Condensed Course of Quantum Mechanics

Pavel Cejnar is professor at the Institute of Particle and Nuclear Physics at Charles University, Prague.

• PrefaceRough guide to notationINTRODUCTION1. FORMALISM - 2. SIMPLE SYSTEMS1.1 Space of quantum statesHilbert space. Rigged Hilbert spaceDirac notationSum & product of spaces2.1 Examples of quantum Hilbert spacesSingle structureless particle with spin 0 or 12 distinguishable/indistinguishable particles. Bosons & fermionsEnsembles of N > 2 particles 1.2 Representation of observables Observables as Hermitian operators. Basic properties Eigenvalues & eigenvectors in finite & infinite dimensionDiscrete & continuous spectrum. Spectral decomposition2.2 Examples of quantum operatorsSpin-1/2 operatorsCoordinate & 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 observablesCompatible 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 particle1.4 Representation of physical transformationsProperties of unitary operators Canonical & symmetry transformations Basics of group theory 2.4 Fundamental spatio-temporal symmetriesSpace translationSpace rotationSpace inversionTime translation & reversal. Galilean transformationsSymmetry & degeneracy1.5 Unitary evolution of quantum systemsNonstationary Schrödinger equation. Flow. Continuity equation.Conservation laws & symmetriesEnergy x time uncertainty. (Non)exponential decayHamiltonians depending on time. Dyson seriesSchrodinger, Heisenberg & Dirac descriptionGreen operator. Single-particle propagator2.5 Examples of quantum evolution Two-level system Free particle Coherent states in harmonic oscillator Spin in rotating magnetic field1.6 Quantum measurement State vector reduction & consequences EPR situation. Interpretation problems2.6 Implications & applications of quantum measurement . .Paradoxes of quantum measurement Applications of quantum measurementHidden variables. Bell inequalities. Nonlocality 1.7 Quantum statistical physicsPure and mixed states. Density operator Entropy. Canonical ensemble Wigner distribution functionDensity operator for open systems Evolution of density operator: closed & open systems2.7 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 CORRESPONDENCE3.1 Classical limit of quantum mechanicsThe 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 trajectoriesApplication to the Aharonov-Bohm effectApplication to the density of states4. ANGULAR MOMENTUM4.1 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 TECHNIQUES5.1 Variational methodDynamical & stationary variational principle. Ritz method5.2 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 THEORY6.1 Elementary description of elastic scatteringScattering by fixed potential. Cross sectionTwo-body problem. Center-of-mass systemEffect of particle indistinguishability in cross section ....6.2 Perturbative approach the scattering problem . . . .Lippmann-Schwinger equationBorn series for scattering amplitude 6.3 Method of partial wavesExpression of elastic scattering in terms of spherical waves .Inclusion of inelastic scattering Low-energy & resonance scattering 7. MANY-BODY SYSTEMS7.1 Formalism of particle creation/annihilation operatorsHilbert space of bosons & fermions Bosonic & fermionic creation/annihilation operatorsOperators in bosonic & fermionic N-particle spacesQuantization of electromagnetic field7.2 Many-body techniquesFermionic mean field & Hartree-Fock method Bosonic condensates & Hartree-Bose method Pairing & BCS method Quantum gases