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Claude Cohen-Tannoudji
Quantum Mechanics, Volume 3
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Оглавление
Страница 1
Table of Contents
List of Illustrations
Guide
Pages
QUANTUM MECHANICS Volume III Fermions, Bosons, Photons, Correlations, and Entanglement
Страница 7
Страница 8
Страница 9
Страница 10
Страница 11
Chapter XV
Creation and annihilation operators for identical particles
Introduction
A. General formalism
A-1. Fock states and Fock space
A-1-a. Fock states for identical bosons
A-1-b. Fock states for identical fermions
A-1-c. Fock space
A-2. Creation operators
a
†
A-2-a. Bosons
A-2-b. Fermions
A-3. Annihilation operators
a
A-3-a. Bosons
A-3-b. Fermions
A-4. Occupation number operators (bosons and fermions)
A-5. Commutation and anticommutation relations
A-5-a. Bosons: commutation relations
A-5-b. Fermions: anticommutation relations
A-5-c. Common relations for bosons and fermions
A-6. Change of basis
B. One-particle symmetric operators
B-1. Definition
B-2. Expression in terms of the operators
a
and
a
†
B-2-a. Action of
F
(N)
on a ket with
N
particles
B-2-b. Expression valid in the entire Fock space
B-3. Examples
B-4. Single particle density operator
C. Two-particle operators
C-1. Definition
C-2. A simple case: factorization
C-3. General case
C-4. Two-particle reduced density operator
C-5. Physical discussion; consequences of the exchange
C-5-a. Two terms in the matrix elements
C-5-b. Particle interaction energy, the direct and exchange terms
Conclusion
COMPLEMENTS OF
CHAPTER XV
, READER’S GUIDE
Страница 48
Страница 49
Complement B
XV
Ideal gas in thermal equilibrium; quantum distribution functions
1. Grand canonical description of a system without interactions
1-a. Density operator
1-b. Grand canonical partition function, grand potential
α. Fermions
β. Bosons
2. Average values of symmetric one-particle operators
2-a. Fermion distribution function
2-b. Boson distribution function
2-c. Common expression
2-d. Characteristics of Fermi-Dirac and Bose-Einstein distributions
3. Two-particle operators
3-a. Fermions
3-b. Bosons
α. Average value calculation
β. Physical discussion: occupation number fluctuations
3-c. Common expression
4. Total number of particles
4-a. Fermions
4-b. Bosons
α. Non-condensed bosons
β. Condensed bosons
5. Equation of state, pressure
5-a. Fermions
5-b. Bosons
Страница 75
Complement C
XV
Condensed boson system, Gross-Pitaevskii equation
1. Notation, variational ket
1-a. Hamiltonian
1-b. Choice of the variational ket (or trial ket)
2. First approach
2-a. Trial wave function for spinless bosons, average energy
2-b. Variational optimization
α. Variation of the wave function
β. Stationary condition: Gross-Pitaevskii equation
ϒ. Zero-range potential
δ. Other normalization
3. Generalization, Dirac notation
3-a. Average energy
3-b. Energy minimization
3-c. Gross-Pitaevskii equation
4. Physical discussion
4-a. Energy and chemical potential
4-b. Healing length
4-c. Another trial ket: fragmentation of the condensate
Страница 95
Complement D
XV
Time-dependent Gross-Pitaevskii equation
1. Time evolution
1-a. Functional variation
1-b. Variational computation: the time-dependent Gross-Pitaevskii equation
1-c. Phonons and Bogolubov spectrum
α. Excitation propagation
β. Discussion
2. Hydrodynamic analogy
2-a. Probability current
2-b. Velocity evolution
3. Metastable currents, superfluidity
3-a. Toroidal geometry, quantization of the circulation, vortex
3-b. Repulsive potential barrier between states of different
l
α. A simple geometry
α. Other geometries, different relaxation channels
3-c. Critical velocity, metastable flow
3-d. Generalization; topological aspects
Страница 113
Complement E
XV
Fermion system, Hartree-Fock approximation
Introduction
1. Foundation of the method
1-a. Trial family and Hamiltonian
1-b. Energy average value
α. Kinetic energy
β. Potential energy
ϒ. Interaction energy
1-c. Optimization of the variational wave function
1-d. Equivalent formulation for the average energy stationarity
1-e. Variational energy
1-f. Hartree-Fock equations
2. Generalization: operator method
2-a. Average energy
α. Kinetic and external potential energy
β. Average interaction energy, Hartree-Fock potential operator
ϒ. Role of the one-particle reduced density operator
2-b. Optimization of the one-particle density operator
2-c. Mean field operator
2-d. Hartree-Fock equations for electrons
2-e. Discussion
Страница 135
Complement F
XV
Fermions, time-dependent Hartree-Fock approximation
1. Variational ket and notation
2. Variational method
2-a. Definition of a functional
2-b. Stationarity
2-c. Particular case of a time-independent Hamiltonian
3. Computing the optimizer
3-a. Average energy
3-b. Hartree-Fock potential
3-c. Time derivative
3-d. Functional value
4. Equations of motion
4-a. Time-dependent Hartree-Fock equations
4-b. Particles in a single spin state
4-c. Discussion
Страница 151
Complement G
XV
Fermions or Bosons: Mean field thermal equilibrium
1. Variational principle
1-a. Notation, statement of the problem
1-b. A useful inequality
1-c. Minimization of the thermodynamic potential
2. Approximation for the equilibrium density operator
2-a. Trial density operators
2-b. Partition function, distributions
α. Variational partition function
β. One particle, reduced density operator
ϒ. Two particles, distribution functions
2-c. Variational grand potential
2-d. Optimization
α. Variations of the eigenstates
β. Variation of the energies
3. Temperature dependent mean field equations
3-a. Form of the equations
3-b. Properties and limits of the equations
α. Using the equations
β. Validity limit
3-c. Differences with the zero-temperature Hartree-Fock equations (fermions)
3-d. Zero-temperature limit (fermions)
3-e. Wave function equations
Conclusion
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