Claude Cohen-Tannoudji, Bernard Diu , Frank Laloë - Quantum Mechanics Volume 1. 2nd Edition. Wiley (2019)
Cover
Title Page
Copyright Page
Directions for Use
Foreword
Acknowledgments
Volume I
Table of contents
Chapter I. Waves and particles. Introduction to the fundamental ideas of quantum mechanics
A. Electromagnetic waves and photons
A-1. Light quanta and the Planck-Einstein relations
A-2. Wave-particle duality
A-3. The principle of spectral decomposition
B. Material particles and matter waves
B-1. The de Broglie relations
B-2. Wave functions. Schrödinger equation
C. Quantum description of a particle. Wave packets
C-1. Free particle
C-2. Form of the wave packet at a given time
C-3. Heisenberg relations
C-4. Time evolution of a free wave packet
D. Particle in a time-independent scalar potential
D-1. Separation of variables. Stationary states
D-2. One-dimensional “square” potentials. Qualitative study
COMPLEMENTS OF CHAPTER I, READER’S GUIDE
Complement AI Order of magnitude of the wavelengths associated with material particles
Complement BI Constraints imposed by the uncertainty relations
1. Macroscopic system
2. Microscopic system
Complement CI Heisenberg relation and atomic parameters
Complement DI An experiment illustrating the Heisenberg relations
Complement EI A simple treatment of a two-dimensional wave packet
1. Introduction
2. Angular dispersion and lateral dimensions
3. Discussion
Complement FI The relationship between one- and three-dimensional problems
1. Three-dimensional wave packet
1-a. Simple case
1-b. General case
2. Justification of one-dimensional models
Complement GI One-dimensional Gaussian wave packet: spreading of the wave packet
1. Definition of a Gaussian wave packet
2. Calculation of and ; uncertainty relation
3. Evolution of the wave packet
3-a. Calculation of ψ(x,t)
3-b. Velocity of the wave packet
3-c. Spreading of the wave packet
Complement HI Stationary states of a particle in one-dimensional square potentials
1. Behavior of a stationary wave function
1-a. Regions of constant potential energy
1-b. Behavior of ϕ(x ) at a potential energy discontinuity
1-c. Outline of the calculation
2. Some simple cases
2-a. Potential steps
2-b. Potential barriers
2-c. Bound states: square well potential
Complement JI Behavior of a wave packet at a potential step
1. Total reflection: E} representation
1-a. The R operator and functions of R
1-b. The P operator and functions of P
1-c. The Schrödinger equation in the {|r>} representation
2. The {|p>} representation
2-a. The P operator and functions of P
2-b. The R operator and functions of R
2-c. The Schrödinger equation in the {|p>} representation
Complement EII Some general properties of two observables, Q and P, whose commutator is equal to iħ
1. The operator S(λ): definition, properties
2. Eigenvalues and eigenvectors of Q
2-a. Spectrum of Q
2-b. Degree of degeneracy
2-c. Eigenvectors
3. The {|q>} representation
3-a. The action of Q in the {|q>} representation
3-b. TThe action of S(λ) in the {|q>} representation; the translation operator
3-c. The action of P in the {|q>} representation
4. The {|p>} representation. The symmetric nature of the P and Q observables
Complement FII The parity operator
1. The parity operator
1-a. Definition
1-b. Simple properties of II
1-c. Eigensubspaces of II
2. Even and odd operators
2-a. Definitions
2-b. Selection rules
2-c. Examples
2-d. Functions of operators
3. Eigenstates of an even observable B+
4. Application to an important special case
Complement GII An application of the properties of the tensor product: the two-dimensional infinite well
1. Definition; eigenstates
2. Study of the energy levels
2-a. Ground state
2-b. First excited states
2-c. Systematic and accidental degeneracies
Complement HII Exercises
Chapter III. The postulates of quantum mechanics
A. Introduction
B. Statement of the postulates
B-1. Description of the state of a system
B-2. Description of physical quantities
B-3. The measurement of physical quantities
