initial con guration; the rst (B), y, z)=px about axes of the fixed coordinate system (O, x, y, z): (A) second (C), and third (D) rotations of the coordinate system are about initial configuration; the first (B), second (C), and third (D) rotations z-axis by angle c=p/4, about y-axis by angle b=p/2, and about z-axis of the space function are about z-axis by angle a=p, about y-axis by by angle a=p, respectively angle b=p/2, and about z-axis by angle c=p/4, respectively

right-handed rotation of coordinate system about fixed axes left-handed rotations of the space function are about z-axis by Euler angles (c, b, a) keeping the space function fixed by angle a=p, about y-axis by angle b=p/2, and about z- is equivalent to right-handed active rotation of space func- axis by angle c=p/4, respectively. Figure 19D shows that tion about fixed axes by Euler angles (a, b, c). the rotated space function lies on the z-axis in the fixed coordinate system, in agreement with Figure 18D. We have checked that right-handed rotation of coordi- 6 | WHY ROTATION ABOUT FIXED nate system about fixed axes by Euler angles (c, b, a) AXES IS RECOMMENDED? keeping the space function fixed is equivalent to left- handed active rotation of space function about fixed axes Consider a right-handed rotation about the Z-axis by angle by Euler angles (a, b, c). The standard formulation77 is: / of a fixed coordinate system (O, X, Y, Z). A new 24 of 64 | MAN

by angle h, then a rotation of the rotated coordinate system about the rotated z-axis by angle /, we obtain

expðihLyÞ expði/LzÞ¼expði/LZ Þ expðihLY Þ: (108)

Subtracting Eq. (108) from Eq. (107), term by term, yields:

expði/LzÞ expðihLyÞexpðihLyÞ expði/LzÞ

¼ expðihLY Þ expði/LZ Þexpði/LZ Þ expðihLY Þ: (109)

If the two angles / and h are small, Eq. (109) becomes

ð1 þ i/LzÞð1 þ ihLyÞð1 þ ihLyÞð1 þ i/LzÞ

¼ð1 þ ihLY Þð1 þ i/LZ Þð1 þ i/LZ Þð1 þ ihLY Þ: (110) FIGURE 20 A rotation about Z-axis by angle / is followed by a rotation about the rotated y-axis by angle h Expanding each product term of Eq. (110) yields coordinate system (O, x, y, z) is generated (Figure 20). As LyLz LzLy ¼ LZ LY LY LZ : (111) the two axes Z and z are parallel, we can always write That is, hjr expði/LzÞ¼hjr expði/LZ Þ: (103) ½Ly; Lz¼½LZ ; LY : (112) The bra position basis hjr represents the rotated coordi- ﬁ nate system (O, x, y, z). Then we apply a right-handed The angular momentum operators in xed coordinate rotation about the y-axis by angle h to the rotated coordi- system and those in rotated one commute differ- 55,77,81–84 nate system (O, x, y, z). A new coordinate system is gener- ently. Therefore, we focus on rotations about ﬁ ated, whose bra position basis is hjr0 . Equation (103) xed axes in the remaining of the article. becomes 7 | WIGNER ACTIVE AND PASSIVE 0 0 hjr expði/LzÞ expðihLyÞ¼hjr expði/LZ Þ expðihLyÞ: (104) ROTATION MATRICES

The left-hand side of Eq. (104) involves angular Wigner rotation matrices realize transformations of covari- momentum operators about rotated axes, whereas the right- ant components of any spherical tensor under coordinate hand side involves angular momentum operators about rotations.55 We distinguish between Wigner active ﬁ ‘; ‘; xed and rotated axes. Dð AÞða; b; cÞ and passive Dð PÞðc; b; aÞ rotation matrices. Figure 20 shows that Ly is deduced from a right-handed However, they are related. Matrix elements with m0=m=0 active rotation of L about Z-axis by angle /: Y are Legendre polynomials P‘ðcos hÞ.

Ly ¼ expði/LZ ÞLY expði/LZ Þ: (105) Applying the transformation formula in Eq. (84) to 7.1 | Wigner active rotation matrix Eq. (105) yields ð‘;AÞ The Wigner active rotation matrix element D 0 ða; b; cÞ h / h / : m m expði LyÞ¼expði LZ Þ expði LY Þ expði LZ Þ (106) is defined by the right-handed active rotation operator fi 9,11,22,29,30,34,37– 28 about axes of xed coordinate system Replacing exp (ihL ) in Eq. (104) by that in Eq. (106) – y 39,61,63,71,73,85 87 provided by Eq. (66): transforms its right-hand side so that only angular momen- fi ð‘;AÞ a;b;c ‘; 0 fixed a;b;c ‘; tum operators about xed axes are involved. Therefore, Dm0m ð Þ¼hjm RA ð Þjim Eq. (104) becomes 0 ¼ hj‘; m expðiaLzÞexpðibLyÞ È 0 0 / h / / expðicLzÞji‘;m ¼expðiam Þhj‘;m expði LzÞ expði LyÞ¼expði LZ Þ expði LÉZ Þ expðibLyÞji‘;m expðicmÞ expðihLY Þ expði/LZ Þ a 0 ð‘;AÞ b c : ¼ expðihLY Þ expði/LZ Þ: (107) ¼ expði m Þdm0m ð Þexpði mÞ (113) ð‘;AÞ b fi Similarly, if we first apply a right-handed rotation of the The reduced matrix element dm0m ð Þ is de ned – fixed coordinate system (O, X, Y, Z) about the fixed Y-axis by9,13,22,34,37,55,56,72,77,88 90 MAN | 25 of 64

X ÂÃ 1 y X 0 0 2 0 fixed 00 00 fixed ‘; ‘ ! ‘ ! ‘ ! ‘ ! d 0 ‘; m R a;b;c ‘; m ‘; m R a;b;c ‘;m ; ð AÞ b w ½ð þmÞ ð mÞ ð þm Þ ð m Þ m m ¼ hjA ð ÞjihjA ð Þ j i d 0 ð Þ¼ ð1Þ 00 m m ð‘þm0 wÞ!ð‘mwÞ!w!ðwþmm0Þ! m w X hi ‘; ‘; 2‘mþm02w 2wm0þm d ð AÞ a; b; c ð AÞ a; b; c : b b m0m ¼ Dm0m00 ð Þ Dmm00 ð Þ cos sin : 00 2 2 m (114) Equations (118) and (119) are orthogonality relations among Wigner rotation matrices. The number w runs over all integer values for which the factorial arguments are non-negative. ð‘;AÞ b 7.2 | Wigner passive rotation matrix Another expression of dm0m ð Þ is also avail- 11,36,38,57,71,91 able: The Wigner passive rotation matrix element ð‘;PÞ 40,65,70,79,97,98 1 D 0 ðc; b; aÞ is defined by the right-handed X ‘ ! ‘ ! ‘ 0 ! ‘ 0 ! 2 m m ð‘;AÞ w ½ð þmÞ ð mÞ ð þm Þ ð m Þ fi d 0 ðbÞ¼ ð1Þ passive rotation operator about axes of xed coordinate sys- m m ‘ 0 ! ‘ ! ! 0 ! w ð m wÞ ð þmwÞ w ðwmþm Þ tem provided by Eq. (95): b 2‘þmm02w b 2wþm0m cos sin : 2 2 ð‘;PÞ c; b; a ‘; 0 fixed c; b; a ‘; Dm0m ð Þ¼hjm RP ð Þjim (115) 0 ¼ hj‘; m expðicLzÞ expðibLyÞ expðiaLzÞji‘; m ð1;AÞ a; b; c ‘ 9,22,24,34,36,38,54– 0 0 The matrix D ð Þ where =1is ¼ expðicm Þhj‘; m expðibLyÞji‘; m expðiamÞ 57,71,87,91–93 c 0 ð‘;PÞ b a : ¼ expði m Þdm0m ð Þ expði mÞ ; Dð1 AÞða;b;cÞ (120) ji1 ji0 ji1 Fano and Racah,78 Fano and Rao,99 and Schwinger12 0 a c a a c 1 1ð1þcosbÞeið þ Þ p1ffiffi sinbei 1ð1cosbÞeið Þ used the Euler angles Ψ, h, and / instead of c, b, and a, hj1 B2 2 2 C ¼ c c : B p1ffiffi sinbei cosb p1ffiffi sinbei C respectively. Otherwise, their Wigner passive rotation hj0 @ 2 2 A a c a a c matrix elements are identical to Eq. (120). hj1 1ð1cosbÞeið Þ p1ffiffi sinbei 1ð1þcosbÞeið þ Þ 2 2 2 Wigner100 and Weissbluth59 exchanged the positions of (116) a and c. This means that the Euler angles defined with ; c fi b The Wigner active rotation matrix Dð2 AÞða; b; cÞ is avail- rotated coordinate systems are (the rst rotation angle), able.85–87,91,94,95 The reduced Wigner active rotation matrix (the second rotation angle), and a (the third rotation angle). ; 96 55 fi dð4 AÞðbÞ is reported in Man. Varshalovich et al. tabulate In other words, the Euler angles in xed coordinate system ‘; a fi b dð AÞðbÞ from ‘=1/2 up to and including ‘=5. are (the rst rotation angle), (the second rotation We mention that Wigner active rotation matrix in angle- angle), and c (the third rotation angle). As a result, the 100 axis of rotation parametrization55,63 is defined by Wigner passive rotation matrix described by Wigner and Weissbluth59 has the following expression: ð‘;AÞ h; ‘; 0 fixed h; ‘; : ‘; ; Dm0m ð nÞ¼hjm RA ð nÞjim (117) ð P WignerÞ a;b;c ‘; 0 a b c ‘; Dm0m ð Þ¼hjm expði LzÞexpði LyÞexpði LzÞjim ‘; ð AÞ 55 0 ð‘;PÞ Explicit forms of D 0 h; n for ‘=1 and 2 are available. a b c : m m ð Þ ¼expði m Þdm0m ð Þexpði mÞ ‘; Because the basis vectors jim are orthonormal and remain (121) soÂÃ on rotation, the WignerÂÃ rotation matrices are unitary, that is, y 1 ð‘;PÞ b Dð‘;AÞða; b; cÞ ¼ Dð‘;AÞða; b; cÞ . As a result,9,88 The reduced Wigner passive rotation matrix d ð Þ is 100 78 59 ÂÃ given by Wigner, Fano and Racah, Weissbluth, ‘; 0 ‘; ‘; 0 fixed a; b; c y fixed a; b; c ‘; ; Scheck,98 and Fano and Rau:99 him j m ¼ hjm RA ð Þ RA ð Þjim 1 X 0 0 2 X ÂÃ ð‘;PÞ w ½ð‘þmÞ!ð‘mÞ!ð‘þm Þ!ð‘m Þ! 0 fixed y 00 00 fixed d b d 0 ‘; m R a;b;c ‘; m ‘; m R a;b;c ‘;m ; m0m ð Þ¼ ð1Þ m m ¼ hjA ð Þ jihjA ð Þji ð‘m0 wÞ!ð‘þmwÞ!w!ðwmþm0Þ! m00 w ‘ 0 0 (118) b 2 þmm 2w b 2wþm m cos sin : hi 2 2 X d ð‘;AÞ a; b; c ð‘;AÞ a; b; c : m0m ¼ Dm00m0 ð Þ Dm00m ð Þ (122) m00 It differs with Eq. (115) by a minus sign for the sinus function. 9,22,88 Similarly, we have In Eqs. [43.8], [43.12], and [43.13] of Davydov,32 the two ÂÃ y indices of Wigner passive rotation matrix element ‘; 0 ‘; ‘; 0 fixed a;b;c fixed a;b;c ‘; ; ‘; ; him j m ¼hjm RA ð Þ RA ð Þ jim (119) ð P DavydovÞ c; b; a Dmm0 ð Þ are exchanged. But the matrix elements 26 of 64 | MAN no fi remain identical those de ned in Eq. (120). In other words, ð‘;AÞ ð‘;AÞ ‘; ; a; b; c a 0 b c ð P DavydovÞ Dm0m ð Þ ¼ expði m Þdm0m ð Þ expði mÞ the matrix elements D 0 ðc; b; aÞ are the transposes of ‘; mm ð PÞ ð‘;PÞ 0 D 0 ðc; b; aÞ: c b a m m ¼ expði mÞdmm0 ð Þ expði m Þ ð‘;P;DavydovÞ c;b;a ‘; 0 c b a ‘; ð‘;PÞ c; b; a : Dmm0 ð Þ¼hjm expði LzÞexpði LyÞexpði LzÞjim ¼ Dmm0 ð Þ (129) 0 0 ¼expðicm Þhj‘;m expðibLyÞji‘;m expðiamÞ For inverse rotation that is accomplished by performing the c 0 ð‘;P;DavydovÞ b a : ¼expði m Þdmm0 ð Þexpði mÞ rotations through negative Euler angles about the same axes (123) but in opposite order, Wigner active rotation matrix becomes The Wigner passive rotation matrix Dð1;PÞðc; b; aÞ for hi ‘ 56,58,65,70 ð‘;AÞ a; b; c 1 ð‘;AÞ c; b; a =1is Dm0m ð Þ ¼ Dm0m ð Þ ‘; ð1;PÞ 0 ð AÞ c;b;a ¼ expðicm Þd 0 ðbÞ expðiamÞ D ð Þ m m (130) ji1 ji0 ji1 c 0 ð‘;PÞ b a 0 1 ¼ expði m Þdm0m ð Þ expði mÞ 1 iðaþcÞ 1 ic 1 iðacÞ ð1þcosbÞe pffiffi sinbe ð1cosbÞe ð‘;PÞ hj1 2 2 2 c; b; a : B C ¼ Dm0m ð Þ ¼ a a : B p1ffiffi sinbei cosb p1ffiffi sinbei C hj0 @ 2 2 A a c c a c Equations (129) and (130) yield hj1 1ð1cosbÞeið Þ p1ffiffisinbei 1ð1þcosbÞeið þ Þ 2 2 2 (124) no ð‘;AÞ ð‘;AÞ 59 D c; b; a D a; b; c : As Weissbluth defined the first Euler angle with c and m0m ð Þ¼ mm0 ð Þ (131) the third Euler angle with a, his Wigner passive rotation Similarly, Wigner active rotation matrix is the com- matrix is Dð1;PÞða; b; cÞ, exchanging the positions of c and a plex conjugate and transpose of Wigner passive rotation in the matrix of Eq. (124). It is worth noting that Wigner100 matrix: altered not only the Euler angles (like Weissbluth) but also no the order of jim in the passive rotation matrix: ð‘;PÞ c; b; a ð‘;AÞ a; b; c : Dm0m ð Þ ¼ Dmm0 ð Þ (132) ; ; Dð1 P WignerÞða;b;cÞ Furthermore, ji1 ji0 j1i 0 a c a a c 1 1ð1þcosbÞeið þ Þ p1ffiffi sinbei 1ð1cosbÞeið Þ ð‘;PÞ ð‘;AÞ hj1 2 2 2 D 0 a; b; c D 0 a; b; c ; (133) B C: m m ð Þ¼ m m ð Þ ¼ B 1ffiffi ic 1ffiffi ic C hj0 @ p sinbe cosb p sinbe A 2 2 and 1 1 b iðacÞ 1ffiffi b ia 1 b iðaþcÞ no hj ð1cos Þe p sin e ð1þcos Þe 2 2 2 ð‘;PÞ ð‘;PÞ D 0 ða; b; cÞ¼ D 0 ðc; b; aÞ : (134) (125) m m mm The Wigner passive rotation matrix Dð2;PÞðc; b; aÞ for – ‘=2 is available in Refs.101 103 Reduced matrices dð‘;AÞðbÞ and dð‘;PÞðbÞ are related:36,97,104 8 | SPHERICAL HARMONICS no ‘; T ‘; ‘; dð AÞðbÞ ¼ dð AÞðbÞ¼dð PÞðbÞ; As the formulation of spherical tensor is based on spherical harmonics, we begin with the presentation of spherical har- no ‘; T ‘; ‘; monics. The latter are extensively used in dð PÞðbÞ ¼ dð PÞðbÞ¼dð AÞðbÞ: (126) NMR.7,74,86,87,102,105 For example, the perturbed dipolar Hamiltonian for a single pair of identical spins can be written in the spherical tensor formalism as106 7.3 | Some properties of Wigner rotation X H ¼ ðÞ1 mF h ; / T ; matrices D ij 2;m ij ij 2;m m rffiffiffiffiffiffiffiffi (135) l c2 p The matrix elements of Wigner rotation matrices are as follows: 0 h 24 F ; h ; / ¼ Y ; h ; / : 2 m ij ij 4pr3 5 2 m ij ij ð‘;AÞ a; b; c a 0 ð‘;AÞ b c ; ij Dm0m ð Þ¼expði m Þdm0m ð Þ expði mÞ (127) Y ; h ; / are spherical harmonics of order 2 and com- and 2 m ij ij ð‘;PÞ 0 ð‘;PÞ ponent m, whereas T2;m are products of spin operators. D 0 ðc; b; aÞ¼expðicm Þd 0 ðbÞ expðiamÞ: (128) m m m m Spherical harmonics are related to Wigner rotation matri- It follows that the complex conjugate of Wigner active rota- ces via simple relations. These relations allow us to deter- tion matrix is the transpose of Wigner passive rotation matrix:97 mine the type (active or passive) of Wigner rotation matrix. MAN | 27 of 64

‘ TABLE 1 First few normalized harmonic polynomials r Y‘ (x, y, 8.1 | Definition of spherical harmonics ,m z) and normalized spherical harmonics Y‘,m(h, /) in Condon and Shortley phase convention and in covariant notation, with x=r sin h Spherical harmonics Y‘; ðh; /Þ normalized to unity are m / h / h common eigen functions expressed in spherical polar coor- cos , y=r sin sin , and z=r cos 2 ‘ dinates of orbital angular momentum operator squared L ‘ mrY‘,m(x, y, z) Y‘,m(h, /) L 31,34,41 qffiffiffiffi q ffiffiffiffi and z. They may be written as the product of two 00 1 1 p p functions, one that depends only on the polar angle h is a q4ffiffiffiffi q4ffiffiffiffi complex-valued function and the other that depends only 10 3 z 3 cos h q4pffiffiffiffi q4pffiffiffiffi on the azimuthal angle / is a real-valued func- 3 3 i/ 1 1 pðÞx iy p sin he tion:22,34,38,47,55 8 8 qffiffiffiffiffiffiq qffiffiffiffiffiffiq 20 5 1 2 2 5 1 2h 4p 4ðÞ3z r 4p 4ðÞ3cos 1 qffiffiffiffiffiffiq qffiffiffiffiffiffiq Y‘; ðh; /Þ¼H‘; ðhÞU ð/Þ: (136) m m m 2 1 5 3 5 3 h h i/ 4p 2zxðÞ iy 4p 2 cos sin e These two functions are qffiffiffiffiqffiffi qffiffiffiffiqffiffi 2 2 5 3 2 5 3 2h 2i/ 4p 8ðÞx iy 4p 8sin e

U / im/; mð Þ¼e (137) TABLE 2 Associated Legendre functions P‘ ðxÞ and P‘ ðcos hÞ and for m≥0, m m ‘ h mP‘mðxÞ P‘mðcos Þ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = 2 1 2 h 2‘ þ 1 ð‘ mÞ! 11 1 x sin m = H h 1 P cos h : (138) 2 1 2 ‘;mð Þ¼ð Þ p ‘ ! ‘mð Þ 21 3x 1 x 3 cos h sin h 4 ð þ mÞ 231 x2 3 sin2 h m In quantum mechanics, the phase factor ð1Þ in the 1=2 31 3 5x2 1 1 x2 3 5 cos2 h 1 sin h fi 2 2 de nition of spherical harmonics via Eq. (138) was intro- 2 2 107 215x 1 x 15 cos h sin h duced by Condon and Shortley. Other conventions are = 2 3 2 3 h available for spherical harmonics.49,61,108 As a result, this 3151 x 15 sin phase factor ð1Þm does not appear in the associated Legendre functions11,34,38,54,57,61 defined by the Rodrigues0 TABLE 3 First six Legendre polynomials P‘ðxÞ of degree ‘, 1 formula ≤ x ≤ +1 m d ‘ h m h h ; P‘ðxÞ P‘mðcos Þ¼sin h P‘ðcos Þ (139) dðcos Þ 01 1 x where 1 2 2 2 3x 1 ‘ 1 d ‘ 1 x3 x h 2 h 3 2 5 3 P‘ðcos Þ¼ ‘‘! h ðcos 1Þ (140) 2 dðcos Þ 4 1 35x4 30x2 þ 3 8 5 1 63x5 70x3 þ 15x are the ordinary Legendre polynomials of degree 8 ‘.11,34,38,54,57,109 The associated Legendre functions ‘ m P‘ ðcos hÞ are real and depend only on jjm . h þ h : m P‘mð cos Þ¼ðÞ 1 P‘mðcos Þ (142) Table 1 presents some spherical harmonics 1,31,34,70,110 in Condon and Shortley convention and covariant nota- Table 3 presents some ordinary Legendre polynomials h 11,34,54,57,61,109,110 tion.55,111 They have negative phases for positive odd m, P‘ðcos Þ. These polynomials are either h ‘ but positive phases for all other m. They are extensively even or odd functions of x=cos for even or odd degrees . used in NMR. Furthermore, h ‘ h : Table 2 presents some associated Legendre func- P‘ð cos Þ¼ðÞ 1 P‘ðcos Þ (143) tions,11,54 but P cos h with m=0 are not reported ‘mð Þ The spherical harmonics satisfy the identity because P cos h P cos h . Furthermore, 11,15,22,25,31,36,38,55 ‘0ð Þ¼ ‘ð Þ Y ðh; /Þ ¼ Y ðh; /Þ¼ð1ÞmY ðh; /Þ ðÞ‘ m ! ‘;m ‘;m ‘;m P cos h 1 m P cos h ; (141) ‘mð Þ¼ðÞ ‘ ! ‘mð Þ m h; / : ðÞþ m ¼ð1Þ Y‘;mð Þ (144) and For the special case of m=0,15,22,31,38 28 of 64 | MAN rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffi 2‘ þ 1 3 x 1 Y ðh; /Þ¼H ðhÞ¼ P ðÞcos h : (145) pffiffiffi ¼ Y ; ðrÞY ; ðrÞ ; ‘;0 ‘;0 p ‘ 4p 2 1 1 1 1 4 rffiffiffiffiffiffi2r Because P ðÞ1 vanishes except for m=0, that is, when 3 z ‘m ¼ Y ; ðrÞ; 15,24,38,55,59 p 1 0 h=0, rffiffiffiffiffiffi 4 r 8 < 0 for m 6¼ 0 3 yffiffiffi i p ¼ Y1;1ðrÞþY1;1ðrÞ : (152) ; / qffiffiffiffiffiffiffiffi 4p 2r 2 Y‘;mð0 Þ¼: (146) 2‘þ1 for m 0 : 4p ¼ The set of operator equivalent for normalized spherical harmonics is41,113 This definition is rewritten as11,22,73 rffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffi 4p 2‘ þ 1 T‘; ¼ Y‘; ðh; /Þ: (153) Y ð0; /Þ¼ d : (147) m 2‘ þ 1 m ‘;m 4p m;0 This set is called Racah tensor of rank ‘.

