DOCSLIB.ORG

Spin in Quantum Mechanics 120

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Spin in Quantum Mechanics 120 5.11TheSphericalBasis...............................106 5.11.1 Transformation of a spherical vector under rotation of the coordi- natesystem ...............................107 5.11.2 Rotationofquantummechanicalstates . 109 5.12 Special Galilei Transformations: Determination of the Unitary Operator Representing the Transformation . 111 5.13 Invariance Under Special Galilei Transformations . .112 5.14 GeneralConsiderations. 115 5.14.1 ExponentialFormoftheRepresentation . 115 5.14.2 CasimirOperators . .. .. .116 5.15 ProjectionOperators . .. .. 117 6 Spin in Quantum Mechanics 120 6.1 SpinorsandTheirProperties. .120 6.1.1 Geometric Visualization of Spinors . 121 6.1.2 RotationsofSpinors ..........................122 6.2 Pauli Matrices and Their Eigenvectors . 123 6.3 Finite Rotations and Pauli Matrices . 125 6.3.1 RotationsAboutthe3-Axis . .125 6.3.2 RotationsAboutthe2-Axis . .125 1 6.3.3 Spinor Nature of Spin- 2 Rotations ..................126 1 6.3.4 General Euler-Angle Rotations for Spin- 2 ...............126 6.4 SpinSpace....................................127 iv 6.4.1 Spinor Space and Its Matrix Representation . 127 6.4.2 Angular Momentum Operators in Spinor Representation . 128 6.4.3 IncludingSpinSpace ..........................128 6.5 Single Particle States with Spin . 131 6.5.1 Coordinate-Space-Spin Representation . 131 6.5.2 Helicity Representation . 131 6.6 Isospin......................................133 6.6.1 Nucleon-Nucleon (NN) Scattering . 135 7 Combining Angular Momentum Eigenstates 137 7.1 AdditionofTwoAngularMomenta . .137 7.2 ConstructionoftheEigenstates . 141 7.3 SpecialCases ..................................143 7.3.1 Spin-OrbitInteraction . .143 1 7.3.2 Coupling of Two Spin- 2 Particles ...................145 7.4 PropertiesofClebsch-GordanCoefficients . 146 7.5 Wigner’s 3 − j Coefficients...........................149 7.6 CartesianandSphericalTensors . 152 7.7 TensorOperatorsinQuantumMechanics . 156 7.8 Wigner-EckartTheorem . .. .. .158 7.9 Applications...................................161 8 Symmetries III: Time Reversal 167 v Chapter 6 Spin in Quantum Mechanics 6.1 Spinors and Their Properties Spinors are mathematical entities, which are useful when describing half-integer spins in the context of rotations of physical systems. Definition of Spinors: A mathematical entity S is a spinor if it satisfies the requirement that it changes sign under a 2π rotation: S(θ +2π) = −S(θ) . (6.1) Part of this requirement is the assumption that S can have a sign associated with it. 1 As it turns out, spin- 2 objects (primarily electrons and subatomic particles) have wave functions that change sign, if the object is rotated through 2π and which can, therefore, be represented by spinors. 120 6.1.1 Geometric Visualization of Spinors A unit circle in the x − z plane is centered at the origin O and intercepting the positive z-axis at P . Consider a ray OC making an angle θ with the z-axis, so that θ is the usual polar angle. Now, bisect the angle so that the point S bisects PC, since OP = OC. Thus OS(θ) = cos(θ/2) ; PS(θ) = sin(θ/2) . (6.2) OS and PS are taken to be signed quantities, OS is negative for π <θ < 2π, as is PS for 2π <θ </4π. As the point C goes once around the unit circle, the point S goes from P to O. A second 1 revolution of C brings S back to P . The path of S is a circle of radius 2 . This path is traced out in a period of 4π of θ. Considering (6.2), OS and PS change sign when θ −→ θ +π. Therefore, according to our rule, the pair (OS,PS) forms a spinor associated with C. In the two-dimensional representation shown in Fig. 6.1, one may view C as marking the −→ tip of the vector OC, which has unit magnitude and θ as polar angle. Then OS and PS −→ form the components of a spinor associated with OC. 121 6.1.2 Rotations of Spinors The properties under rotations of the spinor (OS,PS) associated with C can also be worked out from Fig. 6.1. If C is rotated through an angle β from its position at θ, then one has the following transformations: θ −→ θ + β OS −→ OSβ = cos(β/2) OS − sin(β/2) PS PS −→ PSβ = sin(β/2) OS + cos(β/2) PS, (6.3) which follow from the trigonometric identities for the sum of two angles. The last two equations can be written as OS(θ + β) cos(β/2) − sin(β/2) OS(θ) = (6.4) PS(θ + β) sin(β/2) cos(β/2) PS(θ) or cos(β/2) − sin(β/2) S(θ + β) = S(θ) , (6.5) sin(β/2) cos(β/2) where S(θ) represents the spinor. In this matrix representation, it is clear that the basic spinor property, S(θ +2π)= −S(θ), is fulfilled. The matrix in (6.5) is the 2 × 2 matrix 1 that transforms any spin- 2 angular momentum eigenstate under rotation by β about the y-axis. If we want to define rotations of spinors around the z-axis, then we should require that those spinors are eigenfunctions of the angular momentum operator component for this axis, ∂S J S(θ) = −i = ±S(θ) (6.6) 3 ∂θ where we have used the definition (4.24) for a generalized angular momentum J~. The linear combination of spinors that satisfy this equation are complex, namely S(θ) = e±iθ/2 . (6.7) The matrix form of rotations around the z-axis, which also satisfy (6.6) are e+iα/2 0 S(θ + α) = S(θ) . (6.8) 0 e−iα/2 Thus the spinor S(θ) is complex valued. 122 6.2 Pauli Matrices and Their Eigenvectors Pauli matrices are here introduced through their algebraic properties. They describe 1 the simplest, non-trivial spin system, namely spin 2 . In cartesian coordinates the Pauli matrices are given by 0 1 0 −i 1 0 σ = σ = σ = (6.9) 1 1 0 2 i 0 3 0 −1 and they are collectively denoted by ~σ = (σ1, σ2, σ3). In terms of Paul matrices, the ”spin” matrices S~ are given as 1 S~ = ~σ. (6.10) 2 Properties of Pauli Matrices are: 2 σj = 1 (6.11) for j =1, 2, 3. They anticommute [σi, σk]+ = σiσk + σkσi = 0 ; k =6 i . (6.12) 1 For the quantum mechanical description of a spin- 2 particle, one needs obviously the ~ 1 representation of the angular momentum operators J with j = 2 . The most general form 1 of spin- 2 state would be given by 1 1 1 1 α | , + i + β | , − i (6.13) 2 2 2 2 or written as column vector α . (6.14) β 1 1 The two basis vectors are then | 2 , ± 2 i or 1 0 and (6.15) 0 1 Let us consider 1 S = σ , (6.16) 3 2 3 123 the component of the spin along an arbitrarily chosen 3-axis. We need to find eigenvalues and eigenvectors of σ3. The eigenvalue equation for σ3 reads a a σ + = λ + (6.17) 3 a ± a − − where λ± are the eigenvalues. The solutions for the eigenvectors are given by 1 0 χ = ; χ = (6.18) + 0 − 1 with the corresponding eigenvalues λ+ = +1 and λ− = −1. The eigenvectors χ± are orthonormal, as required for the eigenvectors of a hermitian operator. The eigenstates χ± are usually referred to as Pauli spinors and χ+ represents a ”spin-up,” χ− a ”spin-down” state. According to our general considerations in Chapter 3, we introduce the ladder operators 0 1 0 0 S ≡ and S = (6.19) + 0 0 − 1 0 and it can be easily verified that S+χ+ = | 0i S+χ− = χ+ S−χ+ = χ− S−χ− = | 0i (6.20) Furthermore 1 1 S = (S + S ) = σ 1 2 + − 2 1 1 1 S = (S − S ) = σ (6.21) 2 2i + − 2 2 which is exactly the definition given in (6.10). One could have also introduced σ± = (σ1 ± iσ2) (6.22) and then defined the ladder operators as 0 0 0 2 σ = ; σ = σ , σ = , (6.23) −1 2 0 0 3 +1 0 0 which is often referred to as ’spherical representation.’ Here we have 2 2 σ± = 0 and σ0 = 1 . (6.24) 124 From this follows that higher powers of these spherical-basis matrix elements must be 1 zero. Furthermore, the basis vectors χ± span the spin-space for spin- 2 particles. The completeness relation is given by 1 0 χ χ† + χ χ† = = 1 . (6.25) + + − − 0 1 6.3 Finite Rotations and Pauli Matrices As discussed in the previous chapter, J3 describes infinitesimal rotations and the Ji are the generators of a Lie algebra. From the generators all finite rotations can be constructed. 1 For spin- 2 states the rotation operator has the following form (cp. 5.95 g) 1 α β γ D 2 (αβγ) = e−i 2 σ3 e−i 2 σ2 e−i 2 σ3 . (6.26) Here the description of rotations via Euler angles is used. 6.3.1 Rotations About the 3-Axis Expanding the expression e−iασ3/2 and using the properties of the Pauli matrices, namely 2m 2m+1 that for n even, n =2m, σ3 = 1, and m odd, n =2m +1, σ3 = σ3, one obtains −iγσ3/2 e = cos(γ/2) 1 − i sin(γ/2) σ3 . (6.27) By explicitly inserting σ3 this leads to e−iγ/2 0 e−iγσ3/2 = . (6.28) 0 e+iγ/2 Thus, matrix elements of rotations around the 3-axis are given by −imγ e δm′m . (6.29) 6.3.2 Rotations About the 2-Axis Since σ2 is not diagonal, the rotation matrix must have a more complicated form. The rotation matrix for a rotation of a spinor was already heuristically derived in 6.1.2 and is 125 given by cos(β/2) − sin(β/2) e−iβσ2/2 = = d1/2(β) .