B-4. Time evolution of systems
B-5. Quantization rules
C. The physical interpretation of the postulates concerning observables and their measurement
C-1. The quantization rules are consistent with the probabilistic interpretation of the wave function
C-2. Quantization of certain physical quantities
C-3. The measurement process
C-4. Mean value of an observable in a given state
C-5. The root mean square deviation
C-6. Compatibility of observables
D. The physical implications of the Schrödinger equation
D-1. General properties of the Schrödinger equation
D-2. The case of conservative systems
E. The superposition principle and physical predictions
E-1. Probability amplitudes and interference effects
E-2. Case in which several states can be associated with the same measurement result
COMPLEMENTS OF CHAPTER III, READER’S GUIDE
Complement AIII Particle in an infinite potential well
1. Distribution of the momentum values in a stationary state
1-a. Calculation of the function φn(p) of , and, of ∆P
1-b. Discussion
2. Evolution of the particle’s wave function
2-a. Wave function at the instant t
2-b. Evolution of the shape of the wave packet
2-c. Motion of the center of the wave packet
3. Perturbation created by a position measurement
Complement BIII Study of the probability current in some special cases
1. Expression for the current in constant potential regions
2. Application to potential step problems
2-a. Case where E > V0
2-b. Case where E < V0
3. Probability current of incident and evanescent waves, in the case of reflection from a two-dimensional potential step
Complement CIII Root mean square deviations of two conjugate observables
1. The Heisenberg relation for and P and Q
2. The “minimum” wave packet
Complement DIII Measurements bearing on only one part of a physical system
1. Calculation of the physical predictions
2. Physical meaning of a tensor product state
3. Physical meaning of a state that is not a tensor product
Complement EIII The density operator
1. Outline of the problem
2. The concept of a statistical mixture of states
3. The pure case. Introduction of the density operator
3-a. Description by a state vector
3-b. Description by a density operator
3-c. Properties of the density operator in a pure case
4. A statistical mixture of states (non-pure case)
4-a. Definition of the density operator
4-b. General properties of the density operator
4-c. Populations; coherences
5. Use of the density operator: some applications
5-a. System in thermodynamic equilibrium
5-b. Separate description of part of a physical system. Concept of a partial trace
Complement FIII The evolution operator
1. General properties
2. Case of conservative systems
Complement GIII The Schrödinger and Heisenberg pictures
Complement HIII Gauge invariance
1. Outline of the problem: scalar and vector potentials associated with an electromagnetic field; concept of a gauge
2. Gauge invariance in classical mechanics
2-a. Newton’s equations
2-b. The Hamiltonian formalism
3. Gauge invariance in quantum mechanics
3-a. Quantization rules
3-b. Unitary transformation of the state vector; form invariance of the Schrödinger equation
3-c. Invariance of physical predictions under a gauge transformation
Complement JIII Propagator for the Schrödinger equation
1. Introduction
2. Existence and properties of a propagator K(2,1)
2-a. Existence of a propagator
2-b. Physical interpretation of K(2,1)
2-c. Expression for K(2,1) in terms of the eigenstates of H
2-d. Equation satisfied by K(2,1)
3. Lagrangian formulation of quantum mechanics
3-a. Concept of a space-time path
3-b. Decomposition of K(2,1) into a sum of partial amplitudes
3-c. Feynman’s postulates
3-d. The classical limit and Hamilton’s principle
Complement KIII Unstable states. Lifetime
1. Introduction
2. Definition of the lifetime
3. Phenomenological description of the instability of a state
Complement LIII Exercises
Complement MIII Bound states in a “potential well” of arbitrary shape