8.2 | Spherical harmonics as covariant 8.3 | Relations with Wigner active rotation components matrix The Cartesian components of a position vector r=(x, y, z) The relations between spherical harmonics and Wigner h / of length r, whose0 polar1 angles0 are and1 , are rotation matrices allow us to know how active or passive x r sin h cos / rotations are studied. Haeberlen102 and Chandrakumar and @ y A ¼ @ r sin h sin / A: (148) Subramanian101 use these relations in secular NMR Hamil- z r cos h tonians under passive rotations. In Section 5 dealing with rotation of space function, we The covariant spherical components of r are defined show that the quantity hjr Rfixedða; b; cÞjiW has two meanings: by1,28,57,58,112 A (a) right-handed active rotation of space function represented 0 1 0 1 1ffiffi fixed a; b; c W p ðÞx þ iy by RA ð Þji; (b) left-handed rotation of coordinate r1þ1 2 @ A @ A: system represented by hjr Rfixedða; b; cÞ. In other words, the r10 ¼ z (149) A p1ffiffi ðÞx iy bra position basis r represents the coordinate system. r11 2 hj They are also known as the standard components.34 Substituting Eq. (148) into Eq. (149) yields36,82,88 8.3.1 | First relation

0 1 Suppose an initial position vector r1 is along the z-axis of 0 1 / p1ffiffi ðÞx þ iy p1ffiffi r sin hei the fixed coordinate system (O, x, y, z) (Figure 21). We 2 B 2 C B C B C apply an active rotation on it such that the final position @ z A ¼ @ r cos h A / vector r is along some arbitrary direction specified by p1ffiffi ðÞx iy p1ffiffi r sin hei 2 20 1 ðÞh; / . Two procedures are possible for this active rotation. rffiffiffiffiffiffi h; / rY1;þ1ð Þ We rotate r first about the y-axis by angle b=h, then 4pB C 1 ¼ @ rY ; ðh; /Þ A: (150) about the z-axis by angle a=/. The polar angles of the 3 1 0 h; / actively rotated vector r in (O, x, y, z) are ðÞh; / . So the rY1;1ð Þ ket position vectors in this first procedure are related as25 This means Y h; / Y h; / Y h; / 1;þ1ð Þ, 1;0ð Þ, and 1;1ð Þ are also covariant spherical components of spherical harmonics fixed a /; b h; c : jir ¼ RA ð ¼ ¼ ¼ 0Þ r1 (154) with order ‘=1. They are36 fi In contrast, if we rotate r1 rst about the z-axis by angle / rffiffiffiffiffiffi rffiffiffiffiffiffi (r1 remains unchanged), then about the y-axis by angle 3 1 xþiy 3 z h, the actively rotated vector r is in the x; z plane. Its Y ; ðrÞ¼ pffiffiffi ; Y ; ðrÞ¼ ; fg 1 1 4p r 1 0 4pr h / rffiffiffiffiffiffi 2 polar angles are and =0. As one of polar angles is zero, we give up this second procedure. 3 1ffiffiffixiy Y1;1ðrÞ¼ p : (151) 4p 2 r We study the rotations of r1 described by Eq. (154) and shown in Figure 21. We introduce the closure relation We can express them as into Eq. (154): MAN | 29 of 64

X fixed fi ji¼r R ð/; h; 0Þji‘; m h‘; m r : (155) By extension, the rst relation can be rewritten A 1 11,22,24,25,30,34,36,38,39,47,52,55–57,60,61,73,87,88,90,91,111,115,116 m as: rffiffiffiffiffiffiffiffiffiffiffiffiffi Then we multiply both sides of Eq. (155) by hj‘; m0 : hi 2‘ þ 1 ð‘;AÞ Y 0 ðb; aÞ ¼ D 0 ða; b; cÞ; ðindependent of cÞ: X ‘;m 4p m 0 ‘; 0 ‘; 0 fixed /; h; ‘; ‘; h m jir ¼ hjm RA ð 0Þjim h m r1 (161) m X (156) ð‘;AÞ D 0 /; h; 0 ‘; m r : ¼ m m ð Þh 1 ‘ 0 (1,A) m An example of Eq. (161) with =m =1, D in b; a Eq. (116), and Y1;1ð Þ in Table 1 is As spherical harmonic is defined by ! rffiffiffiffiffiffi rffiffiffiffiffiffi rffiffiffiffiffiffi 3 ; 3 3 Dð1 AÞ a; b; c sin beia sin beia 0 0 p 10 ð Þ¼ p ¼ p Y‘; 0 ðh; /Þ¼hrji‘; m ¼ hh; /ji‘; m ; (157) 4 8 8 m hi hi ‘; h; / b; a : we deduce from Figure 21 that h m r1 is Y‘;mð Þ ¼ Y1;1ð Þ evaluated at h=0 and / undetermined:25,36,59 (162) hihi ‘; h ; / h ; / d h m r1 ¼ Y‘;mð ¼ 0 Þ ¼ Y‘;mð ¼ 0 ¼ 0Þ m;0 rffiffiffiffiffiffiffiffiffiffiffiffiffi 8.3.2 | Second relation 2‘ þ 1 ¼ d : 4p m;0 Using Eqs. (126) and (127), it is easy to show that hi (158) ð‘;AÞ /; h; ð‘;AÞ ; h; / : Dm0 ð 0Þ ¼ D0m ð0 Þ (163) Therefore,22,38 The first relation defined in Eq. (160) and the identity rffiffiffiffiffiffiffiffiffiffiffiffiffi 2‘ þ 1 in Eq. (144) yield Y ðh ¼ 0; /Þ¼Y ðh ¼ 0; / ¼ 0Þd ¼ d : ‘;m ‘;m m;0 4p m;0 rffiffiffiffiffiffiffiffiffiffiffiffiffi (159) Â Ã ð‘;AÞ 4p D ð/; h; 0Þ ¼ Y‘; ðh; /Þ As a result,25,37,38,52,56,73,114 Eq. (156) becomes the first m0 2‘ þ 1 m rffiffiffiffiffiffiffiffiffiffiffiffiffi (164) relation connecting spherical harmonic with Wigner active 4p Y h; / : rotation matrix: ¼ ‘ ‘;mð Þ hirffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ 1 2‘ þ 1 ð‘;AÞ Y 0 ðh; /Þ ¼ D 0 ð/; h; 0Þ: (160) ‘;m 4p m 0 Therefore, the second relation connecting spherical harmonics with Wigner active rotation matrix is

rffiffiffiffiffiffiffiffiffiffiffiffiffi 2‘ þ 1 ‘; Y ðh; /Þ¼ Dð AÞð0; h; /Þ: (165) ‘;m 4p 0m By extension, the second relation can be rewritten as11,22,24,38,55,56,60,61,111

rffiffiffiffiffiffiffiffiffiffiffiffiffi 2‘ þ 1 ‘; Y ðb; cÞ¼ Dð AÞða; b; cÞ; ðindependent of aÞ; ‘;m 4p 0m (166) or34,116

hirffiffiffiffiffiffiffiffiffiffiffiffi 2‘þ1 ‘; Y ðb;cÞ ¼ðÞ1 m Dð AÞða;b;cÞ;ðindependent of aÞ: ‘;m 4p 0m (167) FIGURE 21 Right-handed active rotations of position vector from ‘ ð1;AÞ r1 to r in a single coordinate system: vector r1 is first rotated about the An example of Eq. (167) with =m=1, D in fi b h fi a / b; c xed y-axis by angle = , then about the xed z-axis by angle = Eq. (116), and Y1;1ð Þ in Table 1 is 30 of 64 | MAN rffiffiffiffiffiffi rffiffiffiffiffiffi rffiffiffiffiffiffi ! 0 rot a ; b h; c / : 3 ; 3 3 hjr ¼ hjr RP ð ¼ 0 ¼ ¼ Þ (171) Dð1 AÞ a; b; c b ic b ic p 01 ð Þ¼ p sin e ¼ p sin e 4 8 8 Conversely, the ket position vectors are related by hi b; c : ¼ Y1;1ð Þ 0 rot a ; b h; c / : (168) ji¼r RP ð ¼ 0 ¼ ¼ Þjir (172) ð‘;AÞ a; b; c Wigner active rotation matrix elements Dm0m ð Þ The polar angles of the position vector in the rotated coordinate with m0=0 (Eq. (167)) or m=0 (Eq. (161)) are proportional system (O, x0, y0, z0)are(h, /). We replace the passive rota- to the complex conjugate of normalized spherical harmon- tion operator about rotated axes by the passive rotation opera- b; c b; a fi ics Y‘;mð Þ and Y‘;m0 ð Þ in Condon and Shortley con- tor about xed axes in Eq. (172), which becomes vention and covariant notation, respectively. 0 fixed c /; b h; a : jir ¼ RP ð ¼ ¼ ¼ 0Þjir (173) 8.4 | Relations with Wigner passive rotation In contrast, if we rotate (O, x, y, z) first about the fixed matrix z-axis by angle a=/, then about rotated y0-axis by angle In Section 5 dealing with rotation of space function, we show b=h (Figure 23), the passively rotated vector r0 is in the 000 fixed c; b; a W fgx0; z0 plane. The polar angles of r0 in the rotated coordi- that the quantity hjr RP ð Þjihas two meanings: (a) right-handed rotation of coordinate system represented by nate system (O, x0, y0, z0) are (h, /=0). As one of the 000 fixed c; b; a polar angles is zero, we give up this second procedure. hjr RP ð Þ; (b) left-handed active rotation of space fixed c; b; a W We focus on rotations shown in Figure 22 and function represented by RP ð Þji. In other words, the bra position basis hjr000 represents the rotated coordinate Eq. (173) from the first procedure. We introduce the clo- system.12 sure relation in Eq. (173):

8.4.1 | First relation Suppose the position vector r is along the z-axis in the ﬁxed coordinate system (O, x, y, z). We perform a right- handed rotation of coordinate system that transforms (O, x, y, z) into (O, x0, y0, z0). The position vector in (O, x0, y0, z0)isr0. This corresponds to a passive rotation of the position vector, which is viewed in the ﬁxed coordinate system (O, x, y, z) and in the rotated coordinate system (O, x0, y0, z0). The two bra position bases are related by (see Eq. (96))56

0 fixed c; b; a : FIGURE 22 Right-handed rotations of coordinate system from hjr RP ð Þ¼hjr (169) (O, x, y, z)to(O, x0,y0,z0): (O, x, y, z)isﬁrst rotated about the ﬁxed 56 Conversely, the two ket position vectors are related by y-axis by angle b=h, then about the rotated z0-axis by angle c=/

0 fixed c; b; a : jir ¼ RP ð Þjir (170) fixed c; b; a W Contrary to the transformation RP ð Þjiwhere fixed c; b; a RP ð Þ is left-handed active rotation operator for ket state as mentioned above, now the transformation fixed c; b; a RP ð Þjir is right-handed passive rotation operator applied to ket position vector. Two procedures are possible for this right-handed passive rotation of position vector. We rotate (O, x, y, z) first about the fixed y-axis by angle b=h, then about the rotated z0-axis by angle c=/ (Figure 22). As usual, the Euler angles are defined with rotated axes. The passive rotation operator about rotated FIGURE 23 Right-handed rotations of coordinate system from rot a ; b h; c / 0 0 0 fi fi axes RP ð ¼ 0 ¼ ¼ Þ is provided by Eq. (79). (O, x, y, z)to(O, x ,y,z): (O, x, y, z)is rst rotated about the xed The bra position bases in this first procedure are related by z-axis by angle a=/, then about the rotated y0-axis by angle b=h MAN | 31 of 64 "# X hirffiffiffiffiffiffiffiffiffiffiffiffiffi 0 fixed c /; b h; a ‘; ‘; : ‘; p jir ¼ RP ð ¼ ¼ ¼ 0Þjim h mjir (174) ð PÞ m 4 D ð/; h; 0Þ ¼ðÞ1 Y‘; ðh; /Þ m m0 2‘ þ 1 m We multiply both sides of Eq. (174) by hj‘; m0 : rffiffiffiffiffiffiffiffiffiffiffiffiffi p 4 h; / : X ¼ ‘ Y‘;mð Þ ‘; 0 0 ‘; 0 fixed c /; b h; a ‘; ‘; : 2 þ 1 h m jir ¼ hjm RP ð ¼ ¼ ¼0Þjim h mjir m (182) (175) Therefore, the second relation connecting spherical har- 12 fi monics with Wigner passive rotation matrix is By de nition of spherical harmonics, rffiffiffiffiffiffiffiffiffiffiffiffiffi ‘ X 2 þ 1 ð‘;PÞ È É Y h; / D ; h; / : 0‘; 0 ‘; 0 fixed c /; b h; ‘;mð Þ¼ p 0m ð0 Þ (183) r m ¼ m RP ð ¼ ¼ 4 am ‘; ‘; : ¼ 0Þjim fghrjim (176) By extension, the second relation can be rewritten 56,58,80,97,102 In the rotated coordinate system (O, x0, y0, z0) repre- as rffiffiffiffiffiffiffiffiffiffiffiffiffi sented by hjr0 , the polar angles of r0 are ðÞh; / . In the 2‘ þ 1 ð‘;PÞ fixed coordinate system (O, x, y, z) represented by hjr , the Y‘; ðb; aÞ¼ D ðc; b; aÞ; ðindependent of cÞ: m 4p 0m h / polar angles of r are =0 and undetermined. We deduce (184) from Eq. (176) that 32 ÂÃX In Eq. [43.17] of Davydov, the two indices 0 and m in ð‘;PÞ ð‘;PÞ Y‘; 0 h; / D c /; b h; c; b; a m ð Þ ¼ rffiffiffiffiffiffiffiffiffiffiffiffiffim0m ð ¼ ¼ D0m ð Þ are exchanged. m ‘ ‘ 0 (1,P) a 2 þ 1d An example of Eq. (184) with =m =1, D in ¼ 0Þ p mr;0 ffiffiffiffiffiffiffiffiffiffiffiffiffi b; a 4 Eq. (124), and Y1;1ð Þ in Table 1 is ð‘;PÞ 2‘ þ 1 ¼ D 0 ð/; h; 0Þ : (177) m 0 4p rffiffiffiffiffiffi rffiffiffiffiffiffi Using Eq. (144), the latter becomes the first relation 3 ð1;PÞ 3 ia D ðc; b; aÞ¼ sin be ¼ Y ; ðb; aÞ: (185) connecting spherical harmonic with Wigner passive rotation 4p 01 8p 1 1 matrix: rffiffiffiffiffiffiffiffiffiffiffiffiffi ð‘;PÞ Wigner passive rotation matrix elements D 0 ðc; b; aÞ 0 ‘; 2‘ þ 1 m m m h; / ð PÞ /; h; : 0 ðÞ1 Y‘;m0 ð Þ¼D 0 ð 0Þ (178) with m =0 (Eq. (184)) or m=0 (Eq. (179)) are proportional m 0 4p to normalized spherical harmonics Y‘; ðb; aÞ and fi m By extension, the rst relation can be rewritten Y‘;m0 ðb; cÞ in Condon and Shortley convention and covari- 56,58,97,102 as rffiffiffiffiffiffiffiffiffiffiffiffi ant notation, respectively. m0 2‘ þ 1 ð‘;PÞ Y‘; 0 ðb;cÞ¼ðÞ 1 D 0 ðc;b;aÞ;ðindependent of aÞ: m 4p m 0 (179) 9 | ROTATION OF SPHERICAL HARMONIC AND SPHERICAL In Eq. [43.17] of Davydov,32 the two indices m0 and 0 in ð‘;PÞ c; b; a TENSOR Dm00 ð Þ are exchanged. An example of Eq. (179) with ‘=m0=1, Dð1;PÞ in An application of Wigner active rotation matrix is the Eq. (124), and Y ; ðb; cÞ in Table 1 is 1 1 transformation of spherical harmonics induced by an active rffiffiffiffiffiffi rffiffiffiffiffiffi rotation of the physical system. This rotation may occurred 3 ð1;PÞ 3 ic D ðc; b; aÞ¼ sin be ¼ Y ; ðb; cÞ: (180) in the same coordinate system or in two coordinate sys- 4p 10 8p 1 1 tems.61 In the configuration of one coordinate system, the 8.4.2 | Second relation active rotation of physical system is described by the Euler angles ða; b; cÞ. In the configuration of two coordinate sys- Using Eqs. (126) and (128), it is easy to show that tems, the second coordinate system is attached to the physi- hi cal system and rotates with it. It is called body-attached ð‘;PÞ ð‘;PÞ coordinate system. Before rotation, the two coordinate D ð/; h; 0Þ ¼ D ð0; h; /Þ: (181) m0 0m systems coincide. The relative orientation of these two The first relation defined in Eq. (178) and the identity coordinate systems after rotation is described by the Euler in Eq. (144) yield angles. 32 of 64 | MAN

The spherical harmonics, eigen functions of orbital coordinate system ðO; x; y; zÞ. The initial position vector is fi angular momentum, are the coordinate representation of the named r1 and the nal or right-hand actively rotated one is angular momentum eigenket vectors ji‘; m with integer val- r in ðO; x; y; zÞ as shown in Figure 24A,B. The correspond- ues of ‘ and m. The actively rotated eigenket vectors ing ket position vectors are related by ji‘; m 0 and the fixed eigenket vectors ji‘; m are 11,14,15,29,36,47 related: fixed a; b; c : jir ¼ RA ð Þ r1 (187) ‘; 0 fixed a; b; c ‘; jim ¼ RA ð Þjim Xþ‘ Conversely, the bra position bases are related by (see ‘; 0 ‘; 0 fixed a; b; c ‘; ¼ jim hjm RA ð Þjim Eq. (64)) m0¼‘ (186) Xþ‘ ð‘;AÞ fixed a; b; c : ‘; 0 a; b; c : r1 ¼ hjr RA ð Þ (188) ¼ jim Dm0m ð Þ m0¼‘ Multiplying each member of Eq. (188) by ji‘; m and The eigenket vectors are in row matrices. including the closure relation yield The passive rotation of spherical harmonics or the rotation of coordinate system associated with spherical harmonics is ‘; fixed a; b; c ‘; more subtle than the active rotation of spherical harmonics. r1jim ¼ hjr RA ð Þjim We provide simple examples to support our results. Xþ‘ ‘; 0 ‘; 0 fixed a; b; c ‘; Finally, we rely on properties deduced from rotations of ¼ hjr jim hjm RA ð Þjim m0¼‘ (189) spherical harmonics to define the rotation properties of Xþ‘ spherical tensor. ‘; 0 ð‘;AÞ a; b; c : ¼ hrjim Dm0m ð Þ m0¼‘ 9.1 | Right-handed active rotation of The polar angles of the initial position vectorr1 and spherical harmonic in one coordinate system ; ; ; h ; / those of the rotated one r in ðO x y zÞ are 1 1 and h; / Consider the first configuration dealing with the right- ðÞ, respectively. These angles allow us to rewrite handed active rotation of spherical harmonics in a fixed Eq. (189) as