Load more

Recommended publications
  • Lecture Notes: Qubit Representations and Rotations
    Phys 711 Topics in Particles & Fields | Spring 2013 | Lecture 1 | v0.3 Lecture notes: Qubit representations and rotations Jeffrey Yepez Department of Physics and Astronomy University of Hawai`i at Manoa Watanabe Hall, 2505 Correa Road Honolulu, Hawai`i 96822 E-mail: yepez@hawaii.edu www.phys.hawaii.edu/∼yepez (Dated: January 9, 2013) Contents mathematical object (an abstraction of a two-state quan- tum object) with a \one" state and a \zero" state: I. What is a qubit? 1 1 0 II. Time-dependent qubits states 2 jqi = αj0i + βj1i = α + β ; (1) 0 1 III. Qubit representations 2 A. Hilbert space representation 2 where α and β are complex numbers. These complex B. SU(2) and O(3) representations 2 numbers are called amplitudes. The basis states are or- IV. Rotation by similarity transformation 3 thonormal V. Rotation transformation in exponential form 5 h0j0i = h1j1i = 1 (2a) VI. Composition of qubit rotations 7 h0j1i = h1j0i = 0: (2b) A. Special case of equal angles 7 In general, the qubit jqi in (1) is said to be in a superpo- VII. Example composite rotation 7 sition state of the two logical basis states j0i and j1i. If References 9 α and β are complex, it would seem that a qubit should have four free real-valued parameters (two magnitudes and two phases): I. WHAT IS A QUBIT? iθ0 α φ0 e jqi = = iθ1 : (3) Let us begin by introducing some notation: β φ1 e 1 state (called \minus" on the Bloch sphere) Yet, for a qubit to contain only one classical bit of infor- 0 mation, the qubit need only be unimodular (normalized j1i = the alternate symbol is |−i 1 to unity) α∗α + β∗β = 1: (4) 0 state (called \plus" on the Bloch sphere) 1 Hence it lives on the complex unit circle, depicted on the j0i = the alternate symbol is j+i: 0 top of Figure 1.
    [Show full text]
  • Theory of Angular Momentum and Spin
    Chapter 5 Theory of Angular Momentum and Spin Rotational symmetry transformations, the group SO(3) of the associated rotation matrices and the 1 corresponding transformation matrices of spin{ 2 states forming the group SU(2) occupy a very important position in physics. The reason is that these transformations and groups are closely tied to the properties of elementary particles, the building blocks of matter, but also to the properties of composite systems. Examples of the latter with particularly simple transformation properties are closed shell atoms, e.g., helium, neon, argon, the magic number nuclei like carbon, or the proton and the neutron made up of three quarks, all composite systems which appear spherical as far as their charge distribution is concerned. In this section we want to investigate how elementary and composite systems are described. To develop a systematic description of rotational properties of composite quantum systems the consideration of rotational transformations is the best starting point. As an illustration we will consider first rotational transformations acting on vectors ~r in 3-dimensional space, i.e., ~r R3, 2 we will then consider transformations of wavefunctions (~r) of single particles in R3, and finally N transformations of products of wavefunctions like j(~rj) which represent a system of N (spin- Qj=1 zero) particles in R3. We will also review below the well-known fact that spin states under rotations behave essentially identical to angular momentum states, i.e., we will find that the algebraic properties of operators governing spatial and spin rotation are identical and that the results derived for products of angular momentum states can be applied to products of spin states or a combination of angular momentum and spin states.
