Оглавление
Claude Cohen-Tannoudji. Quantum Mechanics, Volume 3
Table of Contents
List of Illustrations
Guide
Pages
QUANTUM MECHANICS Volume III. Fermions, Bosons, Photons, Correlations, and Entanglement
Directions for Use
Foreword
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
Complement AXV Particles and holes
1. Ground state of a non-interacting fermion gas
2. New definition for the creation and annihilation operators
3. Vacuum excitations
Complement BXV 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
Complement CXV 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
Complement DXV 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
Complement EXV 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
Complement FXV 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
Complement GXV 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
Complement HXV Applications of the mean field method for non-zero temperature (fermions and bosons)
1. Hartree-Fock for non-zero temperature, a brief review
2. Homogeneous system
2-a. Calculation of the energies
2-b. Quasi-particules
3. Spontaneous magnetism of repulsive fermions
3-a. A simple model
α. Energy of the quasi-particles; spin state populations
β. Change of variables
3-b. Resolution of the equations by graphical iteration
α. Iterations of a function
β. Form of the function F(1)
γ. Form of the functionF(2)
δ. Influence of the coupling parameter on the fixed points
3-c. Physical discussion
4. Bosons: equation of state, attractive instability
4-a. Repulsive bosons
4-b. Attractive bosons
Chapter XVI. Field operator
Introduction
A. Definition of the field operator
A-1. Definition
A-1-a. Spinless particles
A-1-b. Particles with spin
A-2. Commutation and anticommutation relations
A-2-a. Spinless particles
A-2-b. Particles with spin
B. Symmetric operators
B-1. General expression
B-2. Simple examples
B-3. Field spatial correlation functions
B-3-a. Two-point correlation functions
B-3-b. Higher order correlation functions
B-4. Hamiltonian operator
C. Time evolution of the field operator (Heisenberg picture)
C-1. Contribution of the kinetic energy
C-2. Contribution of the potential energy
C-3. Contribution of the interaction energy
C-4. Global evolution
D. Relation to field quantization
COMPLEMENTS OF Chapter XVI, READER’S GUIDE
Complement AXVI Spatial correlations in an ideal gas of bosons or fermions
1. System in a Fock state
1-a. Two-point correlations
1-b. Four-point correlations
2. Fermions in the ground state
2-a. Two-point correlations
2-b. Correlations between two particles
3. Bosons in a Fock state
3-a. Ground state
3-b. Fragmented state
3-c. Other states
Complement BXVI Spatio-temporal correlation functions, Green’s functions
1. Green’s functions in ordinary space
1-a. Spatio-temporal correlation functions
1-b. Two- and four-point Green’s functions
1-c. An example, the ideal gas
2. Fourier transforms
2-a. General definition
2-b. Ideal gas example
2-c. General expression in the presence of interactions
2-d. Discussion
3. Spectral function, sum rule
3-a. Expression of the one-particle correlation functions
3-b. Sum rule
3-c. Expression of various physical quantities
Complement CXVI Wick’s theorem
1. Demonstration of the theorem
1-a. Statement of the problem
1-b. Recurrence relation
1-c. Contractions
1-d. Statement of the theorem
2. Applications: correlation functions for an ideal gas
2-a. First order correlation function
2-b. Second order correlation functions
2-c. Higher order correlation functions
Chapter XVII. Paired states of identical particles
Introduction
A. Creation and annihilation operators of a pair of particles
A-1. Spinless particles, or particles in the same spin state
A-2. Particles in different spin states
B. Building paired states
B-1. Well determined particle number
B-1-a. Particles in the same spin state