FIGURE 24 (A, B) Right-handed active rotation of spherical harmonics from 0 to in a single coordinate system (O, x, y, z); (C, D) left-handed rotation of coordinate system from (O, x, y, z)to(O, x0,y0,z), the same spherical harmonics is called in (O, x, y, z) and in (O, x0,y0,z) MAN | 33 of 64

Xþ‘ combination of spherical harmonics of the same quantum h ; / ‘; h; / ‘; 0 ð‘;AÞ a; b; c : 1 1jim ¼ h jim Dm0m ð Þ (190) number ‘ before rotation. m0¼‘ h ; / h; / Bear in mind that the two vectors r1ð 1 1Þ and rð Þ Writing this result in terms of spherical harmonics are in a single coordinate system, we may use Eq. (187) to yields11,15,37,56 reformulate Eq. (196) as73 Xþ‘ 5 ð‘;AÞ h ; / h; / a; b; c : þ‘ hi Y‘;mð Þ ¼ Y‘;m0 ð ÞD 0 ð Þ (191) X 1 1 m m fixed ð‘;AÞ 0 ‘ a; b; c a; b; c : m ¼ Y‘;m RA ð Þr1 ¼ Dmm0 ð Þ Y‘;m0 ðr1Þ m0 ‘ h ; / ¼ The spherical harmonics Y‘;mð 1 1Þ whose arguments (197) are the polar angles of the initial position vector r1 are h; / Applying Eq. (129) to Eq. (196) yields expressed in terms of Y‘;m0 ð Þ whose arguments are the polar angles of the final or rotated position vector r. The spherical harmonics in the two terms are identical but their Xþ‘ h; / h ; / ð‘;PÞ c; b; a : arguments are different. Y‘;mð Þ¼ Y‘;m0 ð 1 1ÞDm0m ð Þ (198) 0 ‘ Although it remains uncommon, we may include the m ¼ vector and its polar angles as arguments of spherical har- The notation suggests a passive rotation even though we monics: are studying active rotations. We prefer Eqs. (195) and (196) to Eq. (198). ÂÃXþ‘ h ; / h; / ð‘;AÞ a; b; c : Y‘;m r1ð 1 1Þ ¼ Y‘;m0 ½rð Þ Dm0m ð Þ (192) m0¼‘ 9.2 | Example IX.1 These two vectors r and r are in the same coordinate sys- 1 Example 1 from Man1 is again used to illustrate Eq. (196). tem. Consider a position vector A in a 3-D Cartesian coordinate fi By extension, including the de nition of the right- ; ; ; ; ; system ðO x y zÞ with an orthonormal basis {ex ey ez}. handed active rotation of space function yields A ; ; The Cartesian components of in this basis are (ax ay az). We apply a right-handed active rotation about the z- p 1 2 axis by Euler angles a ¼ b ¼ 0 and c ¼ to A (Fig- Y0 ðh; /Þ ¼ Rfixedða; b; cÞY ðh; /Þ ¼ Y ðh ; / Þ 2 ‘;m A ‘;m ‘;m 1 1 ure 25A). The Cartesian components (a0 ; a0 ; a0 ) of the ‘ x y z Xþ (193) A0 5 h; / ð‘;AÞ a; b; c : rotated vector is given by the direct product of the ¼ Y‘;m0 ð ÞDm0m ð Þ m0¼‘ Cartesian active rotation matrix with the column component matrix of A:3 Sometimes, we need the reverse expression of Eq. (191), 0 1 0 10 1 0 p p which expresses Y ðh; /Þ with respect to Y 0 ðh ; / Þ.We a cos sin 0 a ‘;m ‘;m 1 1 B x C B 2 2 CB x C multiply both sides of Eq. (187) by hj‘; m and include the @ 0 A @ p p A@ A ay ¼ sin 2 cos 2 0 ay closure relation in the right-hand side member: a0 001a z 0 10 1 z0 1 (199) Xþ‘ 0 10 ax ay h‘; mjir ¼ hj‘; m Rfixedða; b; cÞji‘; m0 hj‘; m0 r ; (194) B CB C B C A 1 ¼@ 100A@ a A ¼ @ a A: m0¼‘ y x 001 az az which leads to1,39 There is a second way to solve this problem. We can relate the Cartesian components of A to the spherical har- hi‘ hi Xþ ð‘;AÞ monics of order l=1: Y h; / D 0 a; b; c Y 0 h ; / : (195) ‘;mð Þ ¼ mm ð Þ ‘;m ð 1 1Þ rffiffiffiffiffiffi m0¼‘ 3 ax þ iay Y ; ðh ; / Þ¼ pffiffiffi ; The complex conjugates of spherical harmonics are in col- 1 þ1 1 1 4p rffiffiffiffiffiffi d 2 umn matrices. It is tempting to write Eq. (195) as fol- 3 az 24,26,29,39,75 Y ; h ; / ; (200) lows: 1 0ð 1 1Þ¼ p rffiffiffiffiffiffi4 d Xþ‘ hi 3 ax iay ‘; h ; / ffiffiffi ; A : h; / ð AÞ a; b; c h ; / : Y1;1ð 1 1Þ¼ p d ¼jj jj Y‘;mð Þ¼ Dmm0 ð Þ Y‘;m0 ð 1 1Þ (196) 4p d 2 m0¼‘

The right-hand actively rotated spherical harmonics in a In our case, Dð1;AÞða; b; cÞ deﬁned in Eq. (116) ﬁxed coordinate system are expressed as a linear becomes 34 of 64 | MAN 0 1 i0 0 ð1;AÞ a ; b ; c p @ A: D ð ¼ 0 ¼ 0 ¼ 2Þ¼ 010 (201) 00þi

Replacing covariant spherical harmonics in Eq. (196) by covariant spherical components of A and A0 yields

0 1 0 0 0 10 1 axþffiffiiay a þia p þi0 0 xpffiffi y B 2 C B 2 C B 0 C @ A : @ az A ¼ 010@ az A (202) a0 ia0 axffiffiiay xpffiffi y 00i p 2 2 It follows that

0 ; 0 ; 0 ; ax ¼ay ay ¼ ax az ¼ az (203) in agreement with the direct Cartesian rotation in Eq. (199).

9.3 | Left-handed rotation of coordinate system The right-handed active rotation of spherical harmonics discussed above is equivalent to a left-handed rotation of coordinate system keeping the spherical harmonics ﬁxed. Figure 24A,B shows the right-handed active rotation of spherical harmonics and (C, D) the left-handed rotation of coordinate system all about z-axis by angle a. The position vectors are r1 and r in Figure 24A,B; they are r and r0 in Figure 24C,D. We explore the left-handed rotation of coordinate system with the passive point view of FIGURE 25 (A) Right-handed active rotation of vector from A fixed a; b; c A0 a b c p RA ð Þ. to about the z-axis by Euler angles = =0 and = /2 in a single fixed a; b; c coordinate system (O, x, y, z) is equivalent to (B) left-handed rotation As RA ð Þ allows us to describe a left-handed 0 0 rotation of a coordinate system from hjr to hjr0 , the ket of coordinate system from (O, x, y, z)to(O, x ,y,z) about the z-axis fixed a; b; c by Euler angles a=b=0 and c=p/2, A remains ﬁxed position vectors in passive point of view of RA ð Þ are related by Multiplying each member of Eq. (205) by ji‘; m and including the closure relation yield 0 fixed a; b; c : jir ¼ RA ð Þjir (204) ‘; 0 fixed a; b; c ‘; hrjim ¼ hjr RA ð Þjim Conversely, the bra position bases are related by Xþ‘ (207) 0 ‘; 0 ð‘;AÞ a; b; c : ¼ hr jim Dm0m ð Þ m0¼‘ 0 fixed a; b; c : hj¼r hjr RA ð Þ (205) In spherical harmonics notation, we have

These bra position bases are related (see Eq. (96)) by Xþ‘ h; / 5 h0; /0 ð‘;AÞ a; b; c : Y‘;mð Þ ¼ Y‘;m0 ð ÞDm0m ð Þ (208) m0¼‘ hjr ¼ hjr0 Rfixedðc; b; aÞ (206) P The spherical harmonics are in row matrices. Equa- in a right-handed rotation of a coordinate system from tion (208) describes the left-handed rotation of coordinate hjr to hjr0 . In rotation of coordinate system (Eqs. (205) system despite the presence of Wigner active rotation and (206)), the application of rotation operator to the matrix, whereas Eq. (191) describes the right-handed active rotated bra position basis hjr0 yields the ﬁxed bra position rotation of spherical harmonics in a single coordinate sys- basis hjr . tem. They are two equivalent descriptions. MAN | 35 of 64

By extension, introducing the definition of left-handed The Cartesian components (ax0 ; ay0 ; az0 )ofA in the fixed a; b; c h0; /0 0 0 rotation of coordinate system RA ð ÞY‘;mð Þ in rotated coordinate system (O, x , y , z) are given by the Eq. (96) yields direct product of the Cartesian passive rotation matrix with the column component matrix of A. The left-handed passive rotation matrix is deduced from that of right-handed 0 h0; /0 1 fixed a; b; c h0; /0 2 h; / Y‘;mð Þ ¼ RA ð ÞY‘;mð Þ ¼ Y‘;mð Þ passive rotation by changing the sign of its rotation angle: Xþ‘ (209) 5 0 0 ð‘;AÞ ¼ Y‘; 0 ðh ; / ÞD 0 ða; b; cÞ: 0 1 0 10 1 m m m p p m0¼‘ ax0 cos sin 0 a B C B 2 2 CB x C @ A @ p p A@ A fixed a; b; c ay0 ¼ sin 2 cos 2 0 ay In other words, the operator RA ð Þ in the equality 1 ¼ expresses either the right-handed active rotation of spheri- az0 001az 0 10 1 0 1 (211) cal harmonics in a single coordinate system (Eq. (193)) or 0 10 a a B CB x C B y C the left-handed rotation of coordinate system keeping spheri- : ¼ @ 100A@ ay A ¼ @ ax A cal harmonics fixed (Eq. (209)). Equation (209) can be 001 az az deduced from Eq. (193) in which we replace the polar angles h / h0 /0 h ; / h / As we left-hand rotate a coordinate system, we are ( , )by( , ) and ð 1 1Þ by ( , ). 0 fixed a; b; c The coordinate system is not explicit in the notation studying hjr RA ð Þ. The rotation operator is fixed a; b; c h0; /0 Rfixed a 0; b 0; c p . In our case, Dð1;AÞ a; b; c RA ð ÞY‘;mð Þ. In contrast with Dirac notation A ð ¼ ¼ ¼ 2Þ ð Þ 0 fixed a; b; c ‘; fi defined in Eq. (116) becomes hjr RA ð Þjim , the xed and the rotated coordinate systems are represented by the bra position bases hjr and 0 1 hjr0 , respectively. i0 0 ð1;AÞ a ; b ; c p @ A: We deduce from Eq. (208) the reverse expression: D ð ¼ 0 ¼ 0 ¼ 2Þ¼ 010 (212) 00þi

hi‘ hi Now we explore the complex conjugate of Eq. (210): Xþ h0; /0 ð‘;AÞ a; b; c h; / : Y‘;mð Þ ¼ Dmm0 ð Þ Y‘;m0 ð Þ (210) m0¼‘ 0 1 0 1 a 0 þia 0 0 1 a þia x pffiffi y xpffiffi y Y 0 h; / is the spherical harmonic in the fixed coordinate B C þi0 0 ‘;m ð Þ B 2 C @ AB 2 C h0; /0 @ az0 A ¼ 010@ az A: (213) system, whereas Y‘;mð Þ is the same spherical harmonic a 0 ia 0 axiay x pffiffi y 00i pffiffi in the left-handed rotated coordinate system. Equa- 2 2 tion (210) has the same structure as Eq. (195). It follows that

a 0 ¼a ; a 0 ¼ a ; a 0 ¼ a ; (214) 9.4 | Example IX.2 x y y x z z in agreement with the direct Cartesian rotation in Consider the same vector A used in Example IX.1. Now Eq. (211). The coordinates of A in the left-hand rotated we left-hand rotate the coordinate system (O, x, y, z) about coordinate system are identical to those in Eq. (203) of z-axis by Euler angles a=b=0 and c ¼ p (Figure 25B). 2 Example IX.1 concerning the right-handed active rotation We recall that we use both left- and right-handed rota- of A in a ﬁxed coordinate system. These two examples are tions to visualize rotation of vector and that of coordinate equivalent. system. But for computation we still use mathematical expressions deduced from right-handed rotation. In contrast, the expression of a rotation operator R remains 9.5 | Right-handed active rotation of unchanged. But, we explore its active point of view when spherical harmonic in two coordinate systems it is applied to space function represented by a ket state as —canonical transformation RjiW or its passive point of view when it is applied to 60 coordinate system represented by a bra position basis as Consider two coordinate systems. Initially, the position 0 fixed a; b; c W vector r is in the ﬁxed coordinate system (O, x, y, z) that hjr R. More precisely, RA ð Þjimeans we carry a 1 right-handed active rotation of space function, whereas coincides with its body-attached coordinate system (O, x0, 0 fixed a; b; c y0, z0) as shown in Figure 26A. The polar angles of r are hjr RA ð Þ means we carry a left-handed rotation of 1 0 fixed c; b; a ðh ; / Þ. We right-hand rotate the position vector and its coordinate system. In contrast, hjr RP ð Þ means we 1 1 carry a right-handed rotation of coordinate system, whereas body-attached coordinate system simultaneously by the fixed c; b; a W Euler angles as shown in Figure 26C. The actively rotated RP ð Þjimeans we carry a left-handed active rotation of space function. position vector is named r in (O, x, y, z) and r0 in (O, x0, 36 of 64 | MAN

Xþ‘ h0; /0 ‘; h; / ‘; 0 ð‘;AÞ a; b; c : h jim ¼ h jim Dm0m ð Þ (217) m0¼‘ Writing this result in terms of spherical harmonics, we obtain9,14,34,36,37,40,54,57,60,61,72,73,98,111,116,117 Xþ‘ h0; /0 5 h; / ð‘;AÞ a; b; c : Y‘;mð Þ ¼ Y‘;m0 ð ÞDm0m ð Þ (218) m0¼‘ h0; /0 The spherical harmonics Y‘;mð Þ whose arguments are the polar angles of the position vector r0 in its body-attached coor- h; / dinate system are expressed in terms of Y‘;m0 ð Þ whose arguments are the polar angles of the rotated position vector r in the ﬁxed coordinate system. The two position vectors r and r0 in Figure 26C represent the same physical point.47 As the position vector and its body-attached coordinate system rotate h ; / together, its polar angles ð 1 1Þ before rotation in the initial coordinate system are identical to those (h0, /0) after rotation in the rotated body-attached coordinate system:

h0; /0 h ; / : ð Þ¼ð 1 1Þ (219) The spherical harmonics on each side of Eq. (218) are identical but their arguments are different. We may include the vector and its polar angles in spherical harmonics118: Xþ‘ h0; /0 5 h; / ð‘;AÞ a; b; c : Y‘;m½rð Þ ¼ Y‘;m0 ½rð Þ Dm0m ð Þ (220) m0¼‘ FIGURE 26 Similarities and differences between right-handed The rotated position vector is viewed in two coordinate active rotations (or canonical transformation) of spherical harmonics systems: r(h0, /0)=r0(h0, /0) in its body-attached coordinate and right-handed rotations of coordinate system from (O, x, y, z)to system and r(h, /)inthefixed coordinate system. The posi- (O, x0,y0,z0), these two coordinate systems are related by Euler h ; / fi tion vector before rotation is r1ð Þ in the xed coordinate angles. (A) The position vector is r (h , / ) before the right-handed 1 1 1 1 1 system, which is understood in Eq. (218) due to Eq. (219). active rotations of the spherical harmonics in a single coordinate 2 This corresponds to the equality denoted by ¼ in Eqs. (47) system. (B) The position vector is r(h, /) before the right-handed h0 /0 rotations of coordinate system. (C) The rotated vector is also called r and (49). If we replace the polar angles ( , ) in Eq. (218) fi (h, /) in active rotation and r0(h0, /0) in rotation of coordinate system by those de ned in Eq. (219), Eqs. (191) and (218) become identical. They follow the second interpretation of a rotation operator that only changes the arguments of spherical harmonics, leaving the spherical harmonics unchanged. The spherical y0, z0).11 This is an application of Eq. (68) about the active harmonics transform among themselves with exactly the same rotation of a space function. Now, the state ket W is ji matrix of coefficients as that for 2‘+1 angular momentum ‘; m . Equations (68) and (186) allow us to write: ji eigenket vectors ji‘; m in Eq. (186). Equation (215) in spherical harmonic notation for canonical transformation becomes ‘; 0 fixed a; b; c ‘; 0 ‘; hrjim ¼ hjr RA ð Þji¼m hr jim Xþ‘ 0 1 fixed 4 0 0 (215) h; / a; b; c ‘; h; / ‘; h ; / 0 ð‘;AÞ Y‘;mð Þ ¼ RA ð ÞY mð Þ ¼ Y mð Þ ¼ hrji‘; m D 0 ða; b; cÞ: m m ‘ 0 ‘ Xþ (221) m ¼ 5 h; / ð‘;AÞ a; b; c : ¼ Y‘;m0 ð ÞDm0m ð Þ Therefore, m0¼‘

Xþ‘ 0 ‘; ‘; 0 ð‘;AÞ a; b; c : hr jim ¼ hrjim Dm0m ð Þ (216) m0¼‘ 9.6 | Right-handed rotation of coordinate system The polar angles of the same position vectors r and r0 (Fig- ure 26C) are ðÞh; / in (O, x, y, z)andðÞh0; /0 in (O, x0, y0, z0), This is an application of Eq. (96) about the passive rotation respectively. These angles allow us to rewrite Eq. (216) as of space function about ﬁxed axes. As shown in MAN | 37 of 64

Figure 27A,B, the position vector is named r with polar By extension, introducing the definition of right-handed h; / fi fixed c; b; a h0; /0 angles ðÞin the xed coordinate system (O, x, y, z) and rotation of coordinate system RP ð ÞY‘;mð Þ in r0 with polar angles ðÞh0; /0 in the rotated coordinate sys- Eq. (96) yields tem (O, x0, y0, z). The ket position vectors are related by 12 the passive rotation operator as 0 h0; /0 1 fixed c; b; a h0; /0 2 h; / Y‘;mð Þ ¼ RP ð ÞY‘;mð Þ ¼ Y‘;mð Þ 0 fixed Xþ‘ r R c; b; a r : ‘; (226) ji¼ P ð Þji (222) 5 h0; /0 ð PÞ c; b; a : ¼ Y‘m0 ðÞDm0m ðÞ Therefore, the two bra position bases are related by12 m0¼‘ Equation (226) can be deduced from Eq. (209) about r r0 Rfixed c; b; a : (223) hj¼ hjP ð Þ left-handed rotation of coordinate system by replacing the Equation (96) about the passive rotation of space func- Euler angles a, b, c by c, b, a, respectively (Figure 1 tion about fixed axes allows us to write 28). Indeed, the equality ¼ in Eq. (209) becomes