    [Show full text]
  • Rotation Matrix - Wikipedia, the Free Encyclopedia Page 1 of 22
    Rotation matrix - Wikipedia, the free encyclopedia Page 1 of 22 Rotation matrix From Wikipedia, the free encyclopedia In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space. For example the matrix rotates points in the xy -Cartesian plane counterclockwise through an angle θ about the origin of the Cartesian coordinate system. To perform the rotation, the position of each point must be represented by a column vector v, containing the coordinates of the point. A rotated vector is obtained by using the matrix multiplication Rv (see below for details). In two and three dimensions, rotation matrices are among the simplest algebraic descriptions of rotations, and are used extensively for computations in geometry, physics, and computer graphics. Though most applications involve rotations in two or three dimensions, rotation matrices can be defined for n-dimensional space. Rotation matrices are always square, with real entries. Algebraically, a rotation matrix in n-dimensions is a n × n special orthogonal matrix, i.e. an orthogonal matrix whose determinant is 1: . The set of all rotation matrices forms a group, known as the rotation group or the special orthogonal group. It is a subset of the orthogonal group, which includes reflections and consists of all orthogonal matrices with determinant 1 or -1, and of the special linear group, which includes all volume-preserving transformations and consists of matrices with determinant 1. Contents 1 Rotations in two dimensions 1.1 Non-standard orientation
    [Show full text]
  • Unipotent Flows and Applications
    Clay Mathematics Proceedings Volume 10, 2010 Unipotent Flows and Applications Alex Eskin 1. General introduction 1.1. Values of indefinite quadratic forms at integral points. The Op- penheim Conjecture. Let X Q(x1; : : : ; xn) = aijxixj 1≤i≤j≤n be a quadratic form in n variables. We always assume that Q is indefinite so that (so that there exists p with 1 ≤ p < n so that after a linear change of variables, Q can be expresses as: Xp Xn ∗ 2 − 2 Qp(y1; : : : ; yn) = yi yi i=1 i=p+1 We should think of the coefficients aij of Q as real numbers (not necessarily rational or integer). One can still ask what will happen if one substitutes integers for the xi. It is easy to see that if Q is a multiple of a form with rational coefficients, then the set of values Q(Zn) is a discrete subset of R. Much deeper is the following conjecture: Conjecture 1.1 (Oppenheim, 1929). Suppose Q is not proportional to a ra- tional form and n ≥ 5. Then Q(Zn) is dense in the real line. This conjecture was extended by Davenport to n ≥ 3. Theorem 1.2 (Margulis, 1986). The Oppenheim Conjecture is true as long as n ≥ 3. Thus, if n ≥ 3 and Q is not proportional to a rational form, then Q(Zn) is dense in R. This theorem is a triumph of ergodic theory. Before Margulis, the Oppenheim Conjecture was attacked by analytic number theory methods. (In particular it was known for n ≥ 21, and for diagonal forms with n ≥ 5).
    [Show full text]
  • Linear Operators Hsiu-Hau Lin Hsiuhau@Phys.Nthu.Edu.Tw (Mar 25, 2010)
    Linear Operators Hsiu-Hau Lin hsiuhau@phys.nthu.edu.tw (Mar 25, 2010) The notes cover linear operators and discuss linear independence of func- tions (Boas 3.7-3.8). • Linear operators An operator maps one thing into another. For instance, the ordinary func- tions are operators mapping numbers to numbers. A linear operator satisfies the properties, O(A + B) = O(A) + O(B);O(kA) = kO(A); (1) where k is a number. As we learned before, a matrix maps one vector into another. One also notices that M(r1 + r2) = Mr1 + Mr2;M(kr) = kMr: Thus, matrices are linear operators. • Orthogonal matrix The length of a vector remains invariant under rotations, ! ! x0 x x0 y0 = x y M T M : y0 y The constraint can be elegantly written down as a matrix equation, M T M = MM T = 1: (2) In other words, M T = M −1. For matrices satisfy the above constraint, they are called orthogonal matrices. Note that, for orthogonal matrices, computing inverse is as simple as taking transpose { an extremely helpful property for calculations. From the product theorem for the determinant, we immediately come to the conclusion det M = ±1. In two dimensions, any 2 × 2 orthogonal matrix with determinant 1 corresponds to a rotation, while any 2 × 2 orthogonal HedgeHog's notes (March 24, 2010) 2 matrix with determinant −1 corresponds to a reflection about a line. Let's come back to our good old friend { the rotation matrix, cos θ − sin θ ! cos θ sin θ ! R(θ) = ;RT = : (3) sin θ cos θ − sin θ cos θ It is straightforward to check that RT R = RRT = 1.
