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Kniha: A Condensed Course of Quantum Mechanics - Pavel Cejnar

A Condensed Course of Quantum Mechanics

Kniha: A Condensed Course of Quantum Mechanics

This book represents a concise summary of non-relativistic quantum mechanics on the level suitable for university students of physics. It covers, perhaps even slightly exceeds, a one-year course of ... (celý popis)
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Nakladatelství: » Karolinum
Médium / forma: Tištěná kniha
Rok vydání: 09.02.2018
Úprava: 1 online zdroj (209 pages) : illustrations
Jazyk: anglicky
Vazba: kniha, brožovaná vazba
ISBN: 9788024623214
EAN: 9788024623214
Ukázka: » zobrazit ukázku

This book represents a concise summary of non-relativistic quantum mechanics on the level suitable for university students of physics. It covers, perhaps even slightly exceeds, a one-year course of about 50 lectures, requiring basic knowledge of calculus, algebra, classical mechanics and a bit of motivation for the quantum adventure.
The exposition is succinct, with minimal narration, but with a maximum of explicit and hierarchically structured mathematical derivations. The text covers all essential topics of university courses of quantum mechanics - from general mathematical formalism to specific applications. The formulation of quantum theory is accompanied by illustrations of the general concepts of elementary quantum systems. Some subtleties of mathematical foundations are overviewed, but the formalism is used in an accessible, intuitive way. Besides the traditional topics of non-relativistic quantum mechanics, such as single-particle dynamics, symmetries, semiclassical and perturbative approximations, density-matrix formalism, scattering theory, theory of angular momentum, description of many-particle systems - the course also touches upon some modern issues, including quantum entanglement, decoherence, measurement, nonlocality, and quantum information. Historical context and chronology of basic achievements is outlined in brief remarks. The book is intended for beginners as a supplement to lectures, however, it may also be used by more advanced students as a compact and comprehensible overview of elementary quantum theory.

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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.Singlearticle 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 & fermionicNarticle 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

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