0 fixed fixed a c; b b; c a h0; /0 hjr ‘; mi¼hjr R ðc; b; aÞji‘; m RA ð ! ! ! ÞY‘;mð Þ P (227) Xþ‘ fixed c; b; a h0; /0 : ¼ RP ð ÞY‘;mð Þ ¼ hjr0 ji‘; m0 hj‘; m0 Rfixedðc; b; aÞji‘; m P 1 5 m0¼‘ (224) That is, the equality ¼ in Eq. (226). The equality ¼ in Xþ‘ Eq. (209) becomes 0 ‘; 0 ð‘;PÞ c; b; a : ¼ hjr jim Dm0m ð Þ m0¼‘ Xþ‘ h0; /0 ð‘;AÞ a c; b b; c a In spherical harmonic notation (Eq. (157)), we deduce Y‘;m0 ð ÞDm0m ð ! ! ! Þ from Eq. (224) that40,56,79,80 m0¼‘ Xþ‘ h0; /0 ð‘;PÞ c; b; a : Xþ‘ ¼ Y‘m0 ðÞD 0 ðÞ (228) ‘; m m h; / h0; /0 ð PÞ c; b; a : m0¼‘ Y‘mðÞ¼ Y‘m0 ðÞDm0m ðÞ(225) 0 ‘ 5 m ¼ That is, the equality ¼ in Eq. (226). In right-handed rotation of coordinate system, Eq. (225) Equation (225) looks like Eq. (208) but they differ by expresses the spherical harmonics in the ﬁxed coordinate Wigner rotation matrices. Equation (225) describes the right- system as a linear combination of spherical harmonics of the handed passive rotation of spherical harmonics involving same quantum number ‘ in the rotated coordinate system.56 Wigner passive rotation matrix, whereas Eq. (208) describes

FIGURE 27 (A, B) Right-handed rotation of coordinate system from (O, x, y, z)to(O, x0,y0,z) about z-axis by angle a, the same spherical harmonics is called in (O, x, y, z) and in (O, x0,y0, z); (C, D) left-handed active rotation of spherical harmonics from < r1|lm> to 0 about z-axis by angle a in a single coordinate system (O, x, y, z) 38 of 64 | MAN 0 1 the left-handed passive rotation of spherical harmonics þi0 0 involving Wigner active rotation matrix. ð1;PÞ c ; b ; a p @ A: D ð ¼ 0 ¼ 0 ¼ 2Þ¼ 010 (232) Sometimes, we need the reverse expression of Eq. (225), 00i h0; /0 h; / which expresses Y‘mðÞwith respect to Y‘;m0 ð Þ.A similar procedure that leading to Eq. (195) yields56 Numerical application of Eq. (230) is

hi‘ hi Xþ 0 1 0 0 ð‘;PÞ 0 10 1 Y‘; ðh ; / Þ ¼ D 0 ðc; b; aÞ Y‘; 0 ðh; /Þ ; (229) a 0 þia 0 a þia m mm m x pffiffi y xpffiffi y 0 ‘ B C i0 0 m ¼ B 2 C @ AB 2 C @ az0 A ¼ 010@ az A: (233) or a 0 ia 0 axiay x pffiffi y 00þi pffiffi ‘ hi 2 2 Xþ 0 0 ð‘;PÞ Y ðh ; / Þ¼ D 0 ðc; b; aÞ Y 0 ðh; /Þ: (230) ‘;m mm ‘;m It follows that m0¼‘ The spherical harmonics are in column matrices. The a 0 ¼ a ; a 0 ¼a ; a 0 ¼ a ; (234) spherical harmonics in the right-hand rotated coordinate x y y x z z system are expressed as a linear combination of spherical in agreement with the direct Cartesian rotation in ‘ fi harmonics of the same quantum number in the xed Eq. (231). coordinate system. Figure 26 clarifies polar angle definitions. In active rotation (Figure 26A,C), (h, /) and (h0, /0) are the polar angles 9.8 | Left-handed active rotation of spherical of the actively rotated vector in the fixed coordinate system harmonic in one coordinate system and in its body-attached coordinate system, respectively. In The right-handed rotation of coordinate system discussed its attached coordinate system, the actively rotated vector above is equivalent to a left-handed active rotation of h0 h /0 / has the same polar angles, that is, ¼ 1 and ¼ 1, spherical harmonics in a fixed coordinate system. Fig- h ; / fi ( 1 1) being the polar angles of the vector in the xed ure 27A,B shows right-handed rotation of coordinate sys- coordinate system before rotation (Figure 26A). In passive tem about z-axis by angle a and (C, D) left-handed active rotation (Figure 26B,C), the vector remains unchanged. rotation of spherical harmonics about z-axis by angle a. h / h0 /0 The angles ( , ) and ( , ) are the polar angles of the The position vectors are r and r0 in Figure 27A,B; they are vector in the fixed coordinate system and in the rotated r1 and r in Figure 27C,D. For left-handed active rotation coordinate system, respectively. of spherical harmonics, we explore the left-handed active fixed c; b; a point view of RP ð Þ. The ket position vectors are 9.7 | Example IX.3 related by the left-handed active point view of fixed c; b; a RP ð Þ: Consider the same vector A used in Example IX.1. Now we right-hand rotate the coordinate system (O, x, y, z) a p b c r Rfixed c; b; a r ; (235) about z-axis by Euler angles ¼ 2 and = =0 (Fig- ji¼ P ð Þ 1 ure 29A). The Cartesian components (a 0 ; a 0 ; a 0 )ofA in x y z like in Eq. (187). Conversely, the rotated coordinate system (O, x0, y0, z) are given by the direct product of the Cartesian passive rotation matrix with 3 the column component matrix of A: fixed c; b; a : r1 ¼ hjr RP ð Þ (236) 0 1 0 10 1 p p a 0 cos sin 0 a Multiplying both sides of Eq. (236) by ji‘; m and B x C B 2 2 CB x C @ A @ p p A@ A including the closure relation yield ay0 ¼ sin 2 cos 2 0 ay az0 001az (231) ‘; fixed c; b; a ‘; 0 10 1 0 1 r1 mi ¼ hjr RP ð Þjim 010 a a Xþ‘ B CB x C B y C : ¼ hjr ji‘; m0 hj‘; m0 Rfixedðc; b; aÞji‘; m ¼ @ 100A@ ay A ¼ @ ax A P m0¼‘ (237) 001 a a z z Xþ‘ ‘; 0 ð‘;PÞ c; b; a : As we right-hand rotate a coordinate system, we are ¼ hjr jim Dm0m ð Þ 0 ‘ 0 fixed c; b; a m ¼ studying hjr RP ð Þ. The rotation operator is fixed c ; b ; a p ð1;PÞ c; b; a RP ð ¼ 0 ¼ 0 ¼ 2Þ. In our case, D ð Þ defined in Eq. (124) becomes In spherical harmonics notation, we have MAN | 39 of 64

Xþ‘ ‘ hi ‘; Xþ 5 ð PÞ ð‘;PÞ Y‘; ðh ; / Þ ¼ Y‘ 0 ðh; /ÞD 0 ðc; b; aÞ: (238) h; / c; b; a h ; / : m 1 1 m m m Y‘;mð Þ¼ Dmm0 ð Þ Y‘;m0 ð 1 1Þ (242) 0 ‘ m ¼ m0¼‘ h ; / The spherical harmonics Y‘;mð 1 1Þ, whose arguments are The spherical harmonics are in column matrices. The left-hand the polar angles of the initial position vector r1,are actively rotated spherical harmonics in a fixed coordinate sys- h; / expressed in terms of Y‘;m0 ð Þ, whose arguments are the tem are expressed as a linear combination of spherical har- polar angles of the left-hand actively rotated position vector monics of the same quantum number ‘ before rotation. r (Figure 27C,D). In addition to the canonical transformation, we have dis- By extension, introducing the definition of left-handed cussed four other types of rotation in this section. In fact, fixed c; b; a h; / active rotation of spherical harmonics RP ð ÞY‘;mð Þ these four types are related. Knowing one of them allows in Eq. (67) yields us to deduce the other three. Figure 28 shows these relations. 0 h; / 1 fixed c; b; a h; / 2 h ; / Y‘;mð Þ ¼ RP ð ÞY‘;mð Þ ¼ Y‘;mð 1 1Þ Xþ‘ 9.9 | Example IX.4 5 ð‘;PÞ ¼ Y‘ 0 ðÞh; / D 0 ðÞc; b; a : (239) m m m A m0¼‘ Consider the same vector used in Example IX.1. Now A fixed c; b; a we left-hand rotate the vector about z-axis by Euler In other words, the operator RP ð Þ in the equal- p 1 angles a ¼ and b=c=0 (Figure 29B). The Cartesian com- ity ¼ expresses either the right-handed rotation of coordi- 2 ponents (a0 ; a0 ; a0 ) of the rotated vector A0 in the fixed nate system keeping spherical harmonics fixed (Eq. (226)) x y z coordinate system are given by the direct product of the or the left-handed active rotation of spherical harmonics in Cartesian active rotation matrix with the column compo- a single coordinate system (Eq. (239)). The coordinate sys- nent matrix of A. The left-handed active rotation matrix is tem is not explicit in the notation Rfixedðc; b; aÞY ðh; /Þ. P ‘;m deduced from that of right-handed active rotation by chan- In contrast with Dirac notation hjr Rfixedðc; b; aÞji‘; m , the P ging the sign of its rotation angle: fixed coordinate system is represented by the bra position basis hjr . 0 1 0 10 1 Equation (239) can be deduced from Eq. (193) about 0 p p ax cos 2 sin 2 0 ax right-handed active rotation of spherical harmonic in a sin- B 0 C B p p CB C @ a A ¼ @ sin cos 0 A@ ay A a b c y 2 2 gle coordinate system by replacing the Euler angles , , 0 1 a 001a by c, b, a, respectively. Indeed, the equality ¼ in z z (243) Eq. (193) becomes 0 10 1 0 1 010 ax ay fixed a c; b b; c a h; / B CB C B C R ð ! ! ! ÞY‘;mð Þ ¼ @ 100A@ a A ¼ @ a A: A (240) y x fixed c; b; a h; / : a a ¼ RP ð ÞY‘;mð Þ 001 z z 1 5 That is, the equality ¼ in Eq. (239). The equality ¼ in As we left-hand rotate vector A, we are studying fixed c; b; a W fixed c ; Eq. (193) becomes RP ð Þji. The rotation operator is RP ð ¼ 0

Xþ‘ h; / ð‘;AÞ a c; b b; c a Y‘;m0 ð ÞDm0m ð ! ! ! Þ 0 ‘ m ¼ (241) Xþ‘ h; / ð‘;PÞ c; b; a : ¼ Y‘m0 ðÞDm0m ðÞ m0¼‘ 5 That is, the equality ¼ in Eq. (239). Equation (238) looks like Eq. (191) but they differ by Wigner rotation matrices. Equation (238) describes the left- handed active rotation of spherical harmonics involving Wigner passive rotation matrix, whereas Eq. (191) FIGURE 28 Relations between rotations of spherical harmonic describes the right-handed active rotation of spherical har- and those of coordinate system: (A) left-handed active rotation of monics in a single coordinate system involving Wigner spherical harmonic is described by Eq. (239); (B) right-handed active active rotation matrix. rotation of spherical harmonic is described by Eq. (193); (C) right- h; / Similarly, we may express Y‘;m0 ð Þ in terms of handed rotation of coordinate system is described by Eq. (226); (D) h ; / Y‘;mð 1 1Þ as in Eq. (196): left-handed rotation of coordinate system is described by Eq. (209) 40 of 64 | MAN

those in Eq. (234) of Example IX.3 concerning the right- handed rotation of coordinate system keeping the vector A ﬁxed. These two examples are equivalent.

9.10 | Rotation of spherical tensor Any classical tensor can be decomposed into irreducible parts, each of which transforms under a rotation of the coordinate system in the same way as does one of the spherical harmonics. Two definitions of spherical tensor components for rank r=0 to 2 in terms of the nine Carte- sian tensor components are used86,87,119 in the literature. We previously proved1 that only that provided by Cook and De Lucia119 is used in NMR. Similarly to Cartesian vectors and tensors, one can define the contravariant and covariant components of a spherical tensor. Again, one need not worry about this ter- minology, since the angular momentum algebra in most of the literature is based on the covariant components, but without explicit reference to their covariant nature.120 Among the four types of rotation (right- and left-handed active rotations of spherical harmonics, and right- and left- handed rotations of coordinate system), we previously show that knowing one of them allows us to deduce the other three types. Therefore, we restrict the presentation of spherical tensor to right-handed active and passive rotations. FIGURE 29 (A) Right-handed rotation of coordinate system from A spherical tensor of rank k is defined by a set of 2k + (O, x, y, z)to(O, x0,y0,z) about the z-axis by Euler angles a=p/2 and b=c=0 associated with fixed vector A is equivalent to (B) left-handed 1 space functions transforming under coordinate rotations active rotation of vector from A to A0 about the z-axis by Euler angles according to a=p/2 and b=c=0 in a single coordinate system (O, x, y, z) X ðk;AÞ p ; fixed a; b; c a; b; c ; b ¼ 0; a ¼ Þ. In our case, Dð1 PÞðc; b; aÞ defined in RA ð ÞTkqðrÞ¼ Tkq0 ðrÞDq0q ð Þ (247) 2 q0 Eq. (124) becomes and58 0 1 þi0 0 ð1;PÞ p @ A X D ðc ¼ 0; b ¼ 0; a ¼ Þ¼ 010: (244) fixed c; b; a 000 000 ðk;PÞ c; b; a : 2 RP ð ÞTkqðr Þ¼ Tkq0ðr ÞDq0q ð Þ (248) 00i q0

Numerical application of Eq. (242) is Note that the same actively rotated position vector r in the fixed coordinate system appears on both sides of 0 1 0 1 Eq. (247). The same passively rotated position vector r000 a0 þia0 0 1 x ffiffi y axþiay p i0 0 pffiffi B 2 C B 2 C appears on both sides of Eq. (248). B 0 C @ A : @ az A ¼ 010@ az A (245) 0 0 a ia axiay xpffiffi y 00þi pffiffi 2 2 9.10.1 | Right-handed active rotation of It results that spherical tensor in one coordinate system

0 0 0 Now, we provide the transformation rules for Euler angles a ¼ ay; a ¼ax; a ¼ az; (246) x y z defined with rotated coordinate systems, a being the first in agreement with the direct Cartesian rotation in Euler angle. The right-handed active rotation of spherical Eq. (243). The coordinates of the left-hand actively rotated harmonics of rank ‘=1 are already defined in Eqs. (67) and vector A0 in the fixed coordinate system are identical to (193). They are gathered below in extended notation: MAN | 41 of 64

000 000 000 1 4 ðÞY 11ðrÞ ðÞY 10ðrÞ ðÞY 11ðrÞ T0 r Rfixed a; b; c T r T r0 kqð Þ ¼ A ð Þ kqð Þ ¼ kqð Þ 1 X ¼ RfixedðX Þ Y ðrÞ Y ðrÞ Y ðrÞ (249) 5 ðk;AÞ (252) A 1 11 10 11 T 0 r D a; b; c : ¼ kq ð Þ q0q ð Þ 2 q0 ¼ Y11ðr1Þ Y10ðr1Þ Y11ðr1Þ ÈÉÈÉÈÉ 3 P X P X P X With angle-axis of rotation parameter, Eq. (252) ¼ Y11 ð 2Þr Y10 ð 2Þr Y11 ð 2Þr 55 becomes 5 ð1;AÞ X ; ¼ Y11ðrÞ Y10ðrÞ Y1 1ðrÞ D ð 1Þ X 0 1 fixed 4 0 5 ðk;AÞ T ðrÞ¼R ðh;nÞT ðrÞ ¼T ðr Þ ¼ T 0 ðrÞD 0 ðh;nÞ: with X ðÞa; b; c ; X ðÞc; b; a ; and P P : kq A kq kq kq q q 1 2 rotatedZ3Y2Z1 q0 (253) The first line in Eq. (249) is a row matrix containing 000 actively rotated spherical harmonics ðÞY 1m with components of actively rotated position vector r. The other lines 9.10.3 | Right-handed rotation of coordinate system contain initial spherical harmonics Y1m with components of 1 3 actively rotated position vector r in equalities ( ¼, ¼, and 5 The right-handed passive rotation of spherical harmonics of ¼) or components of initial position vector r1 in equality ‘ fi 2 rank =1 are already de ned in Eqs. (96) and (226). They (¼). are gathered below in extended notation: The right-handed active rotations of spherical tensor T kq written in compact notations are 000 000 000 000 000 000 ðÞY 11ðr Þ ðÞY 10ðr Þ ðÞY 11ðr Þ 1 1 2 fixed X 000 000 000 0 fixed a; b; c ¼ RP ð 2Þ Y11ðr Þ Y10ðr Þ Y11ðr Þ TkqðrÞ ¼ RA ð ÞTkqðrÞ ¼ Tkqðr1Þ X 2 5 ; (250) ðk AÞ a; b; c : ¼ Y11ðrÞ Y10ðrÞ Y11ðrÞ ¼ Tkq0 ðrÞDq0q ð Þ ÈÉÈÉÈÉ q0 3 A X 000 A X 000 A X 000 ¼ Y11 ð 1Þr Y10 ð 1Þr Y11 ð 1Þr 5 000 000 000 ð1;PÞ X ; ¼ Y11ðr Þ Y10ðr Þ Y11ðr Þ D ð 2Þ (254) 9.10.2 Canonical transformation X a; b; c ; X c; b; a ; A A : | with 1 ðÞ2 ðÞand rotatedZ1Y2Z3 The canonical transformations of spherical harmonics of The first line in Eq. (254) is a row matrix containing rank ‘=1 are already defined in Eqs. (68) and (221). They passively rotated spherical harmonics Y000 with compo- are gathered below in extended notation: ðÞ1m nents of passively rotated position vector r000 in the rotated coordinate system. The other lines contain initial spherical 000 000 000 000 ðÞY ðrÞ ðÞY ðrÞ ðÞY ðrÞ harmonics Y1m with components of passively rotated r in 11 10 11 1 3 5 equalities (¼, ¼, and ¼) or components of the position 1 fixed X ¼ RA ð 1Þ Y11ðrÞ Y10ðrÞ Y11ðrÞ fi 2 vector r in the xed coordinate system in equality (¼). 4 0 0 0 ¼ Y11ðr Þ Y10ðr Þ Y11ðr Þ The right-handed passive rotations of spherical tensor ÈÉÈÉÈÉ T written in compact notations79,97 are 3 P X P X P X kq ¼ Y11 ð 2Þr Y10 ð 2Þr Y11 ð 2Þr 5 ; 1 2 ¼ Y ðrÞ Y ðrÞ Y ðrÞ Dð1 AÞðX Þ; (251) T000ðr000Þ ¼ Rfixedðc; b; aÞT ðr000Þ ¼ T ðrÞ 11 10 11 1 kq XP kq kq X a; b; c ; X c; b; a ; P P : 5 ðk;PÞ (255) with 1 ðÞ2 ðÞand rotatedZ3Y2Z1 000 c; b; a : ¼ Tkq0 ðr ÞDq0q ð Þ q0 fi The rst line in Eq. (251) is a row matrix containing 2 5 000 Equalities ¼ and ¼, actively rotated spherical harmonics ðÞY 1m with components of rotated position vector r in the fixed coordinate X 5 000 ðk;PÞ T r T 0 r D c; b; a ; system. The other lines contain initial spherical harmonics kqð Þ¼ kq ð Þ q0q ð Þ (256) q0 Y1m with components of rotated position vector r in the 1 3 5 fixed coordinate system in equalities ( ¼, ¼,and¼) or the are used by Landau and Lifshitz,65 formula (58.7) of their components of the position vector r0 in the body-attached book, but applied to wave functions. 4 coordinate system in equality (¼). The components of posi- Equation (256) is also extensively used in NMR for tion vector r0 in the body-attached coordinate system are second rank tensors such as the electric-field-gradient ten- 0 102 identical to those of the position vector r1 in the fixed sor V. In Haeberlen s book, the position vector r refers coordinate system before rotation. to the NMR observation coordinate system (OBS), whereas 000 The canonical transformations of spherical tensor Tkq the position vector r refers to the principal-axis coordi- written in compact notations9,11 are nate system (PAS). Equation (256) is rewritten as 42 of 64 | MAN X 10 | ROTATION OF VECTOR ðOBSÞ 5 ðPASÞ ðk;PÞ c; b; a : Tkq ¼ Tkq0 Dq0q ð Þ (257) OPERATOR q0

In other words, OBS is the fixed coordinate system in In quantum mechanics, physical observables are repre- fi sented by Hermitian operators. Operators corresponding to which the static magnetic eld B0 is along zOBS-axis, whereas PAS is the rotated coordinate system (Figure 30A). various physical quantities can be classified as scalars, vec- Equation (257) is used by Engelhardt and Michel.103 tors, and tensors. Scalar is a spherical tensor operator of In Mehring0s books,86,121,122 the fixed coordinate system rank-0. Vector operator is a spherical tensor operator of 41 is PAS, whereas the rotated coordinate system is OBS (Fig- rank-1. Spin operators in NMR are vector operators. ure 30B). As a result, Eq. (257) changes to Two approaches for defining an RF pulse propagator X related to the convention of sense of rotation appear in PAS OBS k;P ð Þ ð Þ ð Þ c; b; a : 110 Tkq ¼ Tkq0 Dq0q ð Þ (258) NMR literature: In the right-handed, mathematically pos- 0 q itive convention, a 90° x-pulse rotates a magnetization from The reverse expression of that in Eq. (258) can be the +z direction into the –y direction. In the left-handed deduced from Eq. (230): convention, a 90° x-pulse rotates a magnetization from the hi Xþk +z direction into the +y direction. These propagators are: ðOBSÞ ðk;PÞ c; b; a ðPASÞ Tkq ¼ Dqq0 ð Þ Tkq0 q0¼k 122 PxðtÞ¼expðihIxÞ for right-handed convention; hi Xþk T ðPASÞ ðk;PÞ c; b; a ¼ Tkq0 Dqq0 ð Þ 110 PxðtÞ¼expðihIxÞ for left-handed convention: (260) q0¼k Xþk In the tradition of Haeberlen and others, Schmidt-Rohr and ðPASÞ ðk;AÞ a; b; c : ¼ Tkq0 Dq0q ð Þ (259) Spiess110 adopt the left-handed convention. q0¼k When vector operators are sandwiched by a rotation oper- In the expression of Mehring (Eq. (259)) appears the ator and its transpose, it is the operator located at the left- Wigner active rotation matrix despite the fact we are study- hand side of the vector operator that defines the Euler angles. ing passive rotation of spherical tensor. Equation (259) is Rotations of vector operators and spherical tensor opera- also used by Spiess.91 tors are performed about axes of a fixed coordinate system as discussed in Section 5. For simplicity, in this section, we shorten “axes of fixed coordinate system” to “fixed axes”. In this section, we will see that: 1. in right-handed rotation, vector operator components are in row component matrix and rotation matrix is pre- multiplied by row component matrix; 2. in left-handed rotation, vector operator components are in column component matrix and rotation matrix is post-multiplied by column component matrix.

