Preface Rough guide to notation INTRODUCTION 1. FORMALISM - 2. SIMPLE SYSTEMS 1.1 Space of quantum states Hilbert space. Rigged Hilbert space Dirac notation Sum & product of spaces 2.1 Examples of quantum Hilbert spaces Single structureless particle with spin 0 or 1 2 distinguishable/indistinguishable particles. Bosons & fermions Ensembles of N > 2 particles 1.2 Representation of observables Observables as Hermitian operators.
Basic properties Eigenvalues & eigenvectors in finite & infinite dimension Discrete & continuous spectrum. Spectral decomposition 2.2 Examples of quantum operators Spin-1/2 operators Coordinate & momentum Hamiltonian of free particle & particle in potential Orbital angular momentum. Isotropic Hamiltonians Hamiltonian of a particle in electromagnetic field 1.3 Compatible and incompatible observables Compatible observables. Complete set Incompatible observables. Uncertainty relation Analogy with Poisson brackets Equivalent representations 2.3 Examples of commuting & noncommuting operators .
Coordinate, momentum & associated representations Angular momentum components Complete sets of commuting operators for structureless particle 1.4 Representation of physical transformations Properties of unitary operators Canonical & symmetry transformations Basics of group theory 2.4 Fundamental spatio-temporal symmetries Space translation Space rotation Space inversion Time translation & reversal. Galilean transformations Symmetry & degeneracy 1.5 Unitary evolution of quantum systems Nonstationary Schrödinger equation. Flow. Continuity equation. Conservation laws & symmetries Energy x time uncertainty.
(Non)exponential decay Hamiltonians depending on time. Dyson series Schrodinger, Heisenberg & Dirac description Green operator. Single-particle propagator 2.5 Examples of quantum evolution Two-level system Free particle Coherent states in harmonic oscillator Spin in rotating magnetic field 1.6 Quantum measurement State vector reduction & consequences EPR situation. Interpretation problems 2.6 Implications & applications of quantum measurement . Paradoxes of quantum measurement Applications of quantum measurement Hidden variables.
Bell inequalities. Nonlocality 1.7 Quantum statistical physics Pure and mixed states. Density operator Entropy. Canonical ensemble Wigner distribution function Density operator for open systems Evolution of density operator: closed & open systems 2.7 Examples of statistical description Harmonic oscillator at nonzero temperature Coherent superposition vs. statistical mixture Density operator and decoherence for a two-state system . 3.
QUANTUM-CLASSICAL CORRESPONDENCE 3.1 Classical limit of quantum mechanics The limit h -> 0 Ehrenfest theorem. Role of decoherence 3.2 WKB approximation Classical Hamilton-Jacobi theory WKB equations & interpretation Quasiclassical approximation 3.3 Feynman integral Formulation of quantum mechanics in terms of trajectories Application to the Aharonov-Bohm effect Application to the density of states 4. ANGULAR MOMENTUM 4.1 General features of angular momentum Eigenvalues and ladder operators Addition of two angular momenta Addition of three angular momenta 4.2 Irreducible tensor operators Euler angles.
Wigner functions. Rotation group irreps . Spherical tensors. Wigner-Eckart theorem 5. APPROXIMATION TECHNIQUES 5.1 Variational method Dynamical & stationary variational principle. Ritz method 5.2 Stationary perturbation method General setup & equations Nondegenerate case Degenerate case Application in atomic physics Application to level dynamics Driven systems.
Adiabatic approximation 5.3 Nonstationary perturbation method General formalism Step perturbation Exponential & periodic perturbations Application to stimulated electromagnetic transitions . 6. SCATTERING THEORY 6.1 Elementary description of elastic scattering Scattering by fixed potential. Cross section Two-body problem. Center-of-mass system Effect of particle indistinguishability in cross section . 6.
2 Perturbative approach the scattering problem . Lippmann-Schwinger equation Born series for scattering amplitude 6.3 Method of partial waves Expression of elastic scattering in terms of spherical waves . Inclusion of inelastic scattering Low-energy & resonance scattering 7. MANY-BODY SYSTEMS 7.1 Formalism of particle creation/annihilation operators Hilbert space of bosons & fermions Bosonic & fermionic creation/annihilation operators Operators in bosonic & fermionic N-particle spaces Quantization of electromagnetic field 7.2 Many-body techniques Fermionic mean field & Hartree-Fock method Bosonic condensates & Hartree-Bose method Pairing & BCS method Quantum gases.