1. Quantization of the bound state energies
2. Minimum value of the ground state energy
Complement NIII Unbound states of a particle in the presence of a potential well or barrier of arbitrary shape
1. Transmission matrix M(k)
1-a. Definition of M(k)
1-b. Properties of M(k)
2. Transmission and reflection coefficients
3. Example
Complement OIII Quantum properties of a particle in a one-dimensional periodic structure
1. Passage through several successive identical potential barriers
1-a. Notation
1-b. Matching conditions
1-c. Iteration matrix Q(α)
1-d. Eigenvalues of Q(α)
2. Discussion: the concept of an allowed or forbidden energy band
2-a. Behavior of the wave function φα(x)
2-b. Bragg reflection; possible energies for a particle in a periodic potential
3. Quantization of energy levels in a periodic potential; effect of boundary conditions
3-a. Conditions imposed on the wave function
3-b. Allowed energy bands: stationary states of the particle inside the lattice
3-c. Forbidden bands: stationary states localized on the edges
Chapter IV. Application of the postulates to simple cases: spin 1/2 and two-level systems
A. Spin 1/2 particle: quantization of the angular momentum
A-1. Experimental demonstration
A-2. Theoretical description
B. Illustration of the postulates in the case of a spin 1/2
B-1. Actual preparation of the various spin states
B-2. Spin measurements
B-3. Evolution of a spin 1/2 particle in a uniform magnetic field
C. General study of two-level systems
C-1. Outline of the problem
C-2. Static aspect: effect of coupling on the stationary states of the system
C-3. Dynamical aspect: oscillation of the system between the two unperturbed states
COMPLEMENTS OF CHAPTER IV, READER’S GUIDE
Complement AIV The Pauli matrices
1. Definition; eigenvalues and eigenvectors
2. Simple properties
3. A convenient basis of the 2x2 matrix space
Complement BIV Diagonalization of a 2x2 Hermitian matrix
1. Introduction
2. Changing the eigenvalue origin
3. Calculation of the eigenvalues and eigenvectors
3-a. Angles Φ and φ
3-b. Eigenvalues of K
3-c. Eigenvalues of H
3-d. Normalized eigenvectors of H
Complement CIV Fictitious spin 1/2 associated with a two-level system
1. Introduction
2. Interpretation of the Hamiltonian in terms of fictitious spin
3. Geometrical interpretation of the various effects discussed in § C of Chapter IV
3-a. Fictitious magnetic fields associated with H0, W and H
3-b. Effect of coupling on the eigenvalues and eigenvectors of the Hamiltonian
3-c. Geometrical interpretation of P12(t)
Complement DIV System of two spin 1/2 particles
1. Quantum mechanical description
1-a. State space
1-b. Complete sets of commuting observables
1-c. The most general state
2. Prediction of the measurement results
2-a. Measurements bearing simultaneously on the two spins
2-b. Measurements bearing on one spin alone
Complement EIV Spin 1 2 density matrix
1. Introduction
2. Density matrix of a perfectly polarized spin (pure case)
3. Example of a statistical mixture: unpolarized spin
4. Spin 1/2 at thermodynamic equilibrium in a static field
5. Expansion of the density matrix in terms of the Pauli matrices
Complement FIV Spin 1/2 particle in a static and a rotating magnetic fields: magnetic resonance
1. Classical treatment; rotating reference frame
1-a. Motion in a static field; Larmor precession
1-b. Influence of a rotating field; resonance
2. Quantum mechanical treatment
2-a. The Schrödinger equation
2-b. Changing to the rotating frame
2-c. Transition probability; Rabi’s formula
2-d. Case where the two levels are unstable
3. Relation between the classical treatment and the quantum mechanical treatment: evolution of
4. Bloch equations
4-a. A concrete example
4-b. Solution in the case of a rotating field
Complement GIV A simple model of the ammonia molecule
1. Description of the model
2. Eigenfunctions and eigenvalues of the Hamiltonian
2-a. Infinite potential barrier
2-b. Finite potential barrier
