A Condensed Course of Quantum Mechanics
Distributed for Karolinum Press, Charles University
Translated by Daniel Morgan
216 pages

136 mathematical drawings

7 x 9
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 singleparticle dynamics, symmetries, semiclassical and perturbative approximations, densitymatrix 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.
Contents
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
Spin1/2 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 spatiotemporal 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. Singleparticle propagator
2.5 Examples of quantum evolution
Twolevel 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 twostate system . . . .
3. QUANTUMCLASSICAL CORRESPONDENCE
3.1 Classical limit of quantum mechanics
The limit h > 0
Ehrenfest theorem. Role of decoherence
3.2 WKB approximation
Classical HamiltonJacobi theory
WKB equations & interpretation
Quasiclassical approximation
3.3 Feynman integral
Formulation of quantum mechanics in terms of trajectories
Application to the AharonovBohm 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. WignerEckart 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
Twobody problem. Centerofmass system
Effect of particle indistinguishability in cross section ....
6.2 Perturbative approach the scattering problem . . . .
LippmannSchwinger 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
Lowenergy & resonance scattering
7. MANYBODY SYSTEMS
7.1 Formalism of particle creation/annihilation operators
Hilbert space of bosons & fermions
Bosonic & fermionic creation/annihilation operators
Operators in bosonic & fermionic Nparticle spaces
Quantization of electromagnetic field
7.2 Manybody techniques
Fermionic mean field & HartreeFock method
Bosonic condensates & HartreeBose method
Pairing & BCS method
Quantum gases
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
Spin1/2 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 spatiotemporal 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. Singleparticle propagator
2.5 Examples of quantum evolution
Twolevel 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 twostate system . . . .
3. QUANTUMCLASSICAL CORRESPONDENCE
3.1 Classical limit of quantum mechanics
The limit h > 0
Ehrenfest theorem. Role of decoherence
3.2 WKB approximation
Classical HamiltonJacobi theory
WKB equations & interpretation
Quasiclassical approximation
3.3 Feynman integral
Formulation of quantum mechanics in terms of trajectories
Application to the AharonovBohm 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. WignerEckart 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
Twobody problem. Centerofmass system
Effect of particle indistinguishability in cross section ....
6.2 Perturbative approach the scattering problem . . . .
LippmannSchwinger 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
Lowenergy & resonance scattering
7. MANYBODY SYSTEMS
7.1 Formalism of particle creation/annihilation operators
Hilbert space of bosons & fermions
Bosonic & fermionic creation/annihilation operators
Operators in bosonic & fermionic Nparticle spaces
Quantization of electromagnetic field
7.2 Manybody techniques
Fermionic mean field & HartreeFock method
Bosonic condensates & HartreeBose method
Pairing & BCS method
Quantum gases
Review Quotes
JeanPaul 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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Physical Sciences: Experimental and Applied Physics
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