The Mathematical Principles of Quantum Mechanics Book

	Preface	xi
1	Vector representation of states
1.1	The Leibniz Principle	1
1.2	Heisenberg's Uncertainty Principle	3
1.3	Pure and mixed states	6
1.4	Transitions between pure states	8
1.5	Complex finite-dimensional vector spaces	11
1.6	Unitary transformations	15
1.7	The vector representation of states	19
1.8	Matrices representing observables	21
1.9	Representation of functions of observables	24
1.10	Characteristic equations	30
1.11	The Uncertainty Principle	32
1.12	Compatibility of observables	35
1.13	Combination of systems	38
2	Spin
2.1	The Stern-Gerlach experiment	42
2.2	Spinors. Pauli matrices	44
2.3	Illustration of the theory	47
2.4	Unitary transformation of spinors	50
2.5	A two-particle system	53
2.6	Indistinguishable particles. Exclusion Principle	55
3	Observables having continuous spectra
3.1	Spectra with an infinity of eigenvalues	60
3.2	Continuous representations. Impulse function	61
3.3	Transformation of representations	66
3.4	Representation of observables	69
3.5	Momentum and coordinate representations	71
3.6	Momentum and coordinate representation of observables	75
3.7	Eigenstates for momentum and position of a particle	78
3.8	Energy eigenstates for a particle	79
3.9	Particle in a potential well	81
3.10	The harmonic oscillator	87
3.11	Harmonic oscillator in the energy representation	93
3.12	Two-particle systems	95
	Note on Dirac's Notation	99
4	Time variation of states
4.1	The Hamiltonian operator	102
4.2	Rates of change of expected values	105
4.3	Hamilton's equations	107
4.4	Steady states	108
4.5	Time variation of spin	110
4.6	Conservation of probability	116
4.7	Free rectilinear motion of a particle. Wave packets	117
4.8	Freely moving particle incident upon a potential barrier	121
4.9	The tunnel effect	125
4.10	Transmission resonances	127
4.11	Heisenberg's representation	128
4.12	Heisenberg's treatment of the harmonic oscillator problem	131
4.13	Preservation of symmetry and anti-symmetry of system states	134
5	Angular momentum
5.1	Angular momentum operators	135
5.2	Eigenfunctions and eigenvalues of angular momentum	138
5.3	Central force fields	144
5.4	The hydrogen atom	146
5.5	Particle with spin in a central force field	150
5.6	Charged particle in an electromagnetic field	158
5.7	The Normal Zeeman Effect	160
6	Perturbation methods
6.1	Perturbed steady states	164
6.2	The degenerate case	168
6.3	The Zeeman Effect	170
6.4	Spin-orbit coupling	172
6.5	The Stark Effect	176
6.6	Transition theory	180
6.7	Time-independent perturbing force	182
6.8	Scattering by a centre of force	185
6.9	Oscillatory perturbing force	188
6.10	Perturbation of a charged particle by an electromagnetic wave	189
6.11	The electric dipole approximation	195
6.12	Selection rules for the harmonic oscillator	198
6.13	Selection rules for a particle moving in a spherically symmetric field	199
7	Dirac's relativistic equation
7.1	Dirac's Wave Equation	201
7.2	Conservation of probability	206
7.3	Magnetic moment of a particle	207
7.4	Spin	208
7.5	Free motion of a Dirac particle. The positron	211
7.6	A complete set of commuting operators for a Dirac particle in a central field	213
7.7	Simultaneous angular momentum eigenfunctions for a Dirac particle in a central field	216
7.8	Eigenfunctions of the Dirac Hamiltonian	219
7.9	Energy levels of the hydrogen atom	222
Appendix A	Solution of a tensor equation	226
Appendix B	Theorem on series	228
Appendix C	Hermite polynomials	229
Appendix D	Legendre functions	233
Appendix E	Laguerre polynomials	238
Appendix F	Motion of a charge in an electromagnetic field	244
Appendix G	Schlaff's method	246
	Exercises	249
	Bibliography	275
	Index	276