A Condensed Course of Quantum Mechanics

Pavel Cejnar

A Condensed Course of Quantum Mechanics

Pavel Cejnar

Distributed for the Karolinum Press, Charles University

Translated by Daniel Morgan
216 pages | 136 mathematical drawings | 7 x 9
Paper $20.00 ISBN: 9788024623214 Published May 2014 Not for sale in the Czech Republic and the Slovak Republic
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.

Rough guide to notation


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 . . . .

3.1 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

4.1 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  

5.1 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 . . .

6.1 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  

7.1 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
Review Quotes
Jean-Paul Blaizot, IPhT, Paris
“I enjoyed reading this book. What I found particularly interesting was the style of the presentation, the original and excellent selection of topics, and the numerous brief historical remarks. The text is succinct but no superficial: the deeper one immerses in reading, one finds even more inspiring remarks. The reader is alerted to the subtleties of the mathematical formulation of quantum mechanics, without getting lost in unnecessary formalism.”
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