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Theory of Angular Momentum Rotations About Different Axes Fail to Commute 153 CHAPTER 3 3.1. Rotations and Angular Momentum Commutation Relations followed by a 90° rotation about the z-axis. The net results are different, as we can see from Figure 3.1. Our first basic task is to work out quantitatively the manner in which Theory of Angular Momentum rotations about different axes fail to commute. To this end, we first recall how to represent rotations in three dimensions by 3 X 3 real, orthogonal matrices. Consider a vector V with components Vx ' VY ' and When we rotate, the three components become some other set of numbers, V;, V;, and Vz'. The old and new components are related via a 3 X 3 orthogonal matrix R: V'X Vx V'y R Vy I, (3.1.1a) V'z RRT = RTR =1, (3.1.1b) where the superscript T stands for a transpose of a matrix. It is a property of orthogonal matrices that 2 2 /2 /2 /2 vVx + V2y + Vz =IVVx + Vy + Vz (3.1.2) is automatically satisfied. This chapter is concerned with a systematic treatment of angular momen- tum and related topics. The importance of angular momentum in modern z physics can hardly be overemphasized. A thorough understanding of angu- Z z lar momentum is essential in molecular, atomic, and nuclear spectroscopy; I I angular-momentum considerations play an important role in scattering and I I collision problems as well as in bound-state problems. Furthermore, angu- I lar-momentum concepts have important generalizations-isospin in nuclear physics, SU(3), SU(2)® U(l) in particle physics, and so forth. /' 3.1. ROTATIONS AND ANGULAR MOMENTUM x x x z COMMUTATION RELATIONS Z z I I I I Finite Versus Infinitesimal Rotations I I I I I We recall from elementary physics that rotations about the same axis I I commute, whereas rotations about different axes do not. For instance, a 30° Rz(7T/2) rotation about the z-axis followed by a 60° rotation about the same z-axis is 7 / ------! a22 z obviously equivalent to a 60° rotation followed by a 30° rotation, both "til' P" about the same axis. However, let us consider a 90° rotation about the x x x z-axis, denoted by R z( '7T/2), followed by a 90° rotation about the x-axis, denoted by R x< '7T/2); compare this with a 90° rotation about the x-axis FIGURE 3.1. Example to illustrate the noncommutativity of finite rotations. 154 Theory of Angular NlOmentUm II 1{"lations and Angular Momentum Commutation Relations 155 To be definite, we consider a rotation about the z-axis by angle fl1llll iOIl. Elementary matrix manipulations lead to The convention we follow throughout this book is that a rotation opeptifl": affects a physical system itself, as in Figure 3.1, while the coordinate e2 1-- 0 e remain unchanged. The angle </> is taken to be positive when the rotation in 2 question is counterclockwise in the xy-plane, as viewed from the positi RxCe)Ry(e) = I e 2 (3.1.6a) e2 1-- -e z-side. If we associate a right-handed screw with such a rotation, a 2 </> rotation around the z-axis means that the screw is advancing -e e 1- e2 positive z-direction. With this convention, we easily verify that cos</> -sin</> e2 R z (</»= cos</> 1-- e 2 e ( o 2 Ry(e)Rx(e) = I e2 (3.1.6b) Had we adopted a different convention, in which a physical system reo 0 1-- -e mained fixed but the coordinate axes rotated, this same matrix with a 2 positive </> would have represented a clockwise rotation of the x- and y-axes, -e e 1- e2 when viewed from the positive z-side. It is obviously important not to mix hOIll (3.1.6a) and (3.1.6b) we have the first important result: Infinitesimal the two conventions! Some authors distinguish the two approaches by using 2 "active rotations" for physical systems rotated and "passive rotations" for IlIlations about different axes do commute if terms of order e and higher 11Il' ignored.* The second and even more important result concerns the the coordinate axes rotated. 1I111111lCr in which rotations about different axes fail to commute when terms We are particularly interested in an infinitesimal form of R z : III' urdcr e2 are kept: e2 1-- -e 0 2 2 -e Rz(e) = I e2 R.