B-1-b. Fermions in a singlet state
B-1-c. Consequences of the symmetrization
B-2. Undetermined particle number
B-2-a. Particles in the same spin state
B-2-b. Fermions in a singlet state
B-3. Pairs of particles and pairs of individual states
C. Properties of the kets characterizing the paired states
C-1. Normalization
C-1-a. Fermions in a singlet state
C-1-b. Bosons in the same spin state
C-2. Average value and root mean square deviation of particle number
C-2-a. Fermions in a singlet state
C-2-b. Bosons in the same spin state
C-3. “Anomalous” average values
C-3-a. Fermions in a singlet state
C-3-b. Bosons in the same spin state
D. Correlations between particles, pair wave function
D-1. Particles in the same spin state
D-1-a. Simplifications due to pairing
D-1-b. Expression of the correlation function
D-2. Fermions in a singlet state
E. Paired states as a quasi-particle vacuum; Bogolubov-Valatin transformations
E-1. Transformation of the creation and annihilation operators
E-2. Effect on the kets
E-2-a. Fermions in a singlet state
E-2-b. Bosons in the same spin state
E-3. Basis of excited states, quasi-particles
Conclusion
COMPLEMENTS OF Chapter XVII, READER’S GUIDE
Complement AXVII Pair field operator for identical particles
1. Pair creation and annihilation operators
1-a. Particles in the same spin state
α. Expression of in terms of the particle field operator
β. Pair field
ϒ. Inversion; expression for the interaction energy
1-b. Pairs in a singlet spin state
α. Expression ofin terms of the particle field operator
β. Pair field
2. Average values in a paired state
2-a. Average value of a field operator; pair wave function, and order parameter
α. Particles in the same spin state
β. Singlet pairs
2-b. Average value of a product of two field operators; factorization of the order parameter
α. Particles in the same spin state
Computation of the average value
β. Singlet pairs
ϒ. Link with Bose-Einstein condensation of pairs
2-c. Application to the computation of the correlation function (singlet pairs)
α. Inversion of the relation between fields
β. 4-point correlation function
3. Commutation relations of field operators
3-a. Particles in the same spin state
α. Commutation relations of the AK,n
3-b. Singlet pairs
α. Commutation relations of the AK,n
β. Commutation relations of pair field operators
Conclusion
Complement BXVII Average energy in a paired state
1. Using states that are not eigenstates of the total particle number
1-a. Computation of the average values
1-b. A good approximation
2. Hamiltonian
2-a. Operator expression
2-b. Simplifications due to pairing
3. Spin 1/2 fermions in a singlet state
3-a. Different contributions to the energy
α. Kinetic energy
β. Interaction energy
3-b. Total energy
4. Spinless bosons
4-a. Choice of the variational state
4-b. Different contributions to the energy
α. Kinetic energy
β. Interaction energy
4-c. Total energy
Complement CXVII Fermion pairing, BCS theory
1. Optimization of the energy
1-a. Function to be optimized
1-b. Cancelling the total variation
1-c. Short-range potential, study of the gap
α. Self-consistency condition and divergences
β. Calculation of the gap
2. Distribution functions, correlations
2-a. One-particle distribution
α. Momentum space
β. Position space
2-b. Two-particle distribution, internal pair wave function
α. Momentum space, peak in the distribution
β. Position space, correlations described by the pair wave functions
2-c. Properties of the pair wave function, coherence length
α. Form of the pair wave function
β. Norm of the pair wave function
γ. Link to the interaction energy
δ. Non-diagonal order
3. Physical discussion
3-a. Modification of the Fermi surface and phase locking
α. Compromise between the kinetic and potential energies
β. Phase locking and cooperative effects
3-b. Gain in energy
3-c Non-perturbative character of the BCS theory
