Quantum Physics - Department of Physics and Astronomy

301 Pages·2012·2.48 MB·English

Quantum Physics
Eric D’Hoker
Department of Physics and Astronomy,
University of California, Los Angeles, CA 90095, USA
15 September 2012
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Contents
1 Introduction 12
1.1 Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2 Constants of Nature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.3 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.4 Reductionism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Two-state quantum systems 16
2.1 Polarization of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.2 Polarization of photons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 General parametrization of the polarization of light . . . . . . . . . . . . . . 20
2.4 Mathematical formulation of the photon system . . . . . . . . . . . . . . . . 20
2.5 The Stern-Gerlach experiment on electron spin . . . . . . . . . . . . . . . . . 22
3 Mathematical Formalism of Quantum Physics 26
3.1 Hilbert spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
3.1.1 Triangle and Schwarz Inequalities . . . . . . . . . . . . . . . . . . . . 27
3.1.2 The construction of an orthonormal basis . . . . . . . . . . . . . . . . 27
3.1.3 Decomposition of an arbitrary vector . . . . . . . . . . . . . . . . . . 28
3.1.4 Finite-dimensional Hilbert spaces . . . . . . . . . . . . . . . . . . . . 29
3.1.5 Infinite-dimensional Hilbert spaces . . . . . . . . . . . . . . . . . . . 29
3.2 Linear operators on Hilbert space . . . . . . . . . . . . . . . . . . . . . . . . 31
3.2.1 Operators in finite-dimensional Hilbert spaces . . . . . . . . . . . . . 31
3.2.2 Operators in infinite-dimensional Hilbert spaces . . . . . . . . . . . . 31
3.3 Special types of operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.4 Hermitian and unitary operators in finite-dimension . . . . . . . . . . . . . . 34
3.4.1 Unitary operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.4.2 The exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.5 Self-adjoint operators in infinite-dimensional Hilbert spaces . . . . . . . . . . 37
4 The Principles of Quantum Physics 38
4.1 Conservation of probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.2 Compatible versus incompatible observables . . . . . . . . . . . . . . . . . . 39
4.3 Expectation values and quantum fluctuations . . . . . . . . . . . . . . . . . 41
4.4 Incompatible observables, Heisenberg uncertainty relations . . . . . . . . . . 41
4.5 Complete sets of commuting observables . . . . . . . . . . . . . . . . . . . . 43
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5 Some Basic Examples of Quantum Systems 44
5.1 Propagation in a finite 1-dimensional lattice . . . . . . . . . . . . . . . . . . 44
5.1.1 Diagonalizing the translation operator . . . . . . . . . . . . . . . . . 45
5.1.2 Position and translation operator algebra . . . . . . . . . . . . . . . . 46
5.1.3 The spectrum and generalized Hamiltonians . . . . . . . . . . . . . . 47
5.1.4 Bilateral and reflection symmetric lattices . . . . . . . . . . . . . . . 47
5.2 Propagation in an infinite 1-dimensional lattice . . . . . . . . . . . . . . . . 48
5.3 Propagation on a circle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.4 Propagation on the full line . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
5.4.1 The Dirac δ-function . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.5 General position and momentum operators and eigenstates . . . . . . . . . . 53
5.6 The harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.6.1 Lowering and Raising operators . . . . . . . . . . . . . . . . . . . . . 56
5.6.2 Constructing the spectrum . . . . . . . . . . . . . . . . . . . . . . . . 57
5.6.3 Harmonic oscillator wave functions . . . . . . . . . . . . . . . . . . . 57
5.7 The angular momentum algebra . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.7.1 Complete set of commuting observables . . . . . . . . . . . . . . . . . 59
5.7.2 Lowering and raising operators . . . . . . . . . . . . . . . . . . . . . 59
5.7.3 Constructing the spectrum . . . . . . . . . . . . . . . . . . . . . . . . 60
5.8 The Coulomb problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.8.1 Bound state spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.8.2 Scattering spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5.9 Self-adjoint operators and boundary conditions . . . . . . . . . . . . . . . . . 64