2-c. Evolution of the molecule. Inversion frequency
3. The ammonia molecule considered as a two-level system
3-a. The state space
3-b. Energy levels. Removal of the degeneracy due to the transparency of the potential barrier
3-c. Influence of a static electric field
Complement HIV Effects of a coupling between a stable state and an unstable state
1. Introduction. Notation
2. Influence of a weak coupling on states of different energies
3. Influence of an arbitrary coupling on states of the same energy
Complement JIV Exercises
Chapter V. The one-dimensional harmonic oscillator
A. Introduction
A-1. Importance of the harmonic oscillator in physics
A-2. The harmonic oscillator in classical mechanics
A-3. General properties of the quantum mechanical Hamiltonian
B. Eigenvalues of the Hamiltonian
B-1. Notation
B-2. Determination of the spectrum
B-3. Degeneracy of the eigenvalues
C. Eigenstates of the Hamiltonian
C-1. The representation
C-2. Wave functions associated with the stationary states
D. Discussion
D-1. Mean values and root mean square deviations of X and P in a state |φn)
D-2. Properties of the ground state
D-3. Time evolution of the mean values
COMPLEMENTS OF CHAPITER V, READER’S GUIDE
Complement AV Some examples of harmonic oscillators
1. Vibration of the nuclei of a diatomic molecule
1-a. Interaction energy of two atoms
1-b. Motion of the nuclei
1-c. Experimental observations of nuclear vibration
2. Vibration of the nuclei in a crystal
2-a. The Einstein model
2-b. The quantum mechanical nature of crystalline vibrations
3. Torsional oscillations of a molecule: ethylene
3-a. Structure of the ethylene molecule C2H4
3-b. Classical equations of motion
3-c. Quantum mechanical treatment
4. Heavy muonic atoms
4-a. Comparison with the hydrogen atom
4-b. The heavy muonic atom treated as a harmonic oscillator
4-c. Order of magnitude of the energies and spread of the wave functions
Complement BV Study of the stationary states in the {|x>} representation. Hermite polynomials
1. Hermite polynomials
1-a. Definition and simple properties
1-b. Generating function
1-c. Recurrence relations; differential equation
1-d. Examples
2. The eigenfunctions of the harmonic oscillator Hamiltonian
2-a. Generating function
2-b. φn(x) in terms of the Hermite polynomials
2-c. Recurrence relations
Complement CV Solving the eigenvalue equation of the harmonic oscillator by the polynomial method
1. Changing the function and the variable
2. The polynomial method
2-a. The asymptotic form of φ(x)
2-b. The calculation of h(x) in the form of a series expansion
2-c. Quantization of the energy
2-d. Stationary wave functions
Complement DV Study of the stationary states in the {|p>} representation
1. Wave functions in momentum space
1-a. Changing the variable and the function
1-b. Determination of φn(p)
1-c. Calculation of the phase factor
2. Discussion
Complement EV The isotropic three-dimensional harmonic oscillator
1. The Hamiltonian operator
2. Separation of the variables in Cartesian coordinates
3. Degeneracy of the energy levels
Complement FV A charged harmonic oscillator in a uniform electric field
1. Eigenvalue equation of H' (E) in the {|x>} in the representation
2. Discussion
2-a. Electrical susceptibility of an elastically bound electron
2-b. Interpretation of the energy shift
3. Use of the translation operator
Complement GV Coherent “quasi-classical” states of the harmonic oscillator
1. Quasi-classical states
1-a. Introducing the parameter
to characterize a classical motion
1-b. Conditions defining quasi-classical states
1-c. Quasi-classical states are eigenvectors of the operator
2. Properties of the |α> states
2-a. Expansion of |α> on the basis of the stationary states |φn>
2-b. Possible values of the energy in an |α> state
2-c. Calculation of , , and ΔP in an |α> state
2-d. The operator D(α): the wave functions ψα(x)
2-e. The scalar product of two |α> states. Closure relation
3. Time evolution of a quasi-classical state