10.1 | Active rotation of vector operator In classical system, the three Cartesian components (vx ¼ v1, vy ¼ v2, vz ¼ v3) of a vector v in an orthonormal basis vectors transform like26

X3 A vi ! ½ijvj (261) j¼1

under right-handed active rotation of vector v, A being a 393 real, orthogonal matrix associated with right-handed active rotation.3 The active rotation matrix A is post-multiplied by the column component matrix of v. For example, FIGURE 30 Fixed and rotated coordinate systems for (A) the matrix of right-handed active rotation about z-axis by Haeberlen and (B) Mehring angle h is3,43 MAN | 43 of 64

0 1 0 h h h h: cos h sin h 0 Iz ¼ expði IxÞIz expði IxÞ¼Iz cos þ Iy sin (266) B C A h @ h h A: zð Þ¼ sin cos 0 (262) It shows a left-handed active rotation of spin operator com- 001 ponent Iz about x-axis by a positive angle h. Therefore, the This deﬁnition is not universal. Other deﬁnitions are possi- left-handed active rotation of vector operator is symbolized ble.49 With orthonormal basis vectors, the contravariant by the operator Cartesian components of a vector coincide with the covari- y ½RAðhÞ ¼ expðihIzÞ (267) ant ones.1,55 This is not the case for covariant spherical components of a vector expressed in orthonormal basis at left-hand side of the vector operator in Eq. (265). vectors. Usually, covariant spherical components are used Sakurai and Napolitano26 discuss a similar example: in the literature, because angular momentum coupling the- 90 0 h h h h ory is designed for use with covariant spherical tensors. Ix ¼ expði IzÞIx expði IzÞ¼Ix cos Iy sin (268)

without mentioning the left-handed active rotation of Ix. Therefore, Table 4 from Gerstein and Dybowski,125 10.1.1 | Left-handed active rotation of vector Haeberlen,102 and Zettili39 describes the left-handed active operator rotations of spin operator I. We deduce that the matrices A In quantum systems, the state vector jiW transforms under an (/) in Table 4 and Eq. (265), which are post-multiplied by 0 active rotation RA to state vector jiW as deﬁned in Eq. (5). column component matrix, are left-handed active rotation matrices. They are identical to the right-handed active rota- Experiments that measure the operator Vi involve the matrix W W tion matrices A ðhÞ, A ðhÞ, and A ðhÞ derived previously.3 element 1 Vi 2 . The matrix element of a vector opera- x y z tor V expressed with the rotated state vectors is related to that When Euler angles are involved in an active rotation, of V expressed with the original state vectors39 as Eq. (8): Eq. (265) can be generalized: R W W A W0 W0 W y W : 1 Vi 2 ! 1 V i 2 ¼ 1 ðRAÞ ViRA 2 (263) Equation (263) describes an active rotation of the physical system, keeping V ﬁxed. It is reasonable to demand that the matrix element of a vector operator in quantum mechanics be transformed like a classical vector under an active rotation.26 That is,19,43

X3 W0 W0 W y W W W : 1 Vi 2 ¼ 1 ðRAÞ V iRA 2 ¼ Aij 1 V j 2 j¼1 (264) FIGURE 31 Left-handed active rotation of spin operator

Equation (264) relates the matrix element of a vector component Iz about x-axis by a positive angle h operator in the rotated quantum system to the matrix elements in the original quantum system via the rotation matrix TABLE 4 Left-handed active rotation of Cartesian spin operator A. This relation has to be valid for any W and W . 1 2 components, the rotation angle / being positive (/ [ 0), deduced In short, a vector operator transforms19,20,26,43,73,123,124 from Gerstein and Dybowski,125 Haeberlen,102 and Zettili39 according to P3 / y / A / ½RAð Þ IiRAð Þ ½ð Þ ijIj X3 j¼1 0 1 0 10 1 ½R ðhÞ yV R ðhÞ¼ ½AðhÞ V : (265) A i A ij j Ix cos / sin / 0 Ix j¼1 @ A @ A@ A expði/IzÞ Iy expði/IzÞ sin / cos / 0 Iy Iz 001Iz In Eq. (265), the three components of the vector opera- 0 1 0 10 1 tor are gathered in a column matrix. They are also sand- Ix cos / 0 sin / Ix @ A @ A@ A y expði/IyÞ Iy expði/IyÞ 010 Iy wiched by ½RAðhÞ and RA(h). We have shown in Iz sin / 0 cos / Iz Section 3 that this transformation is a left-handed active 0 1 0 10 1 rotation of “vector” operator in a ﬁxed coordinate system. Ix 10 0 Ix / @ A / @ / / A@ A Figure 31 from Gerstein and Dybowski125 provides a expði IxÞ Iy expði IxÞ 0 cos sin Iy I 0 sin / cos / I geometrical representation of the result of z z 44 of 64 | MAN 0 1 ÂÃVx fixed a; b; c y@ A fixed a; b; c RA ð Þ Vy RA ð Þ Vz 0 1 Vx B C ¼ expðicLzÞ expðibLyÞ expðiaLzÞ@ Vy A

V z

expðiaLzÞ expðibLyÞ expðicLzÞ 0 1 Vx B C c b A a @ A b c ¼ expði LzÞ expði LyÞ zð Þ Vy expði LyÞ expði LzÞ V z (269) 0 1 V x c A a A b @ A c ¼ expði LzÞ zð Þ yð Þ V y expði LzÞ V z

0 1 V x A a A b A c @ A ¼ zð Þ yð Þ zð Þ V y V 0z 1 V x A a; b; c @ A: ¼ fixedZ3Y2Z1ð Þ V y V z The order of the three elementary rotation matrices that fi A a; b; c de ne fixedZ3Y2Z1ð Þ in Eq. (269) and that of Euler angles in the arguments of A ða; b; cÞ are due to the fact that FIGURE 32 Left-handed active rotation of spin operator fixedZ3Y2Z1 fi fi fi rotation operators affect the components of vector operator. component Ix about xed axes: (A) initial con guration; the rst (B), and second (C) rotations are about the z-axis by angle a=p/4 and This means A ða; b; cÞ already defined for right- fixedZ3Y2Z1 about the y-axis by angle b=p/2, respectively handed active rotation1,3 is also the left-handed active rotation matrix for a vector operator whose components are gathered in handed active rotations are about z-axis by angle a=p/4 column matrix. The subscript Z3Y2Z1 of A ða; b; cÞ fixedZ3Y2Z1 and about y-axis by angle b=p/2, respectively. Figure 32A is meaningless in left-handed active rotation, because Eq. (269) shows the spin operator component I in the fixed coordi- shows that the first rotation angle is a located at the left-hand x nate system (O, x, y, z). Figure 32B shows I0 , the left- side among the three arguments of A ða; b; cÞ. x fixedZ3Y2Z1 handed active rotation of I about z-axis by angle a=p/4. Equation (269) shows that the operator for left-handed active x 0 ; 00 fi I x lies in the fgx y plane. Figure 32C shows Ix, the left- rotation of a vector operator about xed axes is 0 handed active rotation of I x about y-axis by angle b=p/2. ÂÃ 00 00 y I lies in the fgy; z plane. The rotated I , I , can be calcu- Rfixedða; b; cÞ ¼ expðicL Þ expðibL Þ expðiaL Þ: (270) x x x A z y z lated using Table 4: fi The rst, second, and third left-handed active rotation p p p p 00 angles for vector operator V about fixed axes are a, b, and I ¼ expði IyÞ expði IzÞIx expði IzÞ expði IyÞ x 2 4 4 2 c, respectively. p p ¼ expði I ÞI0 expði I Þ Figure 32 provides a geometrical representation of the 2 y x 2 y p hip p p result of ¼ expði I Þ I C I S expði I Þ 2 y x 4 y 4 2 y p p expðicI Þ expðibI Þ expðiaI ÞI expðiaI Þ expðibI Þ ¼IyS þ IzC : z y z x z y 4 4 c ; a p= ; b p= ; c : expði IzÞ ¼ 4 ¼ 2 and ¼ 0 (272) (271) For simplicity, we replace cos and sin by C and S. The 00 It describes left-handed active rotations of the spin operator spin operator component I x agrees with that shown in Fig- component Ix about fixed axes. The first and second left- ure 32C. MAN | 45 of 64

We can also determine left-hand actively rotated spin 0 1 hiV 00 p p þ1 operator component, I x, using the rotation matrix ð1;AÞ @ A ¼ D ð ; /; Þ V 0 ; (277) A ða; b; cÞ as in Eq. (269): 2 2 fixedZ3Y2Z1 V1

0 p p p p 1 0 1 i2Iy i4Iz i4Iz i2Iy e e Ixe e Ix B p p p p C p p B C 0 1 i2Iy i4Iz i4Iz i2Iy A ; ; 0 1 2 / S/ 2 / 0 1 @ e e Iye e A ¼ ð 0Þ@ I A pffiffi fixedZ3Y2Z1 y C 2 S 2 p p p p 4 2 V þ1 B 2 C V þ1 i I i I i I i I e 2 y e 4 z I e 4 z e 2 y I B C B / / CB C z 0 10 z 1 expði/L Þ@ V Aexpði/L Þ¼B pS ffiffi C/ pSffiffi C@ V A p p y 0 y @ 2 2 A 0 0 S 4 C 4 Ix / / / B CB C V 1 S2 pS ffiffi C2 V 1 @ p p A@ A: 2 2 2 ¼ 0C4 S 4 Iy 0 1 V þ1 10 0 Iz B C ð1;AÞ ;/; @ A; (273) ¼D ð0 0Þ V0 V 1 We deduce that 0 1 0 10 1 p p i/ 00 ipI ipI ipI ipI V e 00 V I ¼ e 2 y e 4 z I e 4 z e 2 y ¼I S þ I C : (274) þ1 þ1 x x y 4 z 4 B C B CB C expði/LzÞ@ V0 Aexpði/LzÞ¼@ 01 0A@ V0 A Equation (274) is identical to Eq. (272). We have at our V i/ V 1 00e 0 1 1 disposal two methods for determining the left-hand rotated hiVþ1 vector operator V0. They are provided by ; B C ¼ Dð1 AÞð0;0;/Þ @ V A: 0 1 0 1 0 0 V x V x V1 B C ÂÃB C @ 0 A fixed a; b; c y@ A fixed a; b; c V y ¼ RA ð Þ V y RA ð Þ ; 0 ð1 AÞ a; b; c V z V z The 3X3 Wigner active rotation matrix D ð Þ fi (275) de ned in Eq. (116) is for right-handed active rotation. 0 1 Equation (277) can be generalized to Euler angles:26,73 Vx B C A a; b; c @ A: ¼ fixedZ3Y2Z1ð Þ Vy 0 1 V ÂÃVþ1 z fixed a; b; c y@ A fixed a; b; c ÂÃ RA ð Þ V 0 RA ð Þ fixed y T R a; b; c V V V V 1 The transformation A ð Þ ðÞx y z 0 1 fixed a; b; c RA ð Þ describes the left-handed active rotation V þ1 about fixed axes for a vector operator V expressed with B C ¼ expðicLzÞ expðibLyÞ expðiaLzÞ@ V 0 A Cartesian components. A ða; b; cÞ is the corre- fixedZ3Y2Z1 V sponding left-handed active rotation matrix. The first rota- 1 expðiaLzÞ expðibLyÞ expðicLzÞ tion angle is a. Covariant spherical components (V+1, V0, V1) and Cartesian components (Vx, Vy, Vz) are related: 0 10 1 ia e 00 V þ1 1 1 B CB C V ¼pffiffiffi V þ iV ; V ¼ V ; V ¼ pffiffiffi V iV : ¼ expðicLzÞ expðibLyÞ@ 01 0A@ V0 A þ1 2 x y 0 z 1 2 x y 00eia V (276) 1 expðibLyÞ expðicLzÞ (278) In contrast, Wigner100 uses the contravariant spherical components. Table 4 allows us to determine the transformation law for covariant spherical components of vector operator:102 0 1 0 1 2 b Sffiffib 2 b 0 1 ia C p S 0 1 e 00 B 2 2 2 C V þ1 B CB b b CB C V þ1 ¼ expðicL Þ@ AB pS ffiffi Cb pSffiffi C@ V A B C z 01 0 @ 2 2 A 0 expði/LxÞ@ V0 A expði/LxÞ ia b b b 00e S2 pS ffiffi C2 V 1 2 2 2 V 1 expðicLzÞ 0 1 0 1 2 / iS/ 2 / 0 1 0 1 b b b 0 10 1 pffiffi 2 pSffiffi 2 C 2 S 2 ia C S ic B 2 C V þ1 e 00 B 2 2 2 C e 00 Vþ1 B CB C Sb Sb iS/ffiffi / iS/ffiffi @ AB pffiffi b pffiffi C@ A@ A ¼ B p C p C@ V0 A ¼ 01 0 @ C A 01 0 V 0 @ 2 2 A a 2 2 c 00ei 2 b Sbffiffi 2 b 00ei V 2 / iS/ 2 / S p C 1 S pffiffi C V 1 2 2 2 2 2 2 46 of 64 | MAN 0 1 hi derived for right-handed rotation, it is its complex conjugate V þ1 ð1;AÞ @ A 73 ¼ D ða;b;cÞ V 0 that is involved in left-handed active rotation. V 1 The order of the three elementary Wigner rotation 10.1.2 | Right-handed active rotation of ÂÃ matrices that define Dð1;AÞða; b; cÞ in Eq. (278) and that vector operator ÂÃ of Euler angles in the arguments of Dð1;AÞ a; b; c are ð Þ We do not develop this case in details but leave it to the due to the fact that rotation operators affect covariant reader as an exercise. First, change the sign of rotation spherical components of vector operator. The first rotation angle h in Eq. (265):34,37,43,47,56,71,93,123,126 angle is a located at the left-hand side among the three ÂÃ arguments of Dð1;AÞða; b; cÞ . When covariant spherical components are used for left-handed active rotation of X3 R ðhÞV ½R ðhÞ y¼ V ½AðhÞ : (281) vectorÂÃ operator as in Eq. (278), the complex conjugate A i A j ji j Dð1;AÞða; b; cÞ of the Wigner active rotation matrix is ¼1 post-multiplied by the column matrix of these compo- The three Cartesian components of vector operator are

nents. In contrast, when Cartesian components are used as gathered in a row matrix. They are also sandwiched by RA(h) A a; b; c y in Eq. (269), the active rotation matrix fixedZ3Y2Z1ð Þ and ½RAðhÞ . The unconventional summation facilitates is post-multiplied by the column matrix of these compo- comparison with formulas for spherical tensors. We have nents. shown in Section 3 that this transformation is a right-handed Consider the same example about the rotation defined active rotation of “vector” operator in a fixed coordinate sys- in Eq. (271) but with covariant spherical components of tem. Figure 33 shows a right-handed active rotation of spin 127 spin operator. The covariant spherical spin operator compo- operator component Iz about x-axis by a positive angle h: fi nents Iþ1, I0, and I1 are de ned similarly as those of vector operator in Eq. (276). We apply Eq. (278): 0 h h h h: Iz ¼ expði IxÞIz expði IxÞ¼Iz cos Iy sin (282)

0 p p p p 1 0 1 i2Iy i4Iz i4Iz i2Iy Therefore, the right-handed active rotation of vector e e Iþ1e e hiIþ1 B p p p p C p p B C i I i I i I i I ð1;AÞ operator is symbolized by the operator @ e 2 y e 4 z I e 4 z e 2 y A ¼ D ð ; ;0Þ @ I A z 4 2 z ipI ipI ipI ipI e 2 y e 4 z I e 4 z e 2 y I 1 0 1 1 p p p i i i h h ; e 4 e 4 e 4 0 1 RAð Þ¼expði IxÞ (283) pffiffi B 2 2 2 C I B CB þ1 C at left-hand side of the vector operator in Eq. (281). Then, B p1ffiffi 0 p1ffiffi C@ A: ¼ B 2 2 C Iz (279) @ p p p A deduce Table 5 from Table 4 by changing the sign of the i i i e 4 e 4 e 4 I pffiffi 1 rotation angle /. The active rotation matrix A(h) has two 2 2 2 meanings in rotation of vector operator: It is a left-handed We deduce that rotation matrix when it is post-multiplied by a column component matrix. It is a right-handed rotation matrix

p p p p when it is pre-multiplied by a row component matrix. 00 i2Iy i4Iz i4Iz i2Iy ; Iz ¼ e e Ize e ¼Ix When Euler angles are involved in an active rotation, Eq. (281) can be generalized: p p p p 00 00 i2Iy i4Iz i4Iz i2Iy Ix þ iIy ¼ e e Ix þ iIy e e p p p p ¼I S þ I C þ i I C þ I S : (280) y 4 z 4 y 4 z 4 The relations of Eq. (280) agree with those in Eq. (273). In other words, we have at our disposal a third method for determining the left-hand rotated vector operator using the complex conjugateÂÃ of Wigner active rotation matrix. fixed a; b; c y T The transformation RA ð Þ ðÞVþ1 V0 V 1 fixed a; b; c RA ð Þ describes the left-handed active rotation about ﬁxedaxesforavectoroperatorV expressed with covariant spherical components. The superscript T means transposi- FIGURE 33 Right-handed active rotation of spin operator ð1;AÞ tion. As the Wigner active rotation matrix D ða; b; cÞ is component Iz about x-axis by a positive angle h MAN | 47 of 64

TABLE 5 Right-handed active rotation of Cartesian spin operator components, the rotation angle / being positive (/ [ 0), deduced from Mehring and Weberruss,86,121,122 Gunther€ and co-workers,146 Duer,147 Kimmich,7 Slichter,144 Schmidt-Rohr and Spiess,110 and Siemens and co-workers66 / / y P3 RAð ÞIi½RAð Þ A / Ij½ð Þ ji j¼1 0 1 expði/IzÞðÞIx Iy Iz cos / sin / 0 @ / / A expði/IzÞ ðÞIx Iy Iz sin cos 0 001 0 1 expði/IyÞðÞIx Iy Iz cos / 0 sin / @ A expði/IyÞ ðÞIx Iy Iz 010 sin / 0 cos / 0 1 expði/IxÞðÞIx Iy Iz 10 0 @ / / A expði/IxÞ ðÞIx Iy Iz 0 cos sin 0 sin / cos /