4. Excited states
4-a. Bogolubov-Valatin transformation
4-b. Broken pairs and excited pairs
4-c. Stationarity of the energies
α. Broken pair
β. Excited pair
4-d. Excitation energies
α. Broken pair
β. Excited pair
γ. Spectrum of the elementary excitations
Conclusion
Complement DXVII Cooper pairs
1. Cooper model
2. State vector and Hamiltonian
3. Solution of the eigenvalue equation
4. Calculation of the binding energy for a simple case
Complement EXVII Condensed repulsive bosons
1. Variational state, energy
1-a. Variational ket
1-b. Total energy
1-c. N0 ≫ Ne approximation
2. Optimization
2-a. Stationarity conditions
2-b. Solution of the equations
3. Properties of the ground state
3-a. Particle number, quantum depletion
3-b. Energy
3-c. Phase locking; comparison with the BCS mechanism
3-d. Correlation functions
4. Bogolubov operator method
4-a. Variational space, restriction on the Hamiltonian
4-b. Bogolubov Hamiltonian
4-c. Constructing a basis of excited states, quasi-particles
Chapter XVIII. Review of classical electrodynamics
A. Classical electrodynamics. A-1. Basic equations and relations. A-1-a. Maxwell’s equations
A-1-b. Lorentz Equations
A-1-c. Constants of motion
A-1-d. Scalar and vector potentials: gauge transformations
A-2. Description in the reciprocal space
A-2-a. Spatial Fourier transforms
A-2-b. Maxwell’s equations in reciprocal space
A-2-c. Longitudinal electric and magnetic fields
A-2-d. Time evolution of the transverse fields
A-2-e. Potentials
A-2-f. Coulomb gauge
A-3. Elimination of the longitudinal fields from the expression of the physical quantities
A-3-a. Total energy
A-3-b. Total momentum
A-3-c. Total angular momentum
B. Describing the transverse field as an ensemble of harmonic oscillators. B-1. Brief review of the one-dimensional harmonic oscillator
B-2. Normal variables for the transverse field. B-2-a. Vibration eigenmodes of the free transverse field
B-2-b. Definition of the normal variables, free field case
B-2-c. Equation of motion for the normal variables in the presence of sources
B-2-d. Expression of the physical quantities in terms of the normal variables
B-3. Discrete modes in a box
B-4. Generalization of the mode concept
COMPLEMENT OF CHAPTER XVIII, READER’S GUIDE
Complement AXVIII Lagrangian formulation of electrodynamics
Introduction
1. Lagrangian with several types of variables. 1-a. Lagrangian formalism with discrete and real variables
1-b. Extension to complex variables
1-c. Lagrangian with continuous variables
2. Application to the free radiation field
2-a. Lagrangian densities in real and reciprocal spaces
2-b. Lagrange’s equations
2-c. Conjugate momentum of the transverse potential vector
2-d. Hamiltonian; Hamilton-Jacobi equations
2-e. Field commutation relations
2-f. Creation and annihilation operators
2-g. Discrete momentum variables
3. Lagrangian of the global system field + interacting particles
3-a. Choice for the Lagrangian
3-b. Lagrange’s equations
3-c. Conjugate momenta
3-d. Hamiltonian
3-e. Commutation relations
Chapter XIX. Quantization of electromagnetic radiation
Introduction
A. Quantization of the radiation in the Coulomb gauge. A-1. Quantization rules
A-2. Radiation contained in a box
A-3. Heisenberg equations. A-3-a. Heisenberg equations for massive particles
A-3-b. Heisenberg equations for fields
B. Photons, elementary excitations of the free quantum field
B-1. Fock space of the free quantum field
B-2. Corpuscular interpretation of states with fixed total energy and momentum
B-3. Several examples of quantum radiation states
B-3-a. Photon vacuum
B-3-b. Field quasi-classical states
B-3-c. Single photon state
C. Description of the interactions. C-1. Interaction Hamiltonian
C-2. Interaction with an atom. External and internal variables
C-3. Long wavelength approximation
C-4. Electric dipole Hamiltonian