5.9.1 Example 1: One-dimensional Schro¨dinger operator on half-line . . . 64
5.9.2 Example 2: One-dimensional momentum in a box . . . . . . . . . . 65
5.9.3 Example 3: One-dimensional Dirac-like operator in a box . . . . . . 67
6 Quantum Mechanics Systems 68
6.1 Lagrangian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.2 Hamiltonian mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
6.3 Constructing a quantum system from classical mechanics . . . . . . . . . . . 71
6.4 Schro¨dinger equation with a scalar potential . . . . . . . . . . . . . . . . . . 72
6.5 Uniqueness questions of the correspondence principle . . . . . . . . . . . . . 73
7 Charged particle in an electro-magnetic field 74
7.1 Gauge transformations and gauge invariance . . . . . . . . . . . . . . . . . . 74
7.2 Constant Magnetic fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
7.2.1 Map onto harmonic oscillators . . . . . . . . . . . . . . . . . . . . . . 76
7.3 Landau Levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
7.3.1 Complex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
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7.4 The Aharonov-Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
7.4.1 The scattering Aharonov-Bohm effect . . . . . . . . . . . . . . . . . . 79
7.4.2 The bound state Aharonov-Bohm effect . . . . . . . . . . . . . . . . . 81
7.5 The Dirac magnetic monopole . . . . . . . . . . . . . . . . . . . . . . . . . . 83
8 Theory of Angular Momentum 86
8.1 Rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
8.2 The Lie algebra of rotations – angular momentum . . . . . . . . . . . . . . . 88
8.3 General Groups and their Representations . . . . . . . . . . . . . . . . . . . 88
8.4 General Lie Algebras and their Representations . . . . . . . . . . . . . . . . 89
8.5 Direct sum and reducibility of representations . . . . . . . . . . . . . . . . . 90
8.6 The irreducible representations of angular momentum . . . . . . . . . . . . . 90
8.7 Addition of two spin 1/2 angular momenta . . . . . . . . . . . . . . . . . . . 92
8.8 Addition of a spin 1/2 with a general angular momentum . . . . . . . . . . . 94
8.9 Addition of two general angular momenta . . . . . . . . . . . . . . . . . . . 96
8.10 Systematics of Clebsch-Gordan coefficients . . . . . . . . . . . . . . . . . . . 97
8.11 Spin Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.12 The Ising Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
8.13 Solution of the 1-dimensional Ising Model . . . . . . . . . . . . . . . . . . . 100
8.14 Ordered versus disordered phases . . . . . . . . . . . . . . . . . . . . . . . . 102
9 Symmetries in Quantum Physics 104
9.1 Symmetries in classical mechanics . . . . . . . . . . . . . . . . . . . . . . . . 104
9.2 Noether’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
9.3 Group and Lie algebra structure of classical symmetries . . . . . . . . . . . . 107
9.4 Symmetries in Quantum Physics . . . . . . . . . . . . . . . . . . . . . . . . . 108
9.5 Examples of quantum symmetries . . . . . . . . . . . . . . . . . . . . . . . . 110
9.6 Symmetries of the multi-dimensional harmonic oscillator . . . . . . . . . . . 110
9.6.1 The orthogonal group SO(N) . . . . . . . . . . . . . . . . . . . . . . 111
9.6.2 The unitary groups U(N) and SU(N) . . . . . . . . . . . . . . . . . 113
9.6.3 The group Sp(2N) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115
9.7 Selection rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
9.8 Vector Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
9.9 Selection rules for vector observables . . . . . . . . . . . . . . . . . . . . . . 118
9.10 Tensor Observables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
9.11 P, C, and T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
10 Bound State Perturbation Theory 124
10.1 The validity of perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 125
10.1.1 Smallness of the coupling . . . . . . . . . . . . . . . . . . . . . . . . . 125
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10.1.2 Convergence of the expansion for finite-dimensional systems . . . . . 126
10.1.3 The asymptotic nature of the expansion for infinite dimensional systems126
10.2 Non-degenerate perturbation theory . . . . . . . . . . . . . . . . . . . . . . . 128
10.3 Some linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
10.4 The Stark effect for the ground state of the Hydrogen atom . . . . . . . . . . 132
10.5 Excited states and degenerate perturbation theory . . . . . . . . . . . . . . . 133
