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E-grāmata: Symmetries in Quantum Mechanics: From Angular Momentum to Supersymmetry (PBK)

, (University of Helsinki and Helsinki Inst of Physics, Finland)
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Symmetries in Quantum Mechanics: From Angular Momentum to Supersymmetry (PBK)Palielināt
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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.

Recenzijas

" a useful complement to the existing literature." -Mathematical Reviews

Preface xiii
1 Introduction
1(15)
1.1 Notation
1(3)
1.2 Some basic concepts in quantum mechanics
4(3)
1.3 Some basic objects of group theory
7(7)
1.3.1 Groups: finite, infinite, continuous, Abelian, non-Abelian; subgroup of a group, cosets
7(1)
1.3.2 Isomorphism, automorphism, homomorphism
8(1)
1.3.3 Lie groups and Lie algebras
9(1)
1.3.4 Representations: faithful, irreducible, reducible, completely reducible (decomposable), indecomposable, adjoint, fundamental
10(1)
1.3.5 Relation between Lie algebras and Lie groups, Casimir operators, rank of a group
11(1)
1.3.6 Schur's lemmas
12(1)
1.3.7 Semidirect sum of Lie algebras and semidirect product of Lie groups (inhomogeneous Lie algebras and groups)
12(1)
1.3.8 The Haar measure
13(1)
1.4 Remark about the introduction of angular momentum
14(2)
2 Symmetry in Quantum Mechanics
16(37)
2.1 Definition of symmetry
16(5)
2.1.1 General considerations
16(4)
2.1.2 Formal definition of symmetry; ray correspondence
20(1)
2.2 Wigner's theorem: the existence of unitary or anti-unitary representations
21(7)
2.3 Continuous matrix groups and their generators
28(14)
2.3.1 General considerations
28(1)
2.3.2 Continuous matrix groups; decomposition into pieces
29(1)
2.3.3 The Lie algebra (Lie ring, infinitesimal ring)
30(2)
2.3.4 Canonical coordinates
32(4)
2.3.5 The structure of the group and its infinitesimal ring
36(2)
2.3.6 Summary: continuous matrix groups and their Lie algebra
38(1)
2.3.7 Group representations
39(3)
2.4 The physical significance of symmetries
42(11)
2.4.1 Continuous groups connected to the identity; Noether's theorem
42(3)
2.4.2 Pieces not connected to the identity; discrete groups
45(1)
2.4.3 Super-selection rules
45(4)
2.4.4 Complete symmetry group, complete sets of commuting observables, complete sets of states
49(3)
2.4.5 Summary of the chapter
52(1)
3 Rotations in Three-Dimensional Space
53(19)
3.1 General remarks on rotations
53(3)
3.1.1 Interpretation
53(1)
3.1.2 Parameters describing a rotation
54(1)
3.1.3 Representation of a rotation
55(1)
3.2 Sequences of rotations
56(7)
3.2.1 Considering the `abstract' rotations R
57(2)
3.2.2 Considering the 3 x 3 rotation matrices M (R)
59(4)
3.3 The Lie algebra and the local group
63(6)
3.3.1 The rotation matrix M p(n)
63(2)
3.3.2 The generators of the rotation group
65(2)
3.3.3 The local group
67(1)
3.3.4 Canonical parameters of the first and the second kind
68(1)
3.4 The unitary representation U (R) induced by the three-dimensional rotation R
69(3)
4 Angular Momentum Operators and Eigenstates
72(49)
4.1 The operators of angular momentum J(1), J(2) and J(3)
72(3)
4.1.1 The physical significance of J
72(3)
4.1.2 The angular momentum component in a direction n
75(1)
4.2 Commutation relations for angular momenta
75(5)
4.3 Direct sum and direct product
80(5)
4.4 Angular momenta of interacting systems
85(2)
4.5 Irreducible representations; Schur's lemma
87(4)
4.6 Eigenstates of angular momentum
91(8)
4.7 Orbital angular momentum
99(12)
4.7.1 Angular momentum operators in polar coordinates
100(2)
4.7.2 Construction of the eigenfunctions
102(2)
4.7.3 Orbital angular momenta have only integer eigenvalues
104(1)
4.7.4 Spherical harmonics
105(3)
4.7.5 The phase convention
108(1)
4.7.6 Parity
108(1)
4.7.7 Particular cases
108(3)
4.7.8 Further formulae
111(1)
4.8 Spin-1/2 eigenstates and operators
111(3)
4.9 Double-valued representations; the covering group SU(2)
114(2)
4.10 Construction of the general j, m-state from spin-1/2 states
116(5)
5 Addition of Angular Momenta
121(49)
5.1 The general problem
121(1)
5.2 Complete sets of mutually commuting (angular momentum) observables
122(6)
5.3 Combining two angular momenta; Clebsch-Gordan (Wigner) coefficients
128(31)
5.3.1 Notation
128(3)
5.3.2 Definition and some properties of the Clebsch-Gordan coefficients
131(2)