3-a. A quasi-classical state always remains an eigenvector of a
3-b. Evolution of physical properties
3-c. Motion of the wave packet
4. Example: quantum mechanical treatment of a macroscopic oscillator
Complement HV Normal vibrational modes of two coupled harmonic oscillators
1. Vibration of the two coupled in classical mechanics
1-a. Equations of motion
1-b. Solving the equations of motion
1-c. The physical meaning of each of the modes
1-d. Motion of the system in the general case
2. Vibrational states of the system in quantum mechanics
2-a. Commutation relations
2-b. Transformation of the Hamiltonian operator
2-c. Stationary states of the system
2-d. Evolution of the mean values
References and suggestions for further reading:
Complement JV Vibrational modes of an infinite linear chain of coupled harmonic oscillators; phonons
1. Classical treatment
1-a. Equations of motion
1-b. Simple solutions of the equations of motion
1-c. Normal variables
1-d. Total energy and energy of each of the modes
2. Quantum mechanical treatment
2-a. Stationary states in the absence of coupling
2-b. Effects of the coupling
2-c. Normal operators. Commutation relations
2-d. Stationary states in the presence of coupling
3. Application to the study of crystal vibrations: phonons
3-a. Outline of the problem
3-b. Normal modes. Speed of sound in the crystal
Complement KV Vibrational modes of a continuous physical system. Application to radiation; photons
1. Outline of the problem
2. Vibrational modes of a continuous mechanical system: example of a vibrating string
2-a. Notation. Dynamical variables of the system
2-b. Classical equations of motion
2-c. Introduction of the normal variables
2-d. Classical Hamiltonian
2-e. Quantization
3. Vibrational modes of radiation: photons
3-a. Notation. Equations of motion
3-b. Introduction of the normal variables
3-c. Classical Hamiltonian
3-d. Quantization
References and suggestions for further reading:
Complement LV One-dimensional harmonic oscillator in thermodynamic equilibrium ata temperature T
1. Mean value of the energy
1-a. Partition function
1-b. Calculation of
2. Discussion
2-a. Comparison with the classical oscillator
2-b. Comparison with a two-level system
3. Applications
3-a. Blackbody radiation
3-b. Bose-Einstein distribution law
3-c. Specific heats of solids at constant volume
4. Probability distribution of the observable X
4-a. Definition of the probability density p(x)
4-b. Calculation of p(x)
4-c. Discussion
4-d. Bloch’s theorem
Complement MV Exercises
Chapter VI. General properties of angular momentum in quantum mechanics
A. Introduction: the importance of angular momentum
B. Commutation relations characteristic of angular momentum
B-1. Orbital angular momentum
B-2. Generalization: definition of an angular momentum
B-3. Statement of the problem
C. General theory of angular momentum
C-1. Definitions and notation
C-2. Eigenvalues of J2 and Jz
C-3. “Standard” {|k, j, m>} representations
D. Application to orbital angular momentum
D-1. Eigenvalues and eigenfunctions of L2 and Lz
D-2. Physical considerations
COMPLEMENTS OF CHAPTER VI, READER’S GUIDE
Complement AVI Spherical harmonics
1. Calculation of spherical harmonics
1-a. Determination of YlI (θ, φ)
1-b. General expression for Ylm (θ, φ)
1-c. Explicit expressions for l=0,1 and 2
2. Properties of spherical harmonics
2-a. Recurrence relations
2-b. Orthonormalization and closure relations
2-c. Parity
2-d. Complex conjugation
2-e. Relation between the spherical harmonics and the Legendre polynomials and associated Legendre functions
Complement BVI Angular momentum and rotations
1. Introduction
2. Brief study of geometrical rotations R
2-a. Definition. Parametrization
2-b. Infinitesimal rotations
3. Rotation operators in state space. Example: a spinless particle
3-a. Existence and definition of rotation operators
3-b. Properties of rotation operators
3-c. Expression for rotation operators in terms of angular momentum observables
4. Rotation operators in the state space of an arbitrary system