10.6 The Zeeman effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
10.7 Spin orbit coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
10.8 General development of degenerate perturbation theory . . . . . . . . . . . . 135
10.8.1 Solution to first order . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
10.8.2 Solution to second order . . . . . . . . . . . . . . . . . . . . . . . . . 137
10.9 Periodic potentials and the formation of band structure . . . . . . . . . . . . 138
10.10Level Crossing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
11 External Magnetic Field Problems 142
11.1 Landau levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
11.2 Complex variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
11.3 Calculation of the density of states in each Landau level . . . . . . . . . . . 145
11.4 The classical Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
11.5 The quantum Hall effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
12 Scattering Theory 151
12.1 Potential Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
12.2 The iε prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
12.3 The free particle propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
12.4 The Lippmann-Schwinger equation in position space . . . . . . . . . . . . . . 155
12.5 Short range versus long range V and massless particles . . . . . . . . . . . . 156
12.6 The wave-function solution far from the target . . . . . . . . . . . . . . . . . 157
12.7 Calculation of the cross section . . . . . . . . . . . . . . . . . . . . . . . . . 157
12.8 The Born approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
12.8.1 The case of the Coulomb potential . . . . . . . . . . . . . . . . . . . 160
12.8.2 The case of the Yukawa potential . . . . . . . . . . . . . . . . . . . . 161
12.9 The optical Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
12.10Spherical potentials and partial wave expansion . . . . . . . . . . . . . . . . 162
12.10.1Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
12.10.2Partial wave expansion of wave functions . . . . . . . . . . . . . . . . 164
12.10.3Calculating the radial Green function . . . . . . . . . . . . . . . . . . 165
12.11Phase shifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
12.12The example of a hard sphere . . . . . . . . . . . . . . . . . . . . . . . . . . 168
12.13The hard spherical shell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
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12.14Resonance scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
13 Time-dependent Processes 173
13.1 Magnetic spin resonance and driven two-state systems . . . . . . . . . . . . . 173
13.2 The interaction picture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
13.3 Time-dependent perturbation theory . . . . . . . . . . . . . . . . . . . . . . 177
13.4 Switching on an interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
13.5 Sinusoidal perturbation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
14 Path Integral Formulation of Quantum Mechanics 181
14.1 The time-evolution operator . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
14.2 The evolution operator for quantum mechanical systems . . . . . . . . . . . 182
14.3 The evolution operator for a free massive particle . . . . . . . . . . . . . . . 183
14.4 Derivation of the path integral . . . . . . . . . . . . . . . . . . . . . . . . . . 184
14.5 Integrating out the canonical momentum p . . . . . . . . . . . . . . . . . . . 187
14.6 Dominant paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
14.7 Stationary phase approximation . . . . . . . . . . . . . . . . . . . . . . . . . 188
14.8 Gaussian fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
14.9 Gaussian integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
14.10Evaluating the contribution of Gaussian fluctuations . . . . . . . . . . . . . 191
15 Applications and Examples of Path Integrals 193
15.1 Path integral calculation for the harmonic oscillator . . . . . . . . . . . . . . 193
15.2 The Aharonov-Bohm Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
15.3 Imaginary time path Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . 197
15.4 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 198
15.5 Path integral formulation of quantum statistical mechanics . . . . . . . . . . 199
15.6 Classical Statistical Mechanics as the high temperature limit . . . . . . . . . 200