(,)R,(')- R,(,)R,(') ( o e 1-- 0 2 o 0 0 1 2 = R z ( e )-I, (3.1.7) where terms of order e3 and higher are ignored. Likewise, we have where all terms of order higher than e2 have been ignored throughout this 1 0 0 dnivation. We also have e2 0 1-- -e 1 = Rany(O) (3.1.8) RxCe) = I 2 e2 0 e 1-- wherc any stands for any rotation axis. Thus the final result can be written 2 liS 2 and = (3.1.9) RxCe)Ry(e)-Ry(e)RxCe). Rz(e )-Rany(O). e2 1-- 0 e Ihis is an example of the commutation relations between rotation oper- 2 ations about different axes, which we will use later in deducing the angular- 0 1 0 (3.1.5b) Ry(e) = I I, IIHllI1Cntum commutation relations in quantum mechanics. e2 -e 0 1-- 2 +Actually thcrc is a familiar example of this in elementary mechanics. The angular velocity which may be read from (3.1.4) by cyclic permutations of x, y, z-that is, \"','Ior w that characterizes an infinitesimal change in rotation angle during an infinitesimal x y, y z, z x. Compare now the effect of a y-axis rotation followed IIIIll" interval follows the usual rule of vector addition, including commutativity of vector by an x-axis rotation with that of an x-axis rotation followed by a y-axis "ddilion. llowcvcr, we cannot ascribe a vectorial property to ajinite angular change. 156 Theory of Angular Momentum H. Rotations and Angular Momentum Commutation Relations 157 Infinitesimal Rotations in Quantum Mechanics (lotained by letting Jk So far we have not used quantum-mechanical concepts. The matrix G-'> - E-'> R is just a 3 X 3 orthogonal matrix acting on a vector V written in column Ii ' matrix form. We must now understand how to characterize rotations in in (3.1.11). With Jk taken to be Hermitian, the infinitesimal rotation quantum mechanics. operator is guaranteed to be unitary and reduces to the identity operator in Because rotations affect physical systems, the state ket corresponding the limit dq, -'> O. More generally, we have to a rotated system is expected to look different from the state ket corre- PIJ(ii, = i Jon) (3.1.15) sponding to the original unrotated system. Given a rotation operation R, dq,) 1- (h dq, characterized by a 3 X 3 orthogonal matrix R, we associate an operator for a rotation about the direction characterized by a unit vector it by an in the appropriate ket space such that infinitesimal angle dq,. R) (3.1 We stress that in this book we do not define the angular-momentum operator to be x X p. This is important because spin angular momentum, to where IO:)R and 10:) stand for the kets of the rotated and original system, which our general formalism also applies, has nothing to do with Xi and respectively.* Note that the 3 X 3 orthogonal matrix R acts on a column IJut in another way, in classical mechanics one can prove that the angular matrix made up of the three components of a classical vector, while the momentum defined to be x X p is the generator of a rotation; in contrast, in operator PIJ (R) acts on state vectors in ket space. The matrix representation 4l1antum mechanics we define J so that the operator for an infinitesimal of PIJ(R), which we will study in great detail in the subsequent sections, rotation takes form (3.1.15). depends on the dimension1!:!ity N of the For A finite rotation can be obtained by compounding successively _ N = 2, fo!.,..descrihing a spin t system with no other degrees of inlinitesimal rotations about the same axis. For instance, if we are interested freedom, is represented by a 2x2 matrix; for a spin 1 system, the in a finite rotation about the z-axis by angle q" we consider appropriate representation is a 3 X 3 unitary matrix, and so on. To construct the rotation operator it is again fruitful to I PlJz(q, ) = [ i ( ) ( t)r examine first its properties under an infinitesimal rotation. We can almost guess how we must proceed by analogy. In both translations and time evolution, which we studied in Sections 1.6 and 2.1, respectively, the appropriate infinitesimal operators could be written as =1- iJzq, Ii - 21i2 + ... (3.1 U. I-iGe (3.1.11) In order to obtain the angular-momentum commutation relations, we with a Hermitian operator G. Specifically, need one more concept. As we remarked earlier, for every rotation R represented by a 3 X 3 orthogonal matrix R there exists a rotation operator E -'> dx' (3.1.12) G-'>Ii' !7)(R) in the appropriate ket space. We further postulate that PIJ(R) has the group properties as R: for an infinitesimal translation by a displacement dx' in the x-direction, and Identity: R·1 R=;oPIJ(R)·l (3.1.17a) H E -'> Closure: G-'>Ii' dt RIR2 = R3 =;0 = PIJ(R]) (3.1.17b) Inverses: RR-1 l=;oPIJ(R)PlJl(R)=l for an infinitesimal time evolution with time displacement dt.
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