4-a. System of several spinless particles
4-b. An arbitrary system
5. Rotation of observables
5-a. General transformation law
5-b. Scalar observables
5-c. Vector observables
6. Rotation invariance
6-a. Invariance of physical laws
6-b. Consequence: conservation of angular momentum
6-c. Applications
Complement CVI Rotation of diatomic molecules
1. Introduction
2. Rigid rotator. Classical study
2-a. Notation
2-b. Motion of the rotator. Angular momentum and energy
2-c. The fictitious particle associated with the rotator
3. Quantization of the rigid rotator
3-a. The quantum mechanical state and observables of the rotator
3-b. Eigenstates and eigenvalues of the Hamiltonian
3-c. Study of the observable Z
4. Experimental evidence for the rotation of molecules
4-a. Heteropolar molecules. Pure rotational spectrum
4-b. Homopolar molecules. Raman rotational spectra
Complement EVI A charged particle in a magnetic field: Landau levels
Complement DVI Angular momentum of stationary states of a two-dimensional harmonic oscillator
1. Introduction
1-a. Review of the classical problem
1-b. The problem in quantum mechanics
2. Classification of the stationary states by the quantum numbers and nx and ny
2-a. Energies; stationary states
2-b. does not constitue a C.S.C.O. in Exy
3. Classification of the stationary states in terms of their angular momenta
3-a. Significance and properties of the operator Lz
3-b. Right and left circular quanta
3-c. Stationary states of well-defined angular momentum
3-d. Wave functions associated with the eigenstates common to and Hxy and Lz
4. Quasi-classical states
4-a. Definition of the states and |αx, αy> and |αr, αl>
4-b. Mean values and root mean square deviations of the various observables
Complement EVI A charged particle in a magnetic field: Landau levels
1. Review of the classical problem
1-a. Motion of the particle
1-b. The vector potential. The classical Lagrangian and Hamiltonian
1-c. Constants of the motion in a uniform field
2. General quantum mechanical properties of a particle in a magnetic field
2-a. Quantization. Hamiltonian
2-b. Commutation relations
2-c. Physical consequences
3. Case of a uniform magnetic field
3-a. Eigenvalues of the Hamiltonian
3-b. The observables in a particular gauge
3-c. The stationary states
3-d. Time evolution
Complement FVI Exercises
Chapter VII. Particle in a central potential. The hydrogen atom
A. Stationary states of a particle in a central potential
A-1. Outline of the problem
A-2. Separation of variables
A-3. Stationary states of a particle in a central potential
B. Motion of the center of mass and relative motion for a system of two interacting particles
B-1. Motion of the center of mass and relative motion in classical mechanics
B-2. Separation of variables in quantum mechanics
C. The hydrogen atom
C-1. Introduction
C-2. The Bohr model
C-3. Quantum mechanical theory of the hydrogen atom
C-4. Discussion of the results
COMPLEMENTS OF CHAPTER VII, READER’S GUIDE
Complement AVII Hydrogen-like systems
1. Hydrogen-like systems with one electron
1-a. Electrically neutral systems
1-b. Hydrogen-like ions
2. Hydrogen-like systems without an electron
2-a. Muonic atoms
2-b. Hadronic atoms
Complement BVII A soluble example of a central potential: the isotropic three-dimensional harmonic oscillator
1. Solving the radial equation
2. Energy levels and stationary wave functions
Complement CVII Probability currents associated with the stationary states of the hydrogen atom
1. General expression for the probability current
2. Application to the stationary states of the hydrogen atom
2-a. Structure of the probability current
2-b. Effect of a magnetic field
Complement DVII The hydrogen atom placed in a uniform magnetic field. Paramagnetism and diamagnetism. The Zeeman effect
1. The Hamiltonian of the problem. The paramagnetic term and the diamagnetic term
1-a. Expression for the Hamiltonian
1-b. Order of magnitude of the various terms