    [Show full text]
  • Introduction to Vector and Tensor Analysis
    Introduction to vector and tensor analysis Jesper Ferkinghoff-Borg September 6, 2007 Contents 1 Physical space 3 1.1 Coordinate systems . 3 1.2 Distances . 3 1.3 Symmetries . 4 2 Scalars and vectors 5 2.1 Definitions . 5 2.2 Basic vector algebra . 5 2.2.1 Scalar product . 6 2.2.2 Cross product . 7 2.3 Coordinate systems and bases . 7 2.3.1 Components and notation . 9 2.3.2 Triplet algebra . 10 2.4 Orthonormal bases . 11 2.4.1 Scalar product . 11 2.4.2 Cross product . 12 2.5 Ordinary derivatives and integrals of vectors . 13 2.5.1 Derivatives . 14 2.5.2 Integrals . 14 2.6 Fields . 15 2.6.1 Definition . 15 2.6.2 Partial derivatives . 15 2.6.3 Differentials . 16 2.6.4 Gradient, divergence and curl . 17 2.6.5 Line, surface and volume integrals . 20 2.6.6 Integral theorems . 24 2.7 Curvilinear coordinates . 26 2.7.1 Cylindrical coordinates . 27 2.7.2 Spherical coordinates . 28 3 Tensors 30 3.1 Definition . 30 3.2 Outer product . 31 3.3 Basic tensor algebra . 31 1 3.3.1 Transposed tensors . 32 3.3.2 Contraction . 33 3.3.3 Special tensors . 33 3.4 Tensor components in orthonormal bases . 34 3.4.1 Matrix algebra . 35 3.4.2 Two-point components . 38 3.5 Tensor fields . 38 3.5.1 Gradient, divergence and curl . 39 3.5.2 Integral theorems . 40 4 Tensor calculus 42 4.1 Tensor notation and Einsteins summation rule .
    [Show full text]
  • Beyond Scalar, Vector and Tensor Harmonics in Maximally Symmetric Three-Dimensional Spaces
    Beyond scalar, vector and tensor harmonics in maximally symmetric three-dimensional spaces Cyril Pitrou1, ∗ and Thiago S. Pereira2, y 1Institut d'Astrophysique de Paris, CNRS UMR 7095, 98 bis Bd Arago, 75014 Paris, France. 2Departamento de F´ısica, Universidade Estadual de Londrina, Rod. Celso Garcia Cid, Km 380, 86057-970, Londrina, Paran´a,Brazil. (Dated: October 1, 2019) We present a comprehensive construction of scalar, vector and tensor harmonics on max- imally symmetric three-dimensional spaces. Our formalism relies on the introduction of spin-weighted spherical harmonics and a generalized helicity basis which, together, are ideal tools to decompose harmonics into their radial and angular dependencies. We provide a thorough and self-contained set of expressions and relations for these harmonics. Being gen- eral, our formalism also allows to build harmonics of higher tensor type by recursion among radial functions, and we collect the complete set of recursive relations which can be used. While the formalism is readily adapted to computation of CMB transfer functions, we also collect explicit forms of the radial harmonics which are needed for other cosmological ob- servables. Finally, we show that in curved spaces, normal modes cannot be factorized into a local angular dependence and a unit norm function encoding the orbital dependence of the harmonics, contrary to previous statements in the literature. 1. INTRODUCTION ables. Indeed, on cosmological scales, where lin- ear perturbation theory successfully accounts for the formation of structures, perturbation modes, Tensor harmonics are ubiquitous tools in that is the components in an expansion on tensor gravitational theories. Their applicability reach harmonics, evolve independently from one an- a wide spectrum of topics including black-hole other.