16 Mixtures and Statistical Entropy 202
16.1 Polarized versus unpolarized beams . . . . . . . . . . . . . . . . . . . . . . . 202
16.2 The Density Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
16.2.1 Ensemble averages of expectation values in mixtures . . . . . . . . . . 205
16.2.2 Time evolution of the density operator . . . . . . . . . . . . . . . . . 205
16.3 Example of the two-state system . . . . . . . . . . . . . . . . . . . . . . . . 205
16.4 Non-uniqueness of state preparation . . . . . . . . . . . . . . . . . . . . . . . 207
16.5 Quantum Statistical Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . 208
16.5.1 Generalized equilibrium ensembles . . . . . . . . . . . . . . . . . . . . 209
16.6 Classical information and Shannon entropy . . . . . . . . . . . . . . . . . . 210
16.7 Quantum statistical entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
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16.7.1 Density matrix for a subsystem . . . . . . . . . . . . . . . . . . . . . 213
16.7.2 Example of relations between density matrices of subsystems . . . . . 213
16.7.3 Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
16.7.4 Completing the proof of subadditivity . . . . . . . . . . . . . . . . . . 216
16.8 Examples of the use of statistical entropy . . . . . . . . . . . . . . . . . . . . 217
16.8.1 Second law of thermodynamics . . . . . . . . . . . . . . . . . . . . . 218
16.8.2 Entropy resulting from coarse graining . . . . . . . . . . . . . . . . . 218
17 Entanglement, EPR, and Bell’s inequalities 219
17.1 Entangled States for two spin 1/2 . . . . . . . . . . . . . . . . . . . . . . . 219
17.2 Entangled states from non-entangled states . . . . . . . . . . . . . . . . . . . 221
17.3 The Schmidt purification theorem . . . . . . . . . . . . . . . . . . . . . . . . 222
17.4 Generalized description of entangled states . . . . . . . . . . . . . . . . . . . 223
17.5 Entanglement entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224
17.6 The two-state system once more . . . . . . . . . . . . . . . . . . . . . . . . . 224
17.7 Entanglement in the EPR paradox . . . . . . . . . . . . . . . . . . . . . . . 225
17.8 Einstein’s locality principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
17.9 Bell’s inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
17.10Quantum predictions for Bell’s inequalities . . . . . . . . . . . . . . . . . . . 229
17.11Three particle entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . 232
18 Introductory Remarks on Quantized Fields 234
18.1 Relativity and quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . 235
18.2 Why Quantum Field Theory ? . . . . . . . . . . . . . . . . . . . . . . . . . . 235
18.3 Further conceptual changes required by relativity . . . . . . . . . . . . . . . 236
18.4 Some History and present significance of QFT . . . . . . . . . . . . . . . . . 237
19 Quantization of the Free Electro-magnetic Field 239
19.1 Classical Maxwell theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
19.2 Fourrier modes and radiation oscillators . . . . . . . . . . . . . . . . . . . . 241
19.3 The Hamiltonian in terms of radiation oscillators . . . . . . . . . . . . . . . 242
19.4 Momentum in terms of radiation oscillators . . . . . . . . . . . . . . . . . . . 245
19.5 Canonical quantization of electro-magnetic fields . . . . . . . . . . . . . . . . 245
19.6 Photons – the Hilbert space of states . . . . . . . . . . . . . . . . . . . . . . 246
19.6.1 The ground state or vacuum . . . . . . . . . . . . . . . . . . . . . . . 246
19.6.2 One-photon states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
19.6.3 Multi-photon states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
19.7 Bose-Einstein and Fermi-Dirac statistics . . . . . . . . . . . . . . . . . . . . 249
19.8 The photon spin and helicity . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
19.9 The Casimir Effect on parallel plates . . . . . . . . . . . . . . . . . . . . . . 253
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20 Photon Emission and Absorption 257
20.1 Setting up the general problem of photon emission/absorption . . . . . . . . 257
20.2 Single Photon Emission/Absorption . . . . . . . . . . . . . . . . . . . . . . . 258
20.3 Application to the decay rate of 2p state of atomic Hydrogen . . . . . . . . . 261
20.4 Absorption and emission of photons in a cavity . . . . . . . . . . . . . . . . 261
20.5 Black-body radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263
21 Relativistic Field Equations 266