ÂÃ fixed a; b; c fixed a; b; c y RA ð ÞðÞVx Vy Vz RA ð Þ A a A b A c ¼ ðÞVx Vy V z zð Þ yð Þ zð Þ A a; b; c : ¼ ðÞVx Vy V z fixedZ3Y2Z1ð Þ (284) A a; b; c fixedZ3Y2Z1ð Þ in Eq. (284) is identical to that defined in Eq. (269). Therefore, it also has two meanings: It is a left-handed active rotation matrix when it is post- multiplied by a column component matrix as in Eq. (269). It is a right-handed active rotation matrix when it is pre-multiplied by a row component matrix as in Eq. (284). fi c The latter shows that the rst rotation angle is located FIGURE 34 Right-handed active rotation of spin operator at the right-hand side among the three arguments of component I about fixed axes: (A) initial configuration; the first (B), A a; b; c x fixedZ3Y2Z1ð Þ. Equation (284) shows that the operator and second (C) rotations are about the z-axis by angle c = p/4 and for right-handed active rotation of a vector operator about about the y-axis by angle b = p/2, respectively fixed axes is

fixed R ða; b; cÞ¼expðiaLzÞ expðibLyÞ expðicLzÞ: A a b c c (285) expði IzÞ expði IyÞ expði IzÞIx expði IzÞ expðibIyÞ expðiaIzÞ; a ¼ 0; b ¼ p=2; and c ¼ p=4: (287) Similar to the active rotations of space function and spherical harmonics about fixed axes, the first, second, and It describes right-handed active rotations of the spin operator third right-handed active rotation angles for vector operator component Ix about fixed axes. The first and second right- V about fixed axes are c, b, and a, respectively. The opera- handed active rotations are about z-axis by angle c = p/4 00 tors for right-handed (Eq. (285)) and left-handed (Eq. (270)) and about y-axis by angle b=p/2, respectively. I x lies in the active rotations of a vector operator about fixed axes are Her- fgy; z plane. We have at our disposal three methods for mitian conjugated. determining the right-hand rotated vector operator: Table 5 allows us to determine the transformation p p law126 for covariant spherical components of vector opera- I00 ¼ I S I C : (288) x y 4 z 4 tor defined in Eq. (276):43,128 ÂÃ 1. rotation operator in Table 5; fixed a; b; c fixed a; b; c y 2. rotation matrix A ða; b; cÞ as in Eq. (284); RA ð ÞðÞVþ1 V0 V 1 RA ð Þ fixedZ3Y2Z1 (286) ð1;AÞ ð1;AÞ 3. Wigner rotation matrix D ða; b; cÞ as in Eq. (286). ¼ ðÞV þ1 V0 V1 D ða; b; cÞ:

fi c 00 The rst rotation angle is located at the right-hand side The spin operator component I x agrees with that shown ; among the three arguments of Dð1 AÞða; b; cÞ. Figure 34 in Figure 34C. These results are examples of general provides a geometrical representation of the result of expressions 48 of 64 | MAN ÂÃ y ÂÃ V0 V0 V 0 ¼ Rfixedða;b;cÞ V V V Rfixedða;b;cÞ P c; b; a T x y z A x y z A ¼ ðÞVx Vy Vz fixedZ1Y2Z3ð Þ ¼ V V V A ða;b;cÞ; A a; b; c ; x y z fixedZ3Y2Z1 ¼ ðÞV x V y V z fixedZ3Y2Z1ð Þ (291) (289) and the term on the final right-hand side of Eq. (284). Simi- 0 0 0 fixed V V V ¼ R ða; b; cÞðÞVþ1 V0 V 1 larly, the replacement in the term on the final right-hand þ1 0 1 ÂÃA y side of Eq. (278) gives Rfixed a; b; c A ð Þ 0 1 ð1;AÞ a; b; c : hi ¼ ðÞV þ1 V 0 V1 D ð Þ Vþ1 ð1;AÞ a c; b b; c a @ A (290) D ð ! ! ! Þ V 0 Â V1 fixed a; b; c fixed 0 1 The transformation RA ð ÞðÞV x V y V z RA y hiV þ1 a; b; c describes the right-handed active rotation about ; ð Þ ¼ Dð1 PÞðc; b; aÞ @ V A fi 0 xed axes for a vector operator V expressed with Cartesian V A a; b; c 1 components. fixedZ3Y2Z1ð Þ is the corresponding right- handed active rotation matrix. The first rotation angle is c. hi When covariant spherical components are used for right- T ð1;PÞ handed active rotation of vector operator as in Eq. (286), ¼ ðÞV þ1 V0 V 1 D ðc; b; aÞ (292) Wigner active rotation matrix Dð1;AÞða; b; cÞ is pre-multiplied 71 by the row matrix of these components. When Cartesian ; ¼ ðÞV V V Dð1 AÞða; b; cÞ; A a; b; c þ1 0 1 components are used as in Eq. (284), fixedZ3Y2Z1ð Þ is fi also pre-multiplied by the row matrix of these components.Â the term on the nal right-hand side of Eq. (286). fixed a;b;c fixed The transformation RA ð ÞðÞVþ1 V0 V1 RA a;b;c y fi ð Þ describes the right-handed active rotation about xed 10.2 | Rotation of coordinate system axes for a vector operator V expressed with covariant spherical components. Furthermore, the Wigner active rotation matrix In classical system, the three Cartesian components ð1;AÞ D ða;b;cÞ is that derived with right-handed active rotation. (vx ¼ v1, vy ¼ v2, vz ¼ v3) of a vector v in an orthonormal When Cartesian components are used, the same rotation basis vectors transform like A a; b; c matrix fixedZ3Y2Z1ð Þ is either a left-handed active rota- X3 tion matrix in left-handed active rotation of vector operator or a P vi ! ½ijvj (293) right-handed active rotation matrix in right-handed active rota- j¼1 tion of vector operator. In contrast, when covariant spherical components are used, Wigner active rotation matrix is involved under a right-handed passive rotation of vector v or under in right-handed active rotation of vector operator and its com- a right-handed rotation of coordinate system keeping vector fi P plex conjugate is involved in left-handed active rotation of vec- v xed, being a 3X3 real, orthogonal matrix associated with a right-handed passive rotation.3 P is post-multiplied tor operator. As the ordersÂÃ of Euler angles in the arguments of A a; b; c ð1;AÞ a; b; c ð1;AÞ a; b; c by the column component matrix of v. For example, the fixedZ3Y2Z1ð Þ, D ð Þ ,andD ð Þ are identical, the first rotation angles are a for left-handed and c for matrix of right-handed passive rotation about z-axis by right-handed active rotations of vector operator. angle h is The left-hand members of Eqs. (269) and (284) show that we can deduce expressions of the right-handed active rota- 0 1 cos h sin h 0 tion of vector operator from those of left-handed case by P ðhÞ¼@ sin h cos h 0 A: (294) a b c z replacing the Euler angles , , in left-handed active rota- 001 tion by c, b, a, respectively. Indeed, the replacement in the term on the final right-hand side of Eq. (269) gives 0 1 We will show that left-handed rotation of coordinate V x fi fi B C system about xed axes keeping the vector operator xed A a c; b b; c a @ A fixedZ3Y2Z1ð ! ! ! Þ V y is equivalent to right-handed active rotation of vector oper- V ator about fixed axes. Similarly, right-handed rotation of 0 1 z coordinate system about fixed axes keeping the vector V x B C operator fixed is equivalent to left-handed active rotation of ¼ P ðc; b; aÞ@ V y A fixedZ1Y2Z3 vector operator about fixed axes. V z MAN | 49 of 64

10.2.1 | Left-handed rotation of coordinate system In quantum system, the state vector jiW remains unchanged under a passive rotation RP as deﬁned in Eq. (10). The matrix element of a passively rotated vector operator com- 0 ponent V i is related to that of the original vector operator component V i as in Eq. (12):

W W RP W 0 W W y W : 1 V i 2 ! 1 V i 2 ¼ 1 ðRPÞ ViRP 2 (295) FIGURE 35 Left-handed rotation of coordinate system from (O, Equation (295) describes the rotation of the vector x, y, z)to(O, x0,y0,z0) about x-axis by a positive angle h, the spin operator from V to V0, keeping the physical system ﬁxed. operator component Iz remains ﬁxed It is reasonable to demand that the matrix element of a vector operator component in quantum mechanics be transformed like a classical vector under a passive expðihI ÞI expðihI Þ¼I cos h I sin h: (299) rotation:80 x z x z y Therefore, Table 6, which is deduced from Table 4 by X3 changing the sign of rotation angle /, describes left-handed W V 0 W W R yV R W P W V W : 1 i 2 ¼ 1 ð PÞ i P 2 ¼ ij 1 j 2 passive rotations of spin operator I. We deduce that the j¼1 rotation matrices P(/) in Table 6 and Eq. (298), which are (296) post-multiplied by column component matrix, are left- handed passive rotation matrices. They are identical to the Equation (296) relates the matrix element of a pas- right-handed passive rotation matrices P ð/Þ, P ð/Þ, and sively rotated vector operator component V0 to those x y i P ð/Þ derived previously.3 of the original vector operator components V . This rela- z i In Section 5 about rotation of space function and Sec- tion has to be valid for any W and W . In short, a vec- 1 2 tion 9 about rotation of spherical harmonics, the coordi- tor operator in passive rotation transforms according to80 nate system is represented by bra position basis at the

X3 left-hand side of the rotation operators and the space h y h P h : ½RPð Þ ViRPð Þ¼ ½ð Þ ijVj (297) function is represented by the ket state at the right-hand j¼1 side of the rotation operator. The generalization of Eq. (298) to Euler angles in passive rotation is less obvi- y As RPðhÞ¼½RAðhÞ , we improve Eq. (297): ous than that in active rotation because coordinate system is not explicit in Eq. (298). Fortunately, we can rely on X3 h y h h h y P h : Eq. (296) to introduce the coordinate system ficti- ½RPð Þ V iRPð Þ¼RAð ÞV i½RAð Þ ¼ ½ð Þ ijVj j¼1 tiously as (298) TABLE 6 Left-handed passive rotation of Cartesian spin operator The three components of vector operator are gathered in a components deduced from Table 4, the rotation angle / being column matrix. They are also sandwiched by ½R ðhÞ y and P positive (/ [ 0) RP(h). / y / P3 ½RPð Þ IiRPð Þ P / Figure 35 represents the spin operator component Iz and ½ð Þ ijIj j¼1 a left-handed rotation of the coordinate system about x-axis 0 1 0 10 1 h 0 0 0 Ix cos / sin / 0 Ix by a positive angle , from (O, x, y, z)to(O, x , y , z ). @ A @ A@ A expði/IzÞ Iy expði/IzÞ sin / cos / 0 Iy This is also called a left-handed passive rotation of spin I 001I h z z operator component Iz about x-axis by angle . That is, Iz 0 1 0 10 1 / / remains fixed but the coordinate system is left-hand rotated. Ix cos 0 sin Ix / @ A / @ A@ A In practice, the expression of I in the left-hand rotated expði IyÞ Iy expði IyÞ 01 0 Iy z I sin / 0 cos / I 0 0 0 z z coordinate system (O, x , y , z ) can be determined as a 0 1 0 10 1 right-handed active rotation of Iz about x-axis by angle h Ix 10 0 Ix / @ A / @ / / A@ A (Eq. (282)): expði IxÞ Iy expði IxÞ 0 cos sin Iy Iz 0 sin / cos / Iz 50 of 64 | MAN subscript Z1Y2Z3 of P ðc; b; aÞ is meaningless in R h yV R h R h y V R h : (300) fixedZ1Y2Z3 hj½Pð Þ i Pð Þji ¼ hj½Pð Þ iðÞPð Þji left-handed rotation of coordinate system because Eq. (302) shows that the first rotation angle is a located We deduce that left-handed rotation of coordinate system is at the right-hand side among the three arguments of the transformation: P c; b; a fixedZ1Y2Z3ð Þ. ÂÃ fixed y y We may replace R c; b; a by its equivalent oper- hj½R ðhÞ : (301) P ð Þ P fixed a; b; c ator RA ð Þ: When Euler angles are involved, the left-handed rotation 0 1 of coordinate system about fixed axes, keeping the vector V x Â ÂÃB C fi fixed fixed c; b; a y@ A fixed c; b; a operator xed, is expressed by the operator RP hj RP ð Þ V y RP ð Þ c; b; a y ð Þ : V z 0 1 (304) ÂÃÂÃ V x y ÂÃ fixed c; b; a c b a y B C y hj R ð Þ ¼ hj expði LzÞ expði LyÞ expði LzÞ fixed a; b; c @ A fixed a; b; c : P hjRA ð Þ V y RA ð Þ a b c : ¼ hj expði LzÞ expði LyÞ expði LzÞ V z (302) That is, the first, second, and third right-handed active rota- fi The rst, second, and third left-handed rotations of coordi- tions of Vi in fixed coordinate system are about z-axis by nate system about fixed axes are about z-axis by angle a, angle c, about y-axis by angle b, and about z-axis by angle about y-axis by angle b, and about z-axis by angle c, a, respectively. It is worth noting that in Eq. (304) the vec- respectively. tor operator components are in column matrix, whereas in When Euler angles are involved in a rotation of coordi- Eq. (284) they are in rowÂÃ matrix. nate system, Eq. (298) can be generalized as80 fixed c; b; a y fixed a; b; c As the two operators RP ð Þ and RA ð Þ 0 1 are equivalent, the left-hand member of Eq. (304) shows ÂÃV x that the first rotation angle is a in left-handed rotations of fixed c; b; a y@ A fixed c; b; a RP ð Þ V y RP ð Þ coordinate system (see also Eq. (302)), whereas the right- V z c 0 1 hand member of Eq. (304) shows that it is in right- Vx handed active rotations of vector operator. B C ¼ expðiaLzÞ expðibLyÞ expðicLzÞ@ Vy A Figure 36 provides a geometrical representation of the result of V z c b a ÂÃ expði LzÞ expði LyÞ expði LzÞ fixed c; b; a y fixed c; b; a 0 1 RP ð Þ IxRP ð Þ ÂÃ V x y a b P c @ A b a Rfixedða; b; cÞI Rfixedða; b; cÞ ¼ expði LzÞ expði LyÞ zð Þ V y expði LyÞ expði LzÞ A x A V z ¼ expðiaIzÞ expðibIyÞ expðicIzÞIx expðicIzÞ (303) (305) 0 1 expðibIyÞ expðiaIzÞ; V x a P c P b @ A a a ¼ p=4; b ¼ p=2; and c ¼ 0: ¼ expði LzÞ zð Þ yð Þ V y expði LzÞ V z 0 1 It describes left-handed rotations of coordinate system Vx @ A about fixed axes. The first and second left-handed rotations ¼ P ðcÞP ðbÞP ðaÞ Vy z y z of coordinate system are about z-axis by angle a=p/4 and Vz 0 1 about y-axis by angle b=p/2, respectively. It also describes V x fi P c; b; a @ A: right-handed active rotations of Ix about xed axes. As ¼ fixedZ1Y2Z3ð Þ V y c=0, the first and second right-handed active rotations of Ix V z are about y-axis by angle b=p/2 and about z-axis by angle The order of the three elementary rotation matrices that a=p/4, respectively. fi P c; b; a de ne fixedZ1Y2Z3ð Þ in Eq. (303) and that of Euler Figure 36A shows the spin operator component Ix in P c; b; a fi angles in the arguments of fixedZ1Y2Z3ð Þ are due to the xed coordinate system (O, x, y, z). Figure 36B the fact that rotation operators affect the components of shows Ix in two coordinate systems (O, x, y, z) and P c; b; a 00 00 00 vector operator. This means fixedZ1Y2Z3ð Þ already (O, x , y , z ). The latter is left-handed rotation of (O, x, 1,3 defined for right-handed passive rotation is also the y, z) about z-axis by angle a=p/4. Figure 36C shows Ix left-handed passive rotation matrix for a vector operator in two coordinate systems (O, x, y, z) and whose components are gathered in column matrix. The (O, x000, y000, z000). The latter is left-handed rotation of MAN | 51 of 64

FIGURE 36 Equivalence between (A, B, C) left-handed rotations of coordinate system about fixed axes with spin operator component Ix fixed and (A, D, E) right- handed active rotations of Ix about fixed axes. (A) Initial configuration. The first (B) and second (C) left-handed rotations of the coordinate system are about z-axis by angle a=p/4 and about y-axis by angle b=p/2, respectively. The first (D) and second (E) right-handed active rotations of Ix are about y-axis by angle b=p/2 and about z-axis by angle a=p/4, respectively

00 00 00 (O, x , y , z ) about y-axis by angle b=p/2. Ix lies in the We can also determine the expression of Ix in 000 00 000 000 000 000 negative direction of z -axis. Figure 36D shows I x, the (O, x , y , z ) or equivalently that of I x in (O, x, y, z) P c; b; a right-handed active rotation of Ix about y-axis by angle using the rotation matrix fixedZ1Y2Z3ð Þ as in 00 b=p/2. I x lies in the negative direction of z-axis. Fig- Eq. (303): 000 00 ure 36E shows I x, the right-handed active rotation of I x a p 000 about z-axis by angle = /4. I x remains in the nega- 0 p p p p 1 0 1 i4Iz i2Iy i2Iy i4Iz tive direction of z-axis. In left-handed rotation of coordi- e e Ixe e Ix p p p p p p B i I i I i I i I C B C 000 @ 4 z 2 y 2 y 4 z A P ; ; @ A nate system, Ix lies in the negative direction of z -axis e e Iye e ¼ fixedZ1Y2Z3ð0 Þ Iy 000 000 000 p p p p 2 4 of the rotated coordinate system (O, x , y , z ), whereas i4Iz i2Iy i2Iy i4Iz e e Ize e Iz 000 0 10 1 in right-handed active rotation, the rotated component I x 001 I ﬁ B CB x C lies in the negative direction of z-axis of the xed coordi- @ p p A@ A: ¼ S 4 C 4 0 Iy nate system (O, x, y, z). p p C 4 S 4 0 Iz We can determine the expression of Ix in 000 000 000 000 (307) (O, x , y , z ) or equivalently that of I x in (O, x, y, z) using Table 6: We deduce that

000 ipI ipI ipI ipI I ¼ e 4 z e 2 y I e 2 y e 4 z ¼I : (308) p p p p x x z I000 ¼ expði I Þ expði I ÞI expði I Þ expði I Þ x 4 z 2 y x 2 y 4 z Equation (308) is identical to Eq. (306). These two p p ¼ expði I ÞI00 expði I Þ (306) results verify the relation 4 z x 4 z p p 0 1 0 1 V0 V ¼ expði IzÞðÞIz expði IzÞ¼Iz: x ÂÃx 4 4 @ 0 A fixed c; b; a y@ A fixed c; b; a V y ¼ RP ð Þ Vy RP ð Þ 0 000 V z V z The spin operator component I x agrees with that shown in Figure 36E. 52 of 64 | MAN 0 1 0 1 V V x ÂÃþ1 B CÂÃ yB C fixed a; b; c @ A fixed a; b; c y Rfixedðc; b; aÞ @ V ARfixedðc; b; aÞ RA ð Þ V y RA ð Þ P 0 P V V 1 z 0 1 0 1 V V x þ1 ÂÃ B C B C y P c; b; a @ A: Rfixedða; b; cÞ@ V A Rfixedða; b; cÞ ¼ fixedZ1Y2Z3ð Þ V y (309) A 0 A V Vz 1 0 1 ÂÃ fixed y T fixed Vþ1 The transformation R ðc; b; aÞ ðÞV x V y Vz R B C P P a b c @ A ðc; b; aÞ describes the left-handed rotation of coordi- ¼ expði LzÞ expði LyÞ expði LzÞ V 0 nate system about fixed axes for a vector operator V V1 P c; b; a c b a expressed with Cartesian components. fixedZ1Y2Z3ð Þ expði LzÞ expði LyÞ expði LzÞ is the corresponding left-handed passive rotation matrix. The first left-handed rotation angle of the coordinate sys- 0 10 1 tem is a. ic e 00 V þ1 Table 6 allows us to determine the transformation law B CB C ¼ expðiaLzÞ expðibLyÞ@ 010A@ V 0 A for covariant spherical components of vector operator 00eic V defined in Eq. (276): 1 b a 0 1 expði LyÞ expði LzÞ V þ1 (311) B C expði/LxÞ@ V 0 A expði/LxÞ V 01 1 0 1 / / / 0 1 2 b Sbffiffi 2 b 2 iSffiffi 2 0 1 ic C p S C p S e 00 2 2 2 B 2 2 2 C Vþ1 B C B CB C B CB Sb Sb C iSffiffi/ / iSffiffi/ ¼ expðiaL Þ@ 010AB pffiffi Cb pffiffi C ¼ B p C p C@ V 0 A z @ 2 2 A @ 2 2 A c / / / i 2 b Sffiffib 2 b 2 iSffiffi 2 V 1 00e S p C S p C 2 2 2 2 2 0 2 1 0 1 hi Vþ1 p p V þ1 ð1;PÞ @ A B C ¼ D ð ; /; Þ V ; (310) @ V 0 AexpðiaLzÞ 2 2 0 V 1 V 1 0 1 0 1 2 b Sbffiffi 2 b ic C p S 0 1 e 00B 2 2 2 C 0 1 / / / 0 1 B CB Sffiffib Sbffiffi C C2 pS ffiffi S2 ¼@ 010AB p Cb p C Vþ1 B 2 2 2 C V þ1 @ 2 2 A ic B C B S/ S/ CB C 00e 2 b Sffiffib 2 b expði/L Þ@ V Aexpði/L Þ¼B pffiffi C/ pffiffi C@ V A S p C y 0 y @ 2 2 A 0 0 102 12 2 / / / ia V1 S2 pSffiffi C2 V 1 e 00 V þ1 2 2 02 1 B CB C @ 010A@ V0 A Vþ1 ia ð1;PÞ B C 00e V 1 ¼D ð0;/;0Þ@ V 0 A; 0 1 hi Vþ1 V1 ð1;PÞ @ A ¼ D ðc; b; aÞ V 0 : V1 0 1 0 10 1 i/ V þ1 e 00 Vþ1 The order ofÂÃ the three elementary Wigner rotation matrices B C B CB C ð1;PÞ / @ A / @ A@ A that define D ðc; b; aÞ in Eq. (311) and that of Euler expði LzÞ V 0 expði LzÞ¼ 010 V 0 ÂÃ angles in the arguments of Dð1;PÞðc; b; aÞ are due to the V 00ei/ V 1 0 1 1 fact that rotation operators affect the covariant spherical com- hiV þ1 ponents of vector operator. The first left-handed rotation ; B C ¼ Dð1 PÞð0;0;/Þ @ V A: 0 angle of the coordinate system is aÂÃlocated at the right-hand ð1;PÞ V 1 side among the three arguments of D ðc; b; aÞ . When covariant spherical components are used for The 3X3 Wigner passive rotation matrices in Eq. (310) are left-handed rotation of coordinate system as in Eq. (311), ÂÃ those of Eq. (277), in which we have changed the sign of the complex conjugate Dð1;PÞðc; b; aÞ of the Wigner angle /. passive rotation matrix is post-multiplied by the column Equation (310) can be generalized to Euler angles:80 matrix of these components.80 In contrast, when Cartesian MAN | 53 of 64 components are used as in Eq. (303), the passive rotation P c; b; a matrix fixedZ1Y2Z3ð Þ is post-multiplied by the column matrix of these components.80 Consider the same example about the rotation defined in Eq. (305) but with covariant spherical spin operator components. We apply Eq. (311): 0 1 0 1 Iþ1 hiIþ1 p p ipI ipI B C ipI ipI ð1;PÞ B C e 4 z e 2 y @ I Ae 2 y e 4 z ¼ D ð0; ; Þ @ I A z 2 4 z I1 I1 0 p p 1 i i e 4 e 4 p1ffiffi 0 1 2 2 B 2 C Iþ1 FIGURE 37 Right-handed rotation of coordinate system about p p B i i C 0 0 0 B e 4 e 4 CB C x-axis by a positive angle h from (O, x, y, z)to(O, x ,y,z), the spin ¼ B pffiffi 0 pffiffi C@ Iz A: 2 2 @ A operator component Iz remains fixed p p i i e 4 e 4 I1 p1ffiffi 2 2 2 (312) In practice, the expression of Iz in the right-handed rotated We deduce that coordinate system (O, x0, y0, z0) can be determined as a