5.3.3 Orthogonality of the Clebsch-Gordan coefficients
133(1)
5.3.4 Sketch of the calculation of the Clebsch-Gordan coefficients; phase convention and reality
134(4)
5.3.5 Calculation of [ j(1)m(1)j(2)j -- m(1)|jj]
138(2)
5.3.6 Obvious symmetry relations for CGCs
140(7)
5.3.7 Wigner's 3j-symbol and Racah's V (j(1)j(2)j(3)|m(1)m(2)m(3))-symbol
147(2)
5.3.8 Racah's formula for the CGCs
149(5)
5.3.9 Regge's symmetry of CGCs
154(2)
5.3.10 Collection of formulae for the CGCs; a table of special values
156(3)
5.4 Combining three angular momenta; recoupling coefficients
159(9)
5.4.1 General remarks; statement of the problem
159(3)
5.4.2 The 6j-symbol and the Racah coefficients
162(2)
5.4.3 Collection of formulae for recoupling coefficients
164(4)
5.5 Combining more than three angular momenta
168(1)
5.6 Numerical tables and important references on addition of angular momenta
168(2)
6 Representations of the Rotation Group
170(26)
6.1 Active and passive interpretation; definition of D^(j)(m'm); the invariant subspaces Hj
170(3)
6.2 The explicit form of D^(j)(m'm) (Alpha, Beta, Gamma)
173(4)
6.2.1 The spin-1/2 case
173(2)
6.2.2 The general case
175(2)
6.3 General properties of D(j)
177(19)
6.3.1 Relation to the Clebsch-Gordan coefficients
177(2)
6.3.2 Significance of the relation to the CGCs
179(4)
6.3.3 Relation to the eigenfunctions of angular momentum
183(5)
6.3.4 Orthogonality relations and integrals over D-matrices
188(5)
6.3.5 A projection formula
190(1)
6.3.6 Completeness relation for the D-matrices
191(3)
6.3.7 Symmetry properties of the D-matrices
194(2)
7 The Jordan-Schwinger Construction and Representations
196(11)
7.1 Bosonic operators
196(3)
7.2 Realization of su(2) Lie algebra and the rotation matrix in terms of bosonic operators
199(5)
7.3 A short note about the new field of quantum groups
204(3)
8 Irreducible Tensors and Tensor Operators
207(20)
8.1 Introduction
207(2)
8.2 Definition and properties
209(2)
8.3 Tensor product; irreducible combination of irreducible tensors; scalar product
211(2)
8.4 Invariants and covariant equations
213(2)
8.5 Spinor and vector spherical harmonics
215(4)
8.6 Angular momenta as spherical tensor operators
219(1)
8.7 The Wigner-Eckart theorem
220(2)
8.8 Examples of applications of the Wigner-Eckart theorem
222(1)
8.8.1 The trace of T(kq)
222(1)
8.8.2 Tensors of rank O (scalars, invariants)
223(1)
8.8.3 The angular momentum operators
223(1)
8.9 Projection theorem for irreducible tensor operators of rank 1
223(4)
9 Peculiarities of Two-Dimensional Rotations: Anyons, Fractional Spin and Statistics
227(12)
9.1 Introduction
227(1)
9.2 Properties of rotations in two-dimensional space and fractional statistics
228(4)
9.3 Particle-flux system: example of anyon
232(5)
9.4 Possible role of anyons in physics
237(2)
10 A Brief Glance at Relativistic Problems
239(36)
10.1 Introduction
239(1)
10.2 The generators of the inhomogeneous Lorentz group (Poincare group)
240(8)
10.2.1 Translations; four-momentum
241(1)
10.2.2 The homogeneous Lorentz group; angular momentum
242(6)
10.3 The angular momentum operators
248(1)
10.3.1 Commutation relations of the J^(mv) with each other
248(1)
10.3.2 Commutation relations of the J^(mv) with P^p
249(1)
10.4 A complete set of commuting observables
249(14)
10.4.1 The spin four-vector w^m and the spin tensor S^(mv)
250(2)
10.4.2 Commutation relations for w^m and S^(mv)
252(3)
10.4.3 Construction of a complete set of commuting observables; helicity
255(5)
10.4.4 Zero-mass particles
260(3)
10.5 The use of helicity states in elementary particle physics
263(12)
10.5.1 Construction of one-particle helicity states of arbitrary p
263(2)
10.5.2 Two-particle helicity states
265(1)
10.5.3 Eigenstates of the total angular momentum
265(2)
10.5.4 The S-matrix; cross-sections
267(2)
10.5.5 Evaluation of cross-section formulae
269(1)
10.5.6 Discrete symmetry relations: parity, time reversal, identical particles
270(5)
11 Supersymmetry in Quantum Mechanics and Particle Physics
275(12)
11.1 What is supersymmetry?
275(4)
11.2 SUSY quantum mechanics
279(2)
11.3 Factorization and the hierarchy of Hamiltonians
281(4)
11.4 Broken supersymmetry
285(2)
Appendix A 287(4)
Appendix B 291(2)
Bibliography 293(4)
Index 297


M. Chaichian, R. Hagedorn
Biežāk uzdotie jautājumi par e-grāmatām Biežāk uzdotie jautājumi par e-grāmatām
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