    [Show full text]
  • Inclusion Relations Between Orthostochastic Matrices And
    Linear and Multilinear Algebra, 1987, Vol. 21, pp. 253-259 Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 Photocopying permitted by license only 1987 Gordon and Breach Science Publishers, S.A. Printed in the United States of America Inclusion Relations Between Orthostochastic Matrices and Products of Pinchinq-- Matrices YIU-TUNG POON Iowa State University, Ames, lo wa 5007 7 and NAM-KIU TSING CSty Poiytechnic of HWI~K~tig, ;;e,~g ,?my"; 2nd A~~KI:Lkwe~ity, .Auhrn; Al3b3m-l36849 Let C be thc set of ail n x n urthusiuihaatic natiiccs and ;"', the se! of finite prnd~uctpof n x n pinching matrices. Both sets are subsets of a,, the set of all n x n doubly stochastic matrices. We study the inclusion relations between C, and 9'" and in particular we show that.'P,cC,b~t-~p,t~forn>J,andthatC~$~~fori123. 1. INTRODUCTION An n x n matrix is said to be doubly stochastic (d.s.) if it has nonnegative entries and ail row sums and column sums are 1. An n x n d.s. matrix S = (sij)is said to be orthostochastic (0.s.) if there is a unitary matrix U = (uij) such that 2 sij = luijl , i,j = 1, . ,n. Following [6], we write S = IUI2. The set of all n x n d.s. (or 0s.) matrices will be denoted by R, (or (I,, respectively). If P E R, can be expressed as 1-t (1 2 54 Y. T POON AND N K. TSING Downloaded By: [Li, Chi-Kwong] At: 00:16 20 March 2009 for some permutation matrix Q and 0 < t < I, then we call P a pinching matrix.
    [Show full text]
  • The Grassmannian Sigma Model in SU(2) Yang-Mills Theory
    The Grassmannian σ-model in SU(2) Yang-Mills Theory David Marsh Undergraduate Thesis Supervisor: Antti Niemi Department of Theoretical Physics Uppsala University Contents 1 Introduction 4 1.1 Introduction ............................ 4 2 Some Quantum Field Theory 7 2.1 Gauge Theories .......................... 7 2.1.1 Basic Yang-Mills Theory ................. 8 2.1.2 Faddeev, Popov and Ghosts ............... 10 2.1.3 Geometric Digression ................... 14 2.2 The nonlinear sigma models .................. 15 2.2.1 The Linear Sigma Model ................. 15 2.2.2 The Historical Example ................. 16 2.2.3 Geometric Treatment ................... 17 3 Basic properties of the Grassmannian 19 3.1 Standard Local Coordinates ................... 19 3.2 Plücker coordinates ........................ 21 3.3 The Grassmannian as a homogeneous space .......... 23 3.4 The complexied four-vector ................... 25 4 The Grassmannian σ Model in SU(2) Yang-Mills Theory 30 4.1 The SU(2) Yang-Mills Theory in spin-charge separated variables 30 4.1.1 Variables .......................... 30 4.1.2 The Physics of Spin-Charge Separation ......... 34 4.2 Appearance of the ostensible model ............... 36 4.3 The Grassmannian Sigma Model ................ 38 4.3.1 Gauge Formalism ..................... 38 4.3.2 Projector Formalism ................... 39 4.3.3 Induced Metric Formalism ................ 40 4.4 Embedding Equations ...................... 43 4.5 Hodge Star Decomposition .................... 48 2 4.6 Some Complex Dierential Geometry .............. 50 4.7 Quaternionic Decomposition ................... 54 4.7.1 Verication ........................ 56 5 The Interaction 58 5.1 The Interaction Terms ...................... 58 5.1.1 Limits ........................... 59 5.2 Holomorphic forms and Dolbeault Cohomology ........ 60 5.3 The Form of the interaction ..................