left-handed active rotation of Iz about x-axis by angle h in p p p p p p 000 i4Iz i2Iy i2Iy i4Iz ; (O, x, y, z) (see Eq. (266)): Iz ¼ e e Ize e ¼ IxC þ IyS 4 4 000 000 ipI ipI ipI ipI I þ iI ¼ e 4 z e 2 y I þ iI e 2 y e 4 z x y x y h h h h: p p (313) expði IxÞIz expði IxÞ¼Iz cos þ Iy sin (315) ¼Iz þ i IxS þ IyC : 4 4 Then, deduce Table 7 from Table 5 by changing the The relations of Eq. (313) agree with those in sign of the rotation angle /. This table describes right- Eq. (307). We have at our disposal three methods for deter- handed passive rotations of spin operator I by left-handed mining the expression of vector operator in a left-hand active rotations of the spin operator I. The passive rotation rotated coordinate system: (1) rotation operator as in matrix P(h) has two meanings: It is a left-handed rotation Eq. (306); (2) rotation matrix as in Eq. (307); (3) the com- matrix when it is post-multiplied by a column component plex conjugate of WignerÂÃ rotation matrix as in Eq. (312). matrix as in Table 6. It is a right-handed rotation matrix fixed c; b; a y T when it is pre-multiplied by a row component matrix as in The transformation RP ð Þ ðÞV þ1 V0 V 1 fixed c; b; a “ Table 7. RP ð Þ is a shorthand notation for left-handed rotations of coordinate system about fixed axes by Euler angles, the Introducing the coordinate system fictitiously yields the fi vector operator remaining fixed”ÂÃ. Its equivalent expression right-handed rotations of coordinate system about xed fixed a; b; c T fixed a; b; c y axes, keeping the vector operator fixed, which is expressed RA ð ÞðÞV þ1 V 0 V 1 RA ð Þ allows us to 0 0 0 T fixed c; b; a : visualize ðÞV þ1 V 0 V 1 via a right-handed active rota- by the operator RP ð Þ tion of vector operator about fixed axes. As the Wigner passive rotation matrix Dð1;PÞðc; b; aÞ is derived for right-handed rotation of coordinate system, its complex conjugate is involved in the left-handed rotation of coordinate system.80 TABLE 7 Right-handed passive rotation of Cartesian spin operator components deduced from Table 5, the rotation angle / being positive (/ [ 0)

10.2.2 | Right-handed rotation of coordinate R / I R / y P3 Pð Þ i½Pð Þ P / system Ij½ð Þ ji j¼1 0 1 We do not develop this case in details but let it to reader expði/IzÞðÞIx Iy Iz cos / sin / 0 / @ / / A as exercices. First, change the sign of the rotation angle h expði IzÞ ðÞIx Iy Iz sin cos 0 001 in Eq. (297): 0 1 X3 expði/IyÞðÞIx Iy Iz cos / 0 sin / y y @ A RPðhÞV ½RPðhÞ ¼ ½RAðhÞ V RAðhÞ¼ V ½PðhÞ : expði/IyÞ ðÞIx Iy Iz 01 0 i i j ji / / j¼1 sin 0 cos 0 1 (314) expði/IxÞðÞIx Iy Iz 10 0 @ / / A expði/IxÞ ðÞIx Iy Iz 0 cos sin Figure 37 shows a right-handed passive rotation of Iz 0 sin / cos / about x-axis by angle h from (O, x, y, z)to(O, x0, y0, z0). 54 of 64 | MAN ÂÃ fixed c; b; a c b a : fixed c; b; a fixed c; b; a y hjRP ð Þ¼hj expði LzÞ expði LyÞ expði LzÞ (316) RP ð ÞIx RP ð Þ

fi ¼ expðicLzÞ expðibLyÞ expðiaLzÞIx expðiaIzÞ The rst, second, and third right-handed rotations of coor- (320) dinate systems about fixed axes are about z-axis by angle expðibIyÞ expðicIzÞ; c, about y-axis by angle b, and about z-axis by angle a, a ¼ p=4; b ¼ p=2; and c ¼ 0: respectively. – Equation (314) can be generalized to Euler angles as Figure 38(A C) describe right-handed rotations of coordinate system about fixed axes. The first and second right- ÂÃ fixed c; b; a fixed c; b; a y handed rotations of coordinate system are about y-axis by RP ð ÞðÞVx V y V z RP ð Þ angle b=p/2 and about z-axis by angle a=p/4, respectively.

Figure 38(A,D,E) describe left-handed active rotations of Ix P c P b P a ¼ ðÞV x V y V z zð Þ yð Þ zð Þ about fixed axes. The first and second left-handed active rota- P c; b; a : a p ¼ ðÞV x V y V z fixedZ1Y2Z3ð Þ (317) tions of Ix are about z-axis by angle = /4 and about y-axis by angle b=p/2, respectively. In right-handed rotation of coordi- The passive rotation matrix P ðc; b; aÞ in fixedZ1Y2Z3 nate system, I lies in fgy00; z00 plane of the rotated coordinate Eq. (317) is identical to that defined in Eq. (303). It also x system (O, x00, y00, z00), whereas in left-handed active rotation has two meanings: It is a right-handed rotation matrix of the spin operator component, the rotated component I00 lies when it is pre-multiplied by a row component matrix as in x in fgy; z plane of the fixed coordinate system (O, x, y, z). Eq. (317). It is a left-handed rotation matrix when it is We have three ways to determine the expression of I in post-multiplied by a column component matrix as in x (O, x00, y00, z00) or equivalently that of I00 in (O, x, y, z), Eq. (303). x p p p p Table 7 allows us to determine the transformation law 00 Ix ¼ expði IyÞexpði IzÞIx expði IzÞexpði IyÞ for covariant spherical components of vector operator 2 4 4 2 (321) – p p defined in Eq. (276):129 131 ¼I S þ I C ; y 4 z 4

ÂÃ1. using rotation matrices in Table 7; fixed c; b; a fixed c; b; a y P c; b; a RP ð ÞðÞV þ1 V0 V1 RP ð Þ 2. using the rotation matrix fixedZ1Y2Z3ð Þ as in ÂÃ fixed y fixed Eq. (317); ¼ R ða; b; cÞ ðÞV V V R ða; b; cÞ ; A þ1 0 1 A 3. using Wigner passive rotation matrix Dð1 PÞðc; b; aÞ as ð1;PÞ ¼ ðÞVþ1 V 0 V 1 D ðc; b; aÞ: (318) in Eq. (318). The first right-handed rotation angle of the coordinate sys- The expression of the spin operator component I00 tem is c (see Eq. (316)) located at the left-hand side among x ; agrees with that shown in Figure 38E. These results are the three arguments of Dð1 PÞðc; b; aÞ. examples of general expressions When covariant spherical components are used for ÂÃ right-handed rotation of coordinate system as in Eq. (318), 0 0 0 fixed c;b;a fixed c;b;a y ; ðÞ¼V x V y V z RP ð ÞðÞV x V y V z RP ð Þ the Wigner passive rotation matrix Dð1 PÞðc; b; aÞ is pre- ÂÃ fixed a;b;c y fixed a;b;c multiplied by the row matrix of these components. When ¼ RA ð Þ ðÞV x V y Vz RA ð Þ P c;b;a ; Cartesian components are used as in Eq. (317), the passive ¼ðÞV x Vy V z fixedZ1Y2Z3ð Þ P c; b; a rotation matrix fixedZ1Y2Z3 ð Þ is pre-multiplied by the (322) row matrix of these components. fixed c; b; a and ÂÃWe may replace RP ð Þ by its equivalent operator fixed a; b; c y 0 0 0 fixed RA ð Þ : ðÞ¼V þ1 V 0 V 1 R ðc; b; aÞðÞVþ1 V 0 V 1 ÂÃP y ÂÃ Rfixedðc; b; aÞ y P hjRfixedðc; b; aÞðÞV V V Rfixedðc; b; aÞ ÂÃ P x y z P fixed a; b; c y fixed a; b; c ÂÃ ¼ RA ð Þ ðÞVþ1 V 0 V 1 RA ð Þ fixed y fixed hj R ða; b; cÞ ðÞV x V y V z R ða; b; cÞ: (319) ð1;PÞ A A ¼ ðÞV þ1 V 0 V1 D ðc; b; aÞ: (323) Â fixed c; b; a fixed That is, the first, second, and third left-handed active rota- The transformation RP ð ÞðÞVx Vy V z RP c; b; a y tions of Vi in fixed coordinate system are about z-axis by ð Þ describes the right-handed rotation of coordinate angle a, about y-axis by angle b, and about z-axis by angle system about fixed axes for a fixed vector operator V with P c; b; a c, respectively. Cartesian components. fixedZ1Y2Z3ð Þ is the corre- Figure 38 provides a geometrical representation of the sponding right-handed passive rotation matrix. The first result of right-handed rotation angle of the coordinate system is c. MAN | 55 of 64

FIGURE 38 Equivalence between (A, B, C) right-handed rotations of coordinate system about fixed axes with spin operator component Ix fixed and (A, D, E) left- handed active rotations of Ix about fixed axes. (A) Initial configuration. The first (B) and second (C) right-handed rotations of the coordinate system are about y-axis by angle b=p/2 and about z-axis by angle a=p/ 4, respectively. The first (D) and second (E) left-handed rotations of Ix are about z-axis by angle a=p/4 and about y-axis by angle b=p/2, respectively Â fixed c; b; a fixed The transformation RP ð ÞðÞV þ1 V 0 V1 RP 0 1 y ðc; b; aÞ is a shorthand notation for “right-handed rotations V x B C fi P c a; b b; a c @ A of coordinate system about xed axes by Euler angles, the vec- fixedZ1Y2Z3ð ! ! ! Þ V y fi ” torÂÃ operator remaining xed . Its equivalent expression Vz fixed y fixed 0 1 R ða; b; cÞ ðÞV þ1 V0 V 1 R ða; b; cÞ allows us to A A V x visualize ðÞV0 V0 V0 in fixed coordinate system via a B C þ1 0 1 ¼ A ða; b; cÞ@ V A ¼ left-handed active rotation of vector operator about fixed axes. fixedZ3Y2Z1 y When Cartesian components are used, the same rotation V z P c; b; a matrix fixedZ1Y2Z3ð Þ is either a left-handed passive ÂÃ A a; b; c T rotation matrix in a left-handed rotation of coordinate sys- ¼ ðÞVx Vy Vz fixedZ3Y2Z1ð Þ P c; b; a ; tem or a right-handed passive rotation matrix in a right- ¼ ðÞV x V y V z fixedZ1Y2Z3ð Þ (324) handed rotation of coordinate system. In contrast, when covariant spherical components are used, the Wigner pas- the right-hand member of Eq. (317). Similarly, the replace- sive rotation matrix is involved in a right-handed rotation ment in the right-hand member of Eq. (311) gives of coordinate system and its complex conjugate is involved 0 1 hiV þ1 in the left-handed rotation of coordinate system. As the B C ð1;PÞ c a; b b; a c @ A orders of the EulerÂÃ angles in the arguments of D ð ! ! ! Þ V0 ð1;PÞ ð1;PÞ P ðc; b; aÞ, D ðc; b; aÞ , and D ðc; b; aÞ V 1 fixedZ1Y2Z3 0 1 (325) are identical, the first rotation angles are a for left-handed hiV þ1 and c for right-handed rotations of coordinate system. ð1;AÞ B C ¼ D ða; b; cÞ @ V 0 A The left-hand members of Eqs. (303) and (317) show that V 1 we can deduce expressions of the right-handed of coordinate hi T ð1;AÞ system from those of left-handed case by replacing the Euler ¼ ðÞV þ1 V 0 V1 D ða; b; cÞ angles a, b, c in left-handed rotation of coordinate system ð1;PÞ ¼ ðÞV þ1 V 0 V1 D ðc; b; aÞ; by c, b, a, respectively. Indeed, the replacement in the right-hand member of Eq. (303) gives the right-hand member of Eq. (318). 56 of 64 | MAN ÂÃ fixed c; b; a fixed c; b; a y RP ð ÞðÞV x ÂÃV y V z RP ð Þ or by its equi- 10.3 | Discussion fixed a;b;c y fixed a;b;c valent expression RA ð Þ ðÞV x V y V z RA ð Þ, Given a vector operator in a coordinate system, a left-handedÂ followed by a right-handed active rotation of theÂ vector fixed fixed a;b;c fixed rotation of the coordinate system described by RP operator described by RA ð ÞðÞV x V y Vz RA c; b; a y T fixed c; b; a a;b;c y ð Þ ðÞVx Vy Vz RP ð Þ or byÂÃ its equivalent ð Þ , yield the same vector operator in the same fixed a; b; c T fixed a; b; c y expression RA ð ÞðÞV x V y V z RA ð Þ , coordinate system. We can show this using either vector followed by a left-handedÂÃ active rotation of the vector operators: fixed a; b; c y T fixed operator described by RA ð Þ ðÞV x V y V z RA a; b; c fixed a; b; c fixed c; b; a ð Þ, yield the same vector operator in the same RA ð ÞRP ð ÞðÞVx Vy Vz ÂÃÂÃ (329) coordinate system. We can show this using either vector fixed c; b; a y fixed a; b; c y RP ð Þ RA ð Þ operators: ÂÃ fixed a; b; c fixed a; b; c y ¼ RA ð Þ RA ð Þ ðÞVx Vy Vz 0 1 ÂÃ fixed a; b; c fixed a; b; c y ; V x RA ð Þ RA ð Þ ¼ ðÞVx V y V z ÂÃÂÃB C fixed a; b; c y fixed c; b; a y@ A RA ð Þ RP ð Þ V y or Cartesian rotation matrices: V z fixed c; b; a fixed a; b; c RP ð ÞRA ð Þ fixed fixed 0 1 R ða; b; cÞR ðc; b; aÞðÞVx Vy V z V A ÂÃP ÂÃ ÂÃ x y y y B C fixed c; b; a fixed a; b; c fixed a; b; c fixed a; b; c @ A RP ð Þ RA ð Þ ¼ RA ð Þ RA ð Þ V y ¼ ðÞV x V y V z P ðc; b; aÞA ða; b; cÞ V z fixedZ1Y2Z3 fixedZ3Y2Z1 0 1 (326) ¼ ðÞVx Vy V z P ðcÞP ðbÞP ðaÞA ðaÞA ðbÞA ðcÞ ÂÃ V x z y z z y z y B C fixed a; b; c fixed a; b; c @ A; ¼ ðÞV x V y V z ; (330) RA ð Þ RA ð Þ¼ V y Vz or Cartesian rotation matrices: or Wigner rotation matrices: 0 1 ÂÃÂÃV x y yB C ÂÃÂÃ fixed a; b; c fixed c; b; a @ A y y RA ð Þ RP ð Þ V y fixed a; b; c fixed c; b; a fixed RA ð Þ RP ð Þ ðÞV þ1 V0 V1 RP V z c; b; a fixed a; b; c ð ÞRA ð Þ Rfixedðc; b; aÞRfixedða; b; cÞ P A 0 1 ð1;PÞ c; b; a ð1;AÞ a; b; c V x ¼ ðÞVþ1 V 0 V 1 D ð ÞD ð Þ (331) ¼ A ða; b; cÞP ðc; b; aÞ@ V A (327) fixedZ3Y2Z1 fixedZ1Y2Z3 y : V ¼ ðÞVþ1 V 0 V 1 0 1 z 0 1 Vx V x Throughout the article, we show that left- and right- A a A b A c P c P b P a @ A @ A; ¼ zð Þ yð Þ zð Þ zð Þ yð Þ zð Þ Vy ¼ V y handed rotations of coordinate system are equivalent to V z V z right- and left-handed active rotations of position vector, space function, and spherical harmonics, respectively. or Wigner rotation matrices: 0 1 This obviously remains true for vector operator. We have Vþ1 derived left- and right-handed rotations of coordinate sys- ÂÃÂÃB C fixed a; b; c y fixed c; b; a y@ A RA ð Þ RP ð Þ V 0 tem independently of right- and left-handed active rotations of vector operator. In fact these rotations are V1 related. Rfixedðc; b; aÞRfixedða; b; cÞ P A 0 1 Equation (269), which describes the left-handed active hiV þ1 B C (328) rotation of vector operator with Cartesian components, ð1;AÞ a; b; c ð1;PÞ c; b; a ¼ D ð ÞD ð Þ @ V0 A should be equivalent to Eq. (317), which describes the V 0 1 1 right-handedÂÃ rotation of coordinate system. Indeed, fixed a; b; c y fixed a; b; c V þ1 replacing RA ð Þ and RA ð Þ in the term B C on the initial left-hand side of Eq. (269) by Rfixedðc; b; aÞ ¼ @ V0 A: ÂÃ P fixed c; b; a y and RP ð Þ yield the term on the initial left- V 1 hand side of Eq. (317). The term on the final right-hand Given a vector operator in a coordinate system, a right- side of Eq. (269) can be written as the transposed handed rotation of the coordinate system described by expression: MAN | 57 of 64 0 1 Vx B C 11.1 | Spherical tensor operator A a; b; c @ A fixedZ3Y2Z1ð Þ Vy ¼ ðÞV x V y V z In general, a tensor operator can be constructed from tensor V z ÂÃoperators of lower ranks using the relationship41,122,132 A a; b; c T fixedZ3Y2Z1ð Þ ffiffiffiffiffiffiffiffiffiffiffiffiX p k þk m k1 k2 k P c; b; a ; T 2k 1 1 1 2 T T ¼ ðÞV x V y V z fixedZ1Y2Z3ð Þ (332) kq ¼ þ ðÞ k q k q ; q q q 1 1 2 2 q1 q2 1 2 X which is the term on the final right-hand side of Eq. (317). ¼ k1q1k2q2 kq Tk q Tk q ; 1 1 2 2 Similarly for spherical components of vector operator, q1 q2 Eq. (278) is equivalent to Eq. (318). Indeed, the term on (334) the final right-hand member of Eq. (278) can be written as k k k where 1 2 is the Wigner 3j symbol and the transposed expression: q q q 1 2 k q k q kq the Clebsch-Gordon coefficient. Compatible 0 1 1 1 2 2 with the symmetry properties of the elements of the Wigner hiV þ1 hi T ð1;AÞ B C ð1;AÞ 3j symbol, the set of q1 and q2 values is restricted by the D ða;b;cÞ @ V0 A¼ðÞV þ1 V0 V 1 D ða;b;cÞ selection rule q1 þ q2 ¼ q. Furthermore, k1, k2,andk are V 1 ; ; ...; linked via k ¼ k1 k2 k1 k2 þ 1 k1 þ k2. ð1;PÞ ¼ðÞV þ1 V0 V 1 D ðc;b;aÞ; A spin I, which has 2I + 1 energy levels, is described (333) by spherical tensor operators up to rank k=2I. The k=1 spherical tensor operators in terms of Cartesian which is the term on the final right-hand side of spin operators are38,77,132 Eq. (318). Equation (284), which describes the right-handed active T ¼ I ; 10 z rotation of vector operator with Cartesian components, 1 1 T ¼pffiffiffi I ¼pffiffiffi I þ iI ; (335) should be equivalent to Eq. (303), which describe the left- 1þ1 2 þ 2 x y handed rotation of coordinate system. Similarly for spheri- 1ffiffiffi 1ffiffiffi : cal components of vector operator, Eq. (286) is equivalent T11 ¼ p I ¼ p Ix iIy 2 2 to Eq. (311). We have discussed four types of rotation in this section The k=2 and q=0 spherical tensor operator 1,38,91,94,96,122,132,133,143 (left- and right-handed active rotations of vector operator, is X and left- and right-handed rotations of coordinate system). T ¼ 1q 1q 20 T T In fact, they are related. Knowing one of them allows us to 20 1 2 1q1 1q2 q ;q deduce the other three. The relations between these four r1 ffiffiffi2 rffiffiffi rffiffiffi types of rotation of vector operator are similar to those 1 1 2 ¼ T1þ1T11 þ T11T1þ1 þ T10T10 (336) between the four types of rotation of spherical harmonics rffiffiffi6 6 3 shown in Figure 28. 1 ¼ 3I2 IðI þ 1Þ : 6 z The k=2 and q=1 spherical tensor operator 11 | ROTATION OF SPHERICAL 1,38,91,94,96,122,132,133,143 TENSOR OPERATOR is X T ¼ 1q 1q 2 þ1 T T In quantum mechanics, operators representing various 2þ1 1 2 1q1 1q2 q ;q physical quantities are classified as scalars, vectors, and r1 ffiffiffi2 rffiffiffi tensors according to their behaviors under rotations. The 1 1 ¼ T T þ T T (337) analog of spherical tensor in classical mechanics is the 2 1þ1 10 2 10 1þ1 132 spherical tensor operator in quantum mechanics. 1 – ¼ I I þ I I Spherical tensor operators are widely used in NMR.133 142 2 þ z z þ The spherical vector operator discussed in Section 10 is 1 ¼ 2I Iz þ IzI I Iz readily generalized to spherical tensor operator of arbitrary 2 þ þ þ rank. We restrict the definitions to right-hand active rota- 1 : ¼ Iþ 2Iz þ 1 tion of spherical tensor operator and right-hand rotation of 2 coordinate system. Left-hand rotations can be deduced The k=2 and q=1 spherical tensor operator easily as shown in Section 10. is1,38,91,94,96,122,132,133,143 58 of 64 | MAN X X ðkÞ 1 ðkÞ ðkÞ T ¼ 1q 1q 2 1 T T Rða; b; cÞT ½Rða; b; cÞ ¼ T 0 D 0 ða; b; cÞ; 21 1 2 1q1 1q2 q q q q q ;q q0 r1 ffiffiffi2 rffiffiffi a; b; c fixed a; b; c : 1 1 Rð Þ¼RA ð Þ ¼ T T þ T T (338) 2 11 10 2 10 11 (342) 1 ¼ I I þ I I This is the generalization of our right-handed active rota- 2 z z tion of vector operator (Eq. (286)). 1 ¼ 2I I þ I I I I Sakurai and Napolitano26 provide the following defini- 2 z z z tion of spherical tensor operator in Chapter 3 of their book: 1 ¼ I 2I 1 : A spherical tensor operator of rank k with 2k + 1 compo- 2 z nents transform under rotations as The k=2 and q=2 spherical tensor operator hi 1,38,91,94,96,122,132,133,143 Xk is a; b; c y ðkÞ a; b; c ðkÞ a; b; c ðkÞ; ½Dð Þ Tq Dð Þ¼ Dqq0 ð Þ Tq0 X q0¼k 1 2: T2þ2 ¼ 1q11q2 2 þ2 T1q T1q ¼ T1þ1T1þ1 ¼ Iþ ; 1 2 2 (343) q1 q2 (339) or equivalently The k=2 and q= 2 spherical tensor operator Xk ðkÞ y ðkÞ ðkÞ 1,38,91,94,96,122,132,133,143 D a; b; c T D a; b; c T D a; b; c ; is ð Þ q ½ð Þ ¼ q0 q0qð Þ X q0¼k 1 2 T ¼ 1q 1q 2 2 T T ¼ T T ¼ I : D a; b; c Rfixed a; b; c : 22 1 2 1q1 1q2 11 11 ð Þ¼ A ð Þ q ;q 2 1 2 (344) (340) Equation (343) is the generalization of our left-handed The most important property of irreducible tensor opera- active rotation of vector operator (Eq.(278)). 34 tors is given by the Wigner-Eckart theorem. Irreducible Varshalovich et al.55 provide the following definition of spherical tensor operators of rank k and order q may be irreducible tensor operators in Chapter 3 of their book: fi 134 de ned via the Wigner-Eckart theorem. Under rotations of the coordinate system described by Irreducible spherical tensor operators of rank k and Euler angles, the components of irreducible tensors fi order q can be de ned either by commutation relations of undergo linear transformations. The coefficients of such Tkq with the angular momentum operators J and Jz or by transformation are the Wigner active rotation matrix. their transformation properties under rotation. These two definitions are entirely equivalent.22,29,36,57,68,70,128 The 0 a; b; c a; b; c 1 TJM0 ðX Þ¼Dð ÞTJM0 ðXÞ½Dð Þ commutation relations7,22,34,39,63,71,122,144 correspond to the XJ (345) definition of spherical tensor operator given by Racah:145 ðJÞ a; b; c ; ¼ TJMðXÞDMM0 ð Þ h i pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi M¼J ; fixed J Tkq ¼ Tkq1 ðk qÞðk q þ 1Þ Dða; b; cÞ¼R ða; b; cÞ: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ¼ T kðk þ 1Þqðq 1Þ; (341) In contrast to most definitions of spherical tensor operators, h kq1 i the coordinates X and X0 are provided in Eq. (345). J ; T ¼ qT : z kq kq If contravariant spherical tensor components TðkÞq are 90 Below, we discuss their transformation properties under used, they transform according to active and passive rotations. hi Xk ðkÞq y ðkÞq0 ðkÞ : DðRÞT ½DðRÞ ¼ T Dq0qðRÞ (346) 0 11.2 | Active rotation q ¼k Most definitions (Messiah,34 Baym,52 Rose,36 Brink and We gather four typical definitions about active rotation of Satchler,9 Varshalovich et al.55) provide the following relation: irreducible spherical tensor operators available in the litera- 34 ÂÃ ture. Messiah provides the following definition of irre- 1 Rfixedða; b; cÞT ðrÞ Rfixedða; b; cÞ ducible tensor operator in Appendix C of his book: The A X Ik A ; ðkÞ ðI AÞ a; b; c : (347) 2k + 1 operators Tq ( q ¼k; ...; þk ) are by definition ¼ TIk0 ðrÞDk0k ð Þ the standard components of an irreducible tensor operator k0 of order k, TðkÞ, if they transform under rotation according We need more details to distinguish one and two coor- to the formula dinate systems. In one coordinate system, we have MAN | 59 of 64

