Symmetries in Quantum Mechanics: From Angular Momentum to Supersymmetry (PBK) (Graduate Students Series in Physics)

Symmetries in Quantum Mechanics: From Angular Momentum to Supersymmetry (PBK) provides a thorough, didactic exposition of the role of symmetry, particularly rotational symmetry, in quantum mechanics. The bulk of the book covers the description of rotations (geometrically and group-theoretically) and their representations, and the quantum theory of angular momentum. Later chapters introduce more advanced topics such as relativistic theory, supersymmetry, anyons, fractional spin, and statistics. With clear, in-depth explanations, the book is ideal for use as a course text for postgraduate and advanced undergraduate students in physics and those specializing in theoretical physics. It is also useful for researchers looking for an accessible introduction to this important area of quantum theory. SYMMETRIES IN QUANTUM MECHANICS: FROM ANGULAR MOMENTUM TO SUPERSYMMETRY......Page 1 Title Page......Page 2 Contents......Page 4 Preface......Page 10 1.1 Notation......Page 12 1.2 Some basic concepts in quantum mechanics......Page 15 1.3.1 Groups: finite, infinite, continuous, Abelian, non-Abelian; subgroup of a group, cosets......Page 18 1.3.2 Isomorphism, automorphism, homomorphism......Page 19 1.3.3 Lie groups and Lie algebras......Page 20 1.3.4 Representations: faithful, irreducible, reducible, completely reducible (decomposable), indecomposable, adjoint, fundamental......Page 21 1.3.5 Relation between Lie algebras and Lie groups, Casimir operators, rank of a group......Page 22 1.3.7 Semidirect sum of Lie algebras and semidirect product of Lie groups (inhomogeneous Lie algebras and groups)......Page 23 1.3.8 The Haar measure......Page 24 1.4 Remark about the introduction of angular momentum......Page 25 2.1.1 General considerations......Page 27 2.1.2 Formal definition of symmetry; ray correspondence......Page 31 2.2 Wigner's theorem: the existence of unitary or anti-unitary representations......Page 32 2.3.1 General considerations......Page 39 2.3.2 Continuous matrix groups; decomposition into pieces......Page 40 2.3.3 The Lie algebra (Lie ring, infinitesimal ring)......Page 41 2.3.4 Canonical coordinates......Page 43 2.3.5 The structure of the group and its infinitesimal ring......Page 47 2.3.6 Summary: continuous matrix groups and their Lie algebra......Page 49 2.3.7 Group representations......Page 50 2.4.1 Continuous groups connected to the identity; Noether's theorem......Page 53 2.4.3 Super-selection rules......Page 56 2.4.4 Complete symmetry group, complete sets of commuting observables, complete sets of states......Page 60 2.4.5 Summary of the chapter......Page 63 3.1.1 Interpretation......Page 64 3.1.2 Parameters describing a rotation......Page 65 3.1.3 Representation of a rotation......Page 66 3.2 Sequences of rotations......Page 67 3.2.1 Considering the 'abstract' rotations R......Page 68 3.2.2 Considering the 3 × 3 rotation matrices M(R)......Page 70 3.3.1 The rotation matrix M[sub(p)](η)......Page 74 3.3.2 The generators of the rotation group......Page 76 3.3.3 The local group......Page 78 3.3.4 Canonical parameters of the first and the second kind......Page 79 3.4 The unitary representation U(R) induced by the three-dimensional rotation R......Page 80 4.1.1 The physical significance of J......Page 83 4.2 Commutation relations for angular momenta......Page 86 4.3 Direct sum and direct product......Page 91 4.4 Angular momenta of interacting systems......Page 96 4.5 Irreducible representations; Schur's lemma......Page 98 4.6 Eigenstates of angular momentum......Page 102 4.7 Orbital angular momentum......Page 110 4.7.1 Angular momentum operators in polar coordinates......Page 111 4.7.2 Construction of the eigenfunctions......Page 113 4.7.3 Orbital angular momenta have only integer eigenvalues......Page 115 4.7.4 Spherical harmonics......Page 116 4.7.7 Particular cases......Page 119 4.8 Spin-1/2 eigenstates and operators......Page 122 4.9 Double-valued representations; the covering group SU(2)......Page 125 4.10 Construction of the general j, m-state from spin-1/2 states......Page 127 5.1 The general problem......Page 132 5.2 Complete sets of mutually commuting (angular momentum) observables......Page 133 5.3.1 Notation......Page 139 5.3.2 Definition and some properties of the Clebsch–Gordan coefficients......Page 142 5.3.3 Orthogonality of the Clebsch–Gordan coefficients......Page 144 5.3.4 Sketch of the calculation of the Clebsch–Gordan coefficients; phase convention and reality......Page 145 5.3.5 Calculation of ......Page 149 5.3.6 Obvious symmetry relations for CGCs......Page 151 5.3.7 Wigner's 3j-symbol and Racah's V(j[sub(1)]j[sub(2)]j[sub(3)]|m[sub(1)]m[sub(2)]m[sub(3)])-symbol......Page 158 5.3.8 Racah's formula for the CGCs......Page 160 5.3.9 Regge's symmetry of CGCs......Page 165 5.3.10 Collection of formulae for the CGCs; a table of special values......Page 167 5.4.1 General remarks; statement of the problem......Page 170 5.4.2 The 6j-symbol and the Racah coefficients......Page 173 5.4.3 Collection of formulae for recoupling coefficients......Page 175 5.6 Numerical tables and important references on addition of angular