    [Show full text]
  • 2 Rotation Kinematics
    2 Rotation Kinematics Consider a rigid body with a fixed point. Rotation about the fixed point is the only possible motion of the body. We represent the rigid body by a body coordinate frame B, that rotates in another coordinate frame G, as is shown in Figure 2.1. We develop a rotation calculus based on trans- formation matrices to determine the orientation of B in G, and relate the coordinates of a body point P in both frames. Z ZP z B zP G P r yP y YP X x P P Y X x FIGURE 2.1. A rotated body frame B in a fixed global frame G,aboutafixed point at O. 2.1 Rotation About Global Cartesian Axes Consider a rigid body B with a local coordinate frame Oxyz that is origi- nally coincident with a global coordinate frame OXY Z.PointO of the body B is fixed to the ground G and is the origin of both coordinate frames. If the rigid body B rotates α degrees about the Z-axis of the global coordi- nate frame, then coordinates of any point P of the rigid body in the local and global coordinate frames are related by the following equation G B r = QZ,α r (2.1) R.N. Jazar, Theory of Applied Robotics, 2nd ed., DOI 10.1007/978-1-4419-1750-8_2, © Springer Science+Business Media, LLC 2010 34 2. Rotation Kinematics where, X x Gr = Y Br= y (2.2) ⎡ Z ⎤ ⎡ z ⎤ ⎣ ⎦ ⎣ ⎦ and cos α sin α 0 − QZ,α = sin α cos α 0 .
    [Show full text]
  • Rotation Averaging
    Noname manuscript No. (will be inserted by the editor) Rotation Averaging Richard Hartley, Jochen Trumpf, Yuchao Dai, Hongdong Li Received: date / Accepted: date Abstract This paper is conceived as a tutorial on rota- Single rotation averaging. In the single rotation averaging tion averaging, summarizing the research that has been car- problem, several estimates are obtained of a single rotation, ried out in this area; it discusses methods for single-view which are then averaged to give the best estimate. This may and multiple-view rotation averaging, as well as providing be thought of as finding a mean of several points Ri in the proofs of convergence and convexity in many cases. How- rotation space SO(3) (the group of all 3-dimensional rota- ever, at the same time it contains many new results, which tions) and is an instance of finding a mean in a manifold. were developed to fill gaps in knowledge, answering funda- Given an exponent p ≥ 1 and a set of n ≥ 1 rotations p mental questions such as radius of convergence of the al- fR1;:::; Rng ⊂ SO(3) we wish to find the L -mean rota- gorithms, and existence of local minima. These matters, or tion with respect to d which is defined as even proofs of correctness have in many cases not been con- n p X p sidered in the Computer Vision literature. d -mean(fR1;:::; Rng) = argmin d(Ri; R) : We consider three main problems: single rotation av- R2SO(3) i=1 eraging, in which a single rotation is computed starting Since SO(3) is compact, a minimum will exist as long as the from several measurements; multiple-rotation averaging, in distance function is continuous (which any sensible distance which absolute orientations are computed from several rel- function is).
    [Show full text]
  • Numerically Stable Coded Matrix Computations Via Circulant and Rotation Matrix Embeddings
    1 Numerically stable coded matrix computations via circulant and rotation matrix embeddings Aditya Ramamoorthy and Li Tang Department of Electrical and Computer Engineering Iowa State University, Ames, IA 50011 fadityar,litangg@iastate.edu Abstract Polynomial based methods have recently been used in several works for mitigating the effect of stragglers (slow or failed nodes) in distributed matrix computations. For a system with n worker nodes where s can be stragglers, these approaches allow for an optimal recovery threshold, whereby the intended result can be decoded as long as any (n − s) worker nodes complete their tasks. However, they suffer from serious numerical issues owing to the condition number of the corresponding real Vandermonde-structured recovery matrices; this condition number grows exponentially in n. We present a novel approach that leverages the properties of circulant permutation matrices and rotation matrices for coded matrix computation. In addition to having an optimal recovery threshold, we demonstrate an upper bound on the worst-case condition number of our recovery matrices which grows as ≈ O(ns+5:5); in the practical scenario where s is a constant, this grows polynomially in n. Our schemes leverage the well-behaved conditioning of complex Vandermonde matrices with parameters on the complex unit circle, while still working with computation over the reals. Exhaustive experimental results demonstrate that our proposed method has condition numbers that are orders of magnitude lower than prior work. I. INTRODUCTION Present day computing needs necessitate the usage of large computation clusters that regularly process huge amounts of data on a regular basis. In several of the relevant application domains such as machine learning, arXiv:1910.06515v4 [cs.IT] 9 Jun 2021 datasets are often so large that they cannot even be stored in the disk of a single server.
    [Show full text]