C-4-a. Electric dipole Hamiltonian for a classical field
C-4-b. Electric dipole Hamiltonian for a quantum field
C-5. Matrix elements of the interaction Hamiltonian; selection rules
COMPLEMENTS OF CHAPTER XIX, READER’S GUIDE
Complement AXIX Momentum exchange between atoms and photons
1. Recoil of a free atom absorbing or emitting a photon
1-a. Conservation laws
1-b. Doppler effect, Doppler width
1-c. Recoil energy
1-d. Radiation pressure force in a plane wave
2. Applications of the radiation pressure force: slowing and cooling atoms
2-a. Deceleration of an atomic beam
2-b. Doppler laser cooling of free atoms
2-c. Magneto-optical trap
3. Blocking recoil through spatial confinement
3-a. Assumptions concerning the external trapping potential
3-b. Intensities of the vibrational lines
3-c. Effect of the confinement on the absorption and emission spectra
3-d. Case of a one-dimensional harmonic potential
3-e. Mössbauer effect
4. Recoil suppression in certain multi-photon processes
Complement BXIX Angular momentum of radiation
Introduction
1. Quantum average value of angular momentum for a spin 1 particle
1-a. Wave function, spin operator
1-b. Average value of the spin angular momentum
1-c. Average value of the orbital angular momentum
2. Angular momentum of free classical radiation as a function of normal variables
2-a. Calculation in position space
2-b. Reciprocal space
2-c. Difference between the angular momenta of massive particles and of radiation
3. Discussion. 3-a. Spin angular momentum of radiation
3-b. Experimental evidence of the radiation spin angular momentum
3-c. Orbital angular momentum of radiation
α. Multipolar waves
β. Beams with cylindrical symmetry around one axis
Complement CXIX Angular momentum exchange between atoms and photons
Introduction
1. Transferring spin angular momentum to internal atomic variables. 1-a. Electric dipole transitions
1-b. Polarization selection rules
1-c. Conservation of total angular momentum
2. Optical methods
2-a. Double resonance method
2-b. Optical pumping
2-c. Original features of these methods
3. Transferring orbital angular momentum to external atomic variables. 3-a. Laguerre-Gaussian beams
3-b. Field expansion on Laguerre-Gaussian modes
Chapter XX. Absorption, emission and scattering of photons by atoms
Introduction
A. A basic tool: the evolution operator
A-1. General properties
A-2. Interaction picture
A-3. Positive and negative frequency components of the field
B. Photon absorption between two discrete atomic levels
B-1. Monochromatic radiation. B-1-a. Probability amplitude (absorption)
B-1-b. Energy conservation
B-1-c. Limits of the perturbative treatment
B-2. Non-monochromatic radiation
B-2-a. Absorption of a broadband radiation
α. Transition probability
β Excitation spectrum
γ. Probability per unit time
B-2-b. A specific case: isotropic radiation
C. Stimulated and spontaneous emissions. C-1. Emission rate
C-2. Stimulated emission
C-3. Spontaneous emission
C-4. Einstein coefficients and Planck’s law
D. Role of correlation functions in one-photon processes
D-1. Absorption process
D-2. Emission process
E. Photon scattering by an atom
E-1. Elastic scattering
E-1-a. Two possible types of relay state
E-1-b. Computation of the scattering amplitude
E-1-c. Semiclassical interpretation
E-1-d. Rayleigh scattering
E-2. Resonant scattering
E-3. Inelastic scattering – Raman scattering
E-3-a. Differences with elastic scattering
E-3-b. Scattering amplitude
E-3-c. Semiclassical interpretation
E-3-d. Stimulated Raman scattering. Raman laser
COMPLEMENTS OF CHAPTER XX, READER’S GUIDE
Complement AXX A multiphoton process: two-photon absorption
1. Monochromatic radiation
2. Non-monochromatic radiation
2-a. Probability amplitude, probability
2-b. Probability per unit time when the radiation is in a Fock state
3. Discussion
3-a. Conservation laws. α. Total energy conservation;
