# A Condensed Course of Quantum Mechanics

9788024623214

9788024623498

Distributed for Karolinum Press, Charles University

# A Condensed Course of Quantum Mechanics

This book represents a concise summary of nonrelativistic quantum mechanics for physics students at the university level. The text covers essential topics, from general mathematical formalism to specific applications. The formulation of quantum theory is explained and supported with illustrations of the general concepts of elementary quantum systems. In addition to traditional topics of nonrelativistic quantum mechanics—including single-particle dynamics, symmetries, semiclassical and perturbative approximations, density-matrix formalism, scattering theory, and the theory of angular momentum—the book also covers modern issues, among them quantum entanglement, decoherence, measurement, nonlocality, and quantum information. Historical context and chronology of basic achievements is also outlined in explanatory notes. Ideal as a supplement to classroom lectures, the book can also serve as a compact and comprehensible refresher of elementary quantum theory for more advanced students.

## Reviews

## Table of Contents

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

## Be the first to know

Get the latest updates on new releases, special offers, and media highlights when you subscribe to our email lists!