21.1 A brief review of special relativity . . . . . . . . . . . . . . . . . . . . . . . . 266
21.2 Lorentz vector and tensor notation . . . . . . . . . . . . . . . . . . . . . . . 268
21.3 General Lorentz vectors and tensors . . . . . . . . . . . . . . . . . . . . . . . 269
21.3.1 Contravariant tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
21.3.2 Covariant tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270
21.3.3 Contraction and trace . . . . . . . . . . . . . . . . . . . . . . . . . . 271
21.4 Classical relativistic kinematics and dynamics . . . . . . . . . . . . . . . . . 271
21.5 Particle collider versus fixed target experiments . . . . . . . . . . . . . . . . 272
21.6 A physical application of time dilation . . . . . . . . . . . . . . . . . . . . . 273
21.7 Relativistic invariance of the wave equation . . . . . . . . . . . . . . . . . . . 273
21.8 Relativistic invariance of Maxwell equations . . . . . . . . . . . . . . . . . . 274
21.8.1 The gauge field and field strength . . . . . . . . . . . . . . . . . . . . 274
21.8.2 Maxwell’s equations in Lorentz covariant form . . . . . . . . . . . . . 275
21.9 Structure of the Poincar´e and Lorentz algebras . . . . . . . . . . . . . . . . . 277
21.10Representations of the Lorentz algebra . . . . . . . . . . . . . . . . . . . . . 279
22 The Dirac Field and the Dirac Equation 282
22.1 The Dirac-Clifford algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282
22.2 Explicit representation of the Dirac algebra . . . . . . . . . . . . . . . . . . . 284
22.3 Action of Lorentz transformations on γ-matrices . . . . . . . . . . . . . . . . 285
22.4 The Dirac equation and its relativistic invariance . . . . . . . . . . . . . . . 286
22.5 Elementary solutions to the free Dirac equation . . . . . . . . . . . . . . . . 288
22.6 The conserved current of fermion number . . . . . . . . . . . . . . . . . . . . 289
22.7 The free Dirac action and Hamiltonian . . . . . . . . . . . . . . . . . . . . . 291
22.8 Coupling to the electro-magnetic field . . . . . . . . . . . . . . . . . . . . . . 291
23 Quantization of the Dirac Field 293
23.1 The basic free field solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
23.2 Spinor Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295
23.3 Evaluation of the electric charge operator and Hamiltonian . . . . . . . . . . 297
23.4 Quantization of fermion oscillators . . . . . . . . . . . . . . . . . . . . . . . 298
23.5 Canonical anti-commutation relations for the Dirac field . . . . . . . . . . . 298
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23.6 The fermion propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
23.7 The concept of vacuum polarization . . . . . . . . . . . . . . . . . . . . . . . 300
Acknowledgements
It is a pleasure to thank John Estes, Yu Guo, Robert Maloney, Antonio Russo, Josh
Samani, and Jie Zhang for suggesting various useful corrections and clarifications.
9
Bibliography
Course textbook
Modern Quantum Mechanics, J.J. Sakurai, Revised Edition, Addison-Wesley (1994)
•
Reference on Classical Mechanics
Classical Mechanics, H. Goldstein, Addison Wesley, (1980);
•
References on undergraduate Quantum Mechanics
Introduction to Quantum Mechanics, D.J. Griffiths, Pearson Prentice Hall (2005);
•
The Feynman Lectures of Physics, Vol III, R.P Feynman, R.B. Leighton and M. Sands,
•
Addison-Wesley, (1965);
General References on Quantum Mechanics
Quantum Physics, M. Le Bellac, Cambridge University Press (2006);
•
Principles of Quantum Mechanics, R. Shankar, Plenum (1980);
•
Quantum Mechanics, E. Abers, Pearson Prentice Hall (2004);
•
Classics
The Principles of Quantum Mechanics, P.A.M. Dirac, Oxford (1968);
•
Quantum Mechanics, Non-relativistic Theory, Course in Theoretical Physics, Vol 3,
•
L. Landau and E. Lifschitz, Butterworth Heinemann, (2004);
Quantum Mechanics, C. Cohen-Tannoudji, B. Diu and F. Laloe, I & II, Wiley (1977),
•
[a very thorough classic, with extensive applications];
Quantum Mechanics and Path Integrals, R.P. Feynman and A. Hibbs, MacGraw Hill;
•
Quantum Mechanics, J. Schwinger, Springer Verlag (2001);
•
Introduction to Quantum Mechanics, Vol I, K. Gottfried, Benjamin (1966);
•
Historical Developments
The Conceptual Development of Quantum Mechanics, Max Jammer, McGraw Hill,
•
(1966), [a superb treatise on the foundations of quantum mechanics];
Inward Bound; On Matter and Forces in the Physical World, A. Pais, Oxford, (1986);
•
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