β. Total momentum conservation
γ. Conservation of total angular momentum and parity
3-b. Case where the relay state becomes resonant for one-photon absorption
Complement BXX Photoionization
1. Brief review of the photoelectric effect
1-a. Interpretation in terms of photons
1-b. Photoionization of an atom
2. Computation of photoionization rates
2-a. A single atom in monochromatic radiation
2-b. Stationary non-monochromatic radiation
α. Field and atomic dipole correlation functions
β. Photoionization rate
2-c. Non-stationary and non-monochromatic radiation
2-d. Correlations between photoionization rates of two detector atoms
3. Is a quantum treatment of radiation necessary to describe photoionization?
3-a. Experiments with a single photodetector atom
3-b. Experiments with two photodetector atoms
α. Situations where a semiclassical treatment is adequate
β. Situations requiring a quantum radiation treatment
4. Two-photon photoionization. 4-a. Differences with the one-photon photoionization
4-b. Photoionization rate
4-c. Importance of fluctuations in the radiation intensity
5. Tunnel ionization by intense laser fields
Complement CXX Two-level atom in a monochromatic field. Dressed-atom method
1. Brief description of the dressed-atom method
1-a. State energies of the atom + photon system in the absence of coupling
1-b. Coupling matrix elements
1-c. Outline of the dressed-atom method
α. Dressed states and energies
β. Rabi oscillation
1-d. Physical meaning of photon number
1-e. Effects of spontaneous emission
2. Weak coupling domain
2-a. Eigenvalues and eigenvectors of the effective Hamiltonian
2-b. Light shifts and radiative broadening
2-c. Dependence on incident intensity and detuning
2-d. Semiclassical interpretation in the weak coupling domain
2-e. Some extensions
α. Non-monochromatic incident radiation
β. Degenerate ground state
3. Strong coupling domain
3-a. Eigenvalues and eigenvectors of the effective Hamiltonian
3-b. Variation of dressed state energies with detuning
3-c. Fluorescence triplet
3-d. Temporal correlations between fluorescent photons
α. Radiative cascade of the dressed atom
β. High spectral resolution photodetection
γ. Photodetection with high temporal resolution
4. Modifications of the field. Dispersion and absorption
4-a. Atom in a cavity
4-b. Frequency shift of the field in the presence of the atom
4-c. Field absorption
Complement DXX Light shifts: a tool for manipulating atoms and fields
1. Dipole forces and laser trapping
2. Mirrors for atoms
3. Optical lattices
4. Sub-Doppler cooling. Sisyphus effect
4-a. Laser configurations with space-dependent polarization
4-b. Atomic transition
4-c. Light shifts
4-d. Optical pumping
4-e. Sisyphus effect
5. Non-destructive detection of a photon
Conclusion
Complement EXX Detection of one- or two-photon wave packets, interference
Introduction
1. One-photon wave packet, photodetection probability
1-a. Photoionization of a broadband detector
1-b. Detection probability amplitude
1-c. Temporal variation of the signal
2. One- or two-photon interference signals
2-a. How should one compute photon interference?
2-b. Interference signal for a one-photon wave packet in two modes
α. One-photon wave packets, after passage through a two-slit screen
β. Calculation of the single photodetection signal
γ. Discussion
2-c. Interference signals for a product of two one-photon wave packets
α. State vector for the two photons
β. Single photodetection signal wI(r, t)
γ. Double photodetection signal WII(rB, tB, rA, tA)
3. Absorption amplitude of a photon by an atom
3-a. Computation of the amplitude
3-b. Properties of that amplitude
4. Scattering of a wave packet
4-a. Absorption amplitude by atom B of the photon scattered by atom A
4-b. Wave packet scattered by atom A
α. Summation over all possible directions of the scattered photon
β. Summation over the energies
γ. Spatial and temporal dependence of the scattered wave packet
δ. Study of two limiting cases
5. Example of wave packets with two entangled photons
5-a. Parametric down-conversion
α. Description of the process
β. Scattering amplitude
5-b. Temporal correlations between the two photons generated in parametric down-conversion
α. Double photodetection signal ωII(r1, t; r2, t + τ)
β. Discussion
Chapter XXI. Quantum entanglement, measurements, Bell’s inequalities
A. Introducing entanglement, goals of this chapter
B. Entangled states of two spin-1/2 systems
B-1. Singlet state, reduced density matrices
B-2. Correlations
C. Entanglement between more general systems
C-1. Pure entangled states, notation
C-2. Presence (or absence) of entanglement: Schmidt decomposition
C-3. Characterization of entanglement: Schmidt rank
D. Ideal measurement and entangled states
D-1. Ideal measurement scheme (von Neumann)
D-1-a. Basic process
D-1-b. Dynamics of the entanglement process
D-2. Coupling with the environment, decoherence; “pointer states”