momenta......Page 179 6.1 Active and passive interpretation; definition of D[sup((j))][sub(m'm)]; the invariant subspaces H[sub(j)]......Page 181 6.2.1 The spin-1/2 case......Page 184 6.2.2 The general case......Page 186 6.3.1 Relation to the Clebsch–Gordan coefficients......Page 188 6.3.2 Significance of the relation to the CGCs......Page 190 6.3.3.1 Transformation of eigenfunctions under rotations......Page 194 6.3.3.2 The D[sup((l))][sub(m0)] are spherical harmonics......Page 195 6.3.3.3 The addition theorem and the composition rule of spherical harmonics; integral over three spherical harmonics......Page 197 6.3.4 Orthogonality relations and integrals over D-matrices......Page 199 6.3.5 A projection formula......Page 201 6.3.6 Completeness relation for the D matrices......Page 202 6.3.7 Symmetry properties of the D matrices......Page 205 7.1 Bosonic operators......Page 207 7.2 Realization of su(2) Lie algebra and the rotation matrix in terms of bosonic operators......Page 210 7.3 A short note about the new field of quantum groups......Page 215 8.1 Introduction......Page 218 8.2 Definition and properties......Page 220 8.3 Tensor product; irreducible combination of irreducible tensors; scalar product......Page 222 8.4 Invariants and covariant equations......Page 224 8.5 Spinor and vector spherical harmonics......Page 226 8.6 Angular momenta as spherical tensor operators......Page 230 8.7 The Wigner–Eckart theorem......Page 231 8.8.1 The trace of T(kq)......Page 233 8.9 Projection theorem for irreducible tensor operators of rank 1......Page 234 9.1 Introduction......Page 238 9.2 Properties of rotations in two-dimensional space and fractional statistics......Page 239 9.3 Particle–flux system: example of anyon......Page 243 9.4 Possible role of anyons in physics......Page 248 10.1 Introduction......Page 250 10.2 The generators of the inhomogeneous Lorentz group (Poincaré group)......Page 251 10.2.1 Translations; four-momentum......Page 252 10.2.2.1 Introducing a new notation adapted to space-time......Page 253 10.2.2.2 Extension from space to space-time......Page 255 10.2.2.3 Physical significance of the new generators......Page 257 10.3.1 Commutation relations of the J[sup(μν)] with each other......Page 259 10.4 A complete set of commuting observables......Page 260 10.4.1 The spin four-vector w[sup(μ)] and the spin tensor S[sup(μν)]......Page 261 10.4.2 Commutation relations for w[sup(μ)] and S[sup(μν)]......Page 263 10.4.3 Construction of a complete set of commuting observables; helicity......Page 266 10.4.4 Zero-mass particles......Page 271 10.5.1 Construction of one-particle helicity states of arbitrary p......Page 274 10.5.3 Eigenstates of the total angular momentum......Page 276 10.5.4 The S-matrix; cross-sections......Page 278 10.5.5 Evaluation of cross-section formulae......Page 280 10.5.6.1 Parity......Page 281 10.5.6.2 Time reversal......Page 283 10.5.6.3 Identical particles......Page 284 11.1 What is supersymmetry?......Page 286 11.2 SUSY quantum mechanics......Page 290 11.3 Factorization and the hierarchy of Harniltonians......Page 292 11.4 Broken supersymmetry......Page 296 Appendix A. Remarks on symmetric and self-adjoint operators......Page 298 Appendix B. The distinction between finite and infinite numbers of degrees of freedom in quantum mechanics......Page 302 Bibliography......Page 304 Index......Page 308 Back Cover......Page 316 This Book Provides A Thorough, Didactic Exposition Of The Role Of Symmetry, Particularly Rotational Symmetry, In Quantum Mechanics. The Bulk Of The Book Covers The Description Of Rotations (geometrically And Group-theoretically) And Their Representations, And The Quantum Theory Of Angular Momentum. Later Chapters Introduce More Advanced Topics Like Relativistic Theory, Supersymmetry, Anyons, Fractional Spin And Statistics. Everything Is Explained Clearly And In Depth Making The Book Ideal For Use As A Course Text For Postgraduate And Advanced Undergraduates Students In Physics, And Those Specializing In Theoretical Physics. The Book Will Also Be Useful For Researchers Looking For An Accessible Introduction To This Important Area Of Quantum Theory. 1. Introduction -- 2. Symmetry In Quantum Mechanics -- 3. Rotations In Three-dimensional Space -- 4. Angular Momentum Operators And Eigenstates -- 5. Addition Of Angular Momenta -- 6. Representations Of The Rotation Group -- 7. Jordan-schwinger Construction And Representations -- 8. Irreducible Tensors And Tensor Operators -- 9. Peculiarities Of Two-dimensional Rotations: Anyons, Fractional Spin And Statistics -- 10. Brief Glance At Relativistic Problems -- 11. Supersymmetry In Quantum Mechanics And Particle Physics. Masud Chaichian, Rolf Hagedorn. Includes Bibliographical References (p. 293-296) And Index. These books provide a vast collection of quotations on their respective subjects. Some quotes are profound, others are wise, some are witty but none are frivolous. Here you will find quotations from the most famous to the unknown. You may not find all the quoted 'jewels' that exist, but we are sure you will find a great many of them here. The extensive author and subject indexes provide you with the perfect tool for locating quotations for practical use or pleasure.