ÂÃ 1 1 65 ðÞT000 ðrÞ ¼ Rfixedða; b; cÞT ðrÞ Rfixedða; b; cÞ adopted by Blum is that of Landau and Lifschitz. The Ik A X Ik A relation between the notation of Blum and that of Zare38 is 2 5 ðI;AÞ a; b; c : hi ¼ TIkðr1Þ ¼ TIk0 ðrÞD 0 ð Þ (348) k k ðKÞ ðKÞ 0 c; b; a a; b; c : k DqQ ð Þ¼ DQq ð Þ (352) ZARE In two coordinate systems or canonical transformation, we That is, we can use Zare0s equations by interchanging q have and Q, a and c, and taking the complex conjugate. The ÂÃ 000 1 fixed a; b; c fixed a; b; c 1 xyz coordinate system (a rotated coordinate system) corre- ðÞT IkðrÞ ¼ RA ð ÞTIkðrÞ RA ð Þ X sponds to our (O, x000, y000, z000), whereas the XYZ coordi- 4 0 5 ðI;AÞ a; b; c : ¼ TIkðr Þ ¼ TIk0 ðrÞDk0k ð Þ (349) nate system (a fixed coordinate system) corresponds to our 0 k (O, x, y, z). Our coordinate systems are defined in The only differences between one and two coordinate Figure 11. 2 4 systems are the equalities ¼ and ¼. In one coordinate sys- We add more relations for right-handed passive rotation for spherical tensor component operators, which corre- tem, TIkðr1Þ is the spherical tensor operator component before the active rotation. In two coordinate systems, sponds to right-handed rotation of the coordinate system or 0 left-handed active rotation of spherical tensor component TIkðr Þ is the spherical tensor operator component in its body-attached coordinate system after the active rotation. operators: 2 4 Without the equalities ¼ and ¼, the definitions for one and ðÞT000 ðr000Þ (353) two coordinate systems become identical (Eq. (347)). Ik ÂÃ 1 y ¼ Rfixed c; b; a T r000 Rfixed c; b; a (354) 11.3 | Passive rotation P ð Þ Ikð Þ P ð Þ

0 70 80 Edmonds third edition (1974) book, Dawson, and Sil- 2 ¼ T ðrÞ (355) ver 58 in Chapter 5 of his book provide the following defi- Ik nition of spherical (irreducible) tensor operators: A X 5 000 ðI;PÞ c; b; a ; spherical (irreducible) tensor operator T(k) is a set of 2k + ¼ TIk0 ðr ÞDk0k ð Þ (356) 0 1 operators TðkqÞ ( q ¼k; k þ 1; ...; k 1; k ) which k transform under rotations of coordinate system like the r000 is in the final or rotated coordinate system and r is in components of the spherical tensor namely as the initial or fixed coordinate system. fi Xþk For passive rotation, most de nitions provide the fol- c; b; a c; b; a 1 0 ðkÞ c; b; a ; 131 Dð ÞTðkqÞ½Dð Þ ¼ TðkqÞDq0qð Þ lowing relation: q0¼k X c; b; a 000 c; b; a 1 000 ðI;PÞ c; b; a ; (350) Dð ÞTIkðr Þ½Dð Þ ¼ TIk0 ðr ÞDk0k ð Þ ic ib ia k0 Dðc; b; aÞ¼exp L exp L exp L h z h y h z (357) fixed c;b;a : ¼ RP ð Þ relating Eqs. (354) and (356). In contrast, Blum0sdefinition Equation (350) is a generalization definition of our right- in Eq. (351) relates Eqs. (355) and (356). handed passive rotation of vector operator defined in Eq. (318) and is the main definition found in the literature. 11.4 | Application Blum97 in Page 120 of his book provides the following definition: Operators which transform under rotations We study the off-resonance excitation of an isolated spin according to the following relation, I = 1 system by an x-pulse during a duration tp using Wigner rotation matrices. An isolated spin I = 1 system is XþK considered here because we want to use Dð1;AÞ and Dð1;PÞ. 0 0 ðKÞ c; b; a ; TðJ JÞKQ ¼ TðJ JÞKqDqQ ð Þ (351) In fact, the results are identical to those of an isolated spin q¼K I = 1/2 system not submitted to any internal interaction. In are called spherical (irreducible) tensor operators of rank K other words, the quadrupole interaction is neglected during and components Q. Equation (351) expresses the operators the excitation of the spin I = 1 system by the x-pulse. The 0 fi fi TðJ JÞKQ de ned in the XYZ coordinate system, in terms effective magnetic eld that excites the spin system is 0 fi 0 of the operators TðJ JÞKq de ned in the xyz coordinate sys- along the x -axis in the tilted coordinate system. There are tem. The convention for the DðKÞðc; b; aÞ matrices two coordinate systems as shown Figure 39: the rotating 60 of 64 | MAN

hino X p 1 0 ð1;PÞ ; h ; : T1kðr 1Þ¼ T1k0 ðr1Þ D 0 0 (360) 2 k0k k0

11.4.3 | Nutation of the magnetization vector about Beff In the tilted coordinate system, the rotation of the magnetization vector is about x0-axis (Figure 39B). As Wigner fi rotation matrices are de ned for rotation about the yOBS- axis, we must first rotate our spin system and the NMR 0 c p spectrometer about the z -axis by angle ¼ 2. Then we apply the irradiation to rotate the magnetization vector about B along y -axis by angle b ¼x t ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffieff OBS e p c 2 D 2 0tp B1 þð B0Þ . Finally, we rotate the spin system 0 a p and the spectrometer about the z -axis by angle ¼2. 2 5 Equalities ¼ and ¼ in Eq. (348) allow us to write

X hip p 0 ð1;AÞ ; x ; 0 : T1m0 ðr Þ D etp ¼ T1mðr 1Þ m0m FIGURE 39 X-pulse irradiation of a spin system off-resonance by m0 2 2 D B0 in NMR rotating coordinate system (O, xOBS, yOBS, zOBS) (A) and (361) 0 0 in tilted coordinate system (O, x , yOBS, z ) (B). The magnetization 0 vector M0 is along zOBS-axis The position vector after excitation r is in the tilted coordinate system. Conversely coordinate system (O, xOBS, yOBS, zOBS) and the tilted coor- hi 0 0 X p p 1 dinate system (O, x , yOBS, z ). 0 0 ð1;AÞ ; x ; : T1mðr Þ¼ T1m0 ðr 1Þ D etp m0m m0 2 2 11.4.1 | Initial state (362)

The nuclear magnetization vector is along the zOBS-axis of the rotating coordinate system. 11.4.4 | Passage from the tilted coordinate system to the rotating coordinate system T ðr Þ¼I ðÞ010: (358) 1k 1 0 As NMR signal is detected in the rotating coordinate sys-

The position vector before excitation r1 is in the rotat- tem, we return back to the rotating coordinate system from ing coordinate system. the tilted coordinate system. This is a passive rotation of b p h angle ¼ 2 about yOBS-axis. 11.4.2 | Passage from rotating coordinate X hip system to tilted coordinate system ð1;PÞ ; h; 0 : T1n0 ðrÞ D 0 0 ¼ T1nðr Þ (363) 2 n0n Figure 39A shows that the passage from the rotating coor- n0 dinate system to the tilted coordinate system is a passive The position vector after excitation r is in the rotating b p h rotation about yOBS-axis of angle ¼ 2 . Equalities coordinate system. Conversely, 2 5 ¼ in Eq. (355) and ¼ in Eq. (356) allow us to write X hinop 0 ð1;PÞ X hi T 0 ðr Þ D 0; h ; 0 ¼ T ðr Þ: ; p 1 1k 1 0 1k 1 0 ð1 PÞ ; h; : 0 2 k k T1nðrÞ¼ T1n0 ðr Þ D 0 0 (364) k n0n n0 2 (359) Putting Eqs. (364), (362), (360), and (358) together The position vector before excitation r0 is in the tilted 1 yields coordinate system. Conversely, MAN | 61 of 64 hino XXX p 1 p1ffiffi Ch Sh p1ffiffi Ch ð1;PÞ ðÞ¼T11 T10 T11 I0 2 2 T ðrÞ¼I0ðÞ010 D 0; h ; 0 1n 0 hi 2 k m0 p p 1 (370) n0 m0 k0 ð1;AÞ D ; xetp; (365) hi2 2 ; p 1 Dð1 PÞ 0; h; 0 2 hihi p p 1 p 1 ð1;AÞ ; x ; ð1;PÞ ; h; : D etp D 0 0 p1ffiffi Ch piffiffi ShSx t ShCx t p1ffiffi Ch piffiffi ShSx t 2 2 m0n0 2 n0n ¼ I0 2 2 e p e p 2 2 e p hi ; p 1 In matrix form, Eq. (365) becomes Dð1 PÞ 0; h; 0 2 hino p 1 ð1;PÞ ; h ; 1 i ðÞ¼T11 T10 T11 I0ðÞ010 D 0 0 ffiffiffi h h x ffiffiffi h x 2h 2h x 2 ¼ I0 p C S ðC etp 1Þp S S etp C þ S C etp 2 2 (366) 1 i pffiffiffi ChShðCxetp 1Þpffiffiffi ShSxetp : 2 2 hihi ; p p 1 ; p 1 Dð1 AÞ ; x t ; Dð1 PÞ 0; h; 0 : The three Cartesian components are 2 e p 2 2 These Wigner rotation matrices are T11 T11 hino no Ix ¼ pffiffiffi ¼ I0 cos h sin hð1 cos xetpÞ; (371) ; p 1 ; p Dð1 PÞ 0; h ; 0 ¼ Dð1 AÞ 0; h ; 0 2 2 2 T11 þffiffiffiT11 h x ; (367) Iy ¼ p ¼ I0 sin sin etp i 2 2 2 Iz ¼ T10 ¼ I0ðcos h þ sin h cos xetpÞ: 0 1 1 ð1 þ ShÞ p1ffiffi Ch 1 ð1 ShÞ We do not comment on the spin operator components B 2 2 2 C ¼ @ p1ffiffi Ch Sh p1ffiffi Ch A: in Eq. (371). They have been discussed in detail else- 2 2 3 1 ð1 ShÞp1ffiffi Ch 1 ð1 þ ShÞ where. Cartesian rotation matrices and Wigner rotation 2 2 2 matrices allow us to determine the same results. Short notations with C=cos and S=sin are used. hi p p 1 p p ð1;AÞ ð1;PÞ D ; xetp; ¼ D ; xetp; 12 | CONCLUSION 2 2 2 2 Throughout the article, we have explored the equivalence 0 p 1 1 x p1ffiffi x i2 1 ip 2 ð1 þ C etpÞS etpe ð1 CxetpÞe between the active rotation of a physical quantity in one 2 2 p B 1ffiffi ip 1ffiffi i A @ p Sx t e 2 Cx t p Sxetpe 2 direction and the rotation of the coordinate system in the ¼ 2 e p e p 2 p p 1 1 x i p1ffiffi x i2 x opposite direction in a progressive way. We use both the ð1 C etpÞe S etpe 2 ð1 þ C etpÞ 2 2 left- and right-handed rotation conventions whereas the right-handed rotation convention is mainly used in the liter- 0 1 1 ð1 þ Cx t Þpiffiffi Sx t 1 ð1 Cx t Þ ature. Then we extend this rule to Euler angles. The article 2 e p 2 e p 2 e p B iffiffi iffiffi C presents redundant mathematical demonstrations between ¼ @ p Sxetp Cxetp p Sxetp A: 2 2 active and passive rotations, but they clarify the meanings 1 ð1 Cx t Þpiffiffi Sx t 1 ð1 þ Cx t Þ 2 e p 2 e p 2 e p of some important expressions not well developed in the (368) literature. Now we have at our disposal the mathematical Finally, tools for describing the dynamics of spin with the density 0 1 matrix operator. 1 h p1ffiffi h 1 h hi2 ð1 þ S Þ C 2 ð1 þ S Þ p 1 B 2 C Dð0; h; 0 ¼ @ p1ffiffi Ch Sh p1ffiffi Ch A: 2 2 2 REFERENCES 1 ð1 ShÞ p1ffiffi Ch 1 ð1 þ ShÞ 2 2 2 1. Man PP. Cartesian and spherical tensors in NMR Hamiltonian. (369) Concepts Magn Reson. 2013;42A:197–244. Including these Wigner rotation matrices into Eq. (366) 2. Dirac PAM. A new notation for quantum mechanics. Math Proc – yields Camb Philos Soc. 1939;35:416 418. 62 of 64 | MAN

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