D-3. Uniqueness of the measurement result
D-3-a. The infinite von Neumann chain
D-3-b. Postulate of reduction of the state vector
E. “Which path” experiment: can one determine the path followed by the photon in Young’s double slit experiment?
E-1. Entanglement between the photon states and the plate states
E-2. Prediction of measurements performed on the photon
F. Entanglement, non-locality, Bell’s theorem
F-1. The EPR argument
F-1-a. Exposing the argument
F-1-b. Assumptions and conclusions
F-2. Bohr’s reply, non-separability
F-3. Bell’s inequality
F-3-a. Bell’s theorem
F-3-b. Contradictions
α. Contradictions with quantum mechanics predictions
β. Contradictions with experimental results
Conclusion
COMPLEMENTS OF CHAPTER XXI, READER’S GUIDE
Complement AXXI Density operator and correlations; separability
1. Von Neumann statistical entropy
1-a. General definition
1-b. Physical system composed of two subsystems
α. Pure state
β. Statistical mixture
2. Differences between classical and quantum correlations
2-a. Two levels of correlations
2-b. Quantum monogamy
3. Separability
3-a. Separable density operator
3-b. Two spins in a singlet state
Complement BXXI GHZ states, entanglement swapping
1. Sign contradiction in a GHZ state
1-a. Quantum calculation
1-b. Reasoning in the local realism framework
1-c. Discussion; contextuality
2. Entanglement swapping
2-a. General scheme
2-b. Discussion
Conclusion
Complement CXXI Measurement induced relative phase between two condensates
Introduction
1. Probabilities of single, double, etc. position measurements
1-a. Single measurement (one particle)
1-b. Double measurement (two particles)
1-c. Generalization: measurement of any number of positions
2. Measurement induced enhancement of entanglement
2-a. Measuring the single density
2-b. Entanglement between the two modes after the first detection
2-c. Measuring the double density
2-d. Discussion
3. Detection of a large number Q of particles
3-a. Probability of a multiple detection sequence
α. Simplifying hypothesis
β. Simple expression for the multiple detection probability
3-b. Discussion; emergence of a relative phase
α. Detecting the first particles
β. Emergence of a well-defined relative phase after a large number of detections
Narrowing of the relative phase distribution
Conclusion
Complement DXXI Emergence of a relative phase with spin condensates; macroscopic non-locality and the EPR argument
1. Two condensates with spins
1-a. Spin 1/2: a brief review
1-b. Projectors associated with the measurements
2. Probabilities of the different measurement results
2-a. A first expression for the probability
2-b. Introduction of the phase and the quantum angle
3. Discussion
3-a. Measurement number Q ≪ 2N
3-b. Macroscopic EPR argument
3-c. Measuring all the spins, violation of Bell’s inequalities
Conclusion
Appendix IV. Feynman path integral
1. Quantum propagator of a particle
1-a. Expressing the propagator as a sum of products
1-b. Calculation of the matrix elements
α. Free particle
β. Effects of the external potential
γ. Final expression
2. Interpretation in terms of classical histories
2-a. Expressing the propagator as a function of classical actions
2-b. Generalization: several particles interacting via a potential
3. Discussion; a new quantization rule
3-a. Analogy with classical optics
3-b. A new quantization rule
4. Operators
4-a. One single operator
α. Operator function of the position
β. Velocity operator; canonical commutation relations
4-b. Several operators
α. Several operators at different times
β. Position and velocity operators, symmetrization
Conclusion
Appendix V. Lagrange multipliers
1. Function of two variables
2. Function of N variables
Appendix VI. Brief review of Quantum Statistical Mechanics
1. Statistical ensembles
1-a. Microcanonical ensemble
α. Density operator, entropy
β. Temperature, chemical potential
γ. Entropy maximization
1-b. Canonical ensemble
α. Density operator
β. Minimization of the free energy
1-c. Grand canonical ensemble
α. Density operator
β. Grand potential
2. Intensive or extensive physical quantities
2-a. Microcanonical ensemble
2-b. Canonical ensemble
2-c. Grand canonical ensemble
2-d. Other ensembles
Appendix VII. Wigner transform
Introduction
1. Delta function of an operator
2. Wigner distribution of the density operator (spinless particle)
2-a. Definition of the distribution, Weyl operators
2-b. Expressions for the Wigner transform
α. Position representation
β. Momentum representation
γ. Inverting the relations
2-c. Reality, normalization, operator form
2-d. Gaussian wave packet
2-e. Semiclassical situations
α. Wave packet with slow spatial variations
β. Density operator; link with the probability fluid
2-f. Quantum situations where the Wigner distribution is not a probability distribution
α. Odd wave function
β. Two-peak wave function
3. Wigner transform of an operator
3-a. Average value of a Hermitian operator (observable)
3-b. Special cases
3-c. Wigner transform of an operator product
α. General expression
β. A few simple cases
γ. Gradient expansions
3-d. Evolution of the density operator
α. Classical limit
β. Particle in an external potential
4. Generalizations
4-a. Particle with spin
4-b. Several particles
5. Discussion: Wigner distribution and quantum effects
5-a. An interference experiment
5-b. General discussion; “ghost” component
Conclusion
Bibliography of volume III
Index
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