1-c. Interpretation of the paramagnetic term
1-d. Interpretation of the diamagnetic term
2. The Zeeman effect
2-a. Energy levels of the atom in the presence of a magnetic field
2-b. Electric dipole oscillations
2-c. Frequency and polarization of emitted radiation
Complement EVII Some atomic orbitals. Hybrid orbitals
1. Introduction
2. Atomic orbitals associated with real wave functions
2-a. orbitals (l=1)
2-b. orbitals (l=1)
2-c. Other values of l
3. sp hybridization
3-a. Introduction of sp hybrid orbitals
3-b. Properties of sp hybrid orbitals
3-c. Example: the structure of acetylene
4. sp2 hybridization
4-a. Introduction of sp2 hybrid orbitals
4-b. Properties of sp2 hybrid orbitals
4-c. Example: the structure of ethylene
5. sp3 hybridization
5-a. Introduction of sp3 hybrid orbitals
5-b. Properties of sp3 hybrid orbitals
5-c. Example: The structure of methane
Complement FVII Vibrational-rotational levels of diatomic molecules
1. Introduction
2. Approximate solution of the radial equation
2-a. The zero angular momentum states (l=0)
2-b. General case (l any positive integer)
2-c. The vibrational-rotational spectrum
3. Evaluation of some corrections
3-a. More precise study of the form of the effective potential Veff(r)
3-b. Energy levels and wave functions of the stationary states
3-c. Interpretation of the various corrections
Complement GVII Exercises
Index
EULA
Claude Cohen-Tannoudji, Bernard Diu , Frank Laloë - Quantum Mechanics Volume 2. 2nd Edition. Wiley (2019)
Cover
Title Page
Copyright Page
Directions for Use
Foreword
VOLUME II
Table of contents
Chapter VIII. An elementary approach to the quantum theory of scattering by a potential
A Introduction
A-1. Importance of collision phenomena
A-2. Scattering by a potential
A-3. Definition of the scattering cross section
A-4. Organization of this chapter
B. Stationary scattering states. Calculation of the cross section
B-1. Definition of stationary scattering states
B-2. Calculation of the cross section using probability currents
B-3. Integral scattering equation
B-4. The Born approximation
C. Scattering by a central potential. Method of partial waves
C-1. Principle of the method of partial waves
C-2. Stationary states of a free particle
C-3. Partial waves in the potential V(r)
C-4. Expression of the cross section in terms of phase shifts
COMPLEMENTS OF CHAPTER VIII, READER’S GUIDE
Complement AVIII The free particle: stationary states with well-defined angular momentum
1. The radial equation
2. Free spherical waves
2-a. Recurrence relations
2-b. Calculation of free spherical waves
2-c. Properties
3. Relation between free spherical waves and plane waves
Complement BVIII Phenomenological description of collisions with absorption
1. Principle involved
2. Calculation of the cross sections
2-a. Elastic scattering cross section
2-b. Absorption cross section
2-c. Total cross section. Optical theorem
Complement CVIII Some simple applications of scattering theory
1. The Born approximation for a Yukawa potential
1-a. Calculation of the scattering amplitude and cross section
1-b. The infinite-range limit
2. Low energy scattering by a hard sphere
3. Exercises
3-a. Scattering of the p wave by a hard sphere
3-b. “Square spherical well”: bound states and scattering resonances
Chapter IX. Electron spin
A. Introduction of electron spin
A-1. Experimental evidence
A-2. Quantum description: postulates of the Pauli theory
B. Special properties of an angular momentum 1/2
C. Non-relativistic description of a spin 1/2 particle
C-1. Observables and state vectors
C-2. Probability calculations for a physical measurement
COMPLEMENTS OF CHAPTER IX, READER’S GUIDE
Complement AIX Rotation operators for a spin 1/2 particle
1. Rotation operators in state space
1-a. Total angular momentum
1-b. Decomposition of rotation operators into tensor products
2. Rotation of spin states
2-a. Explicit calculation of the rotation operators in
2-b. Operator associated with a rotation through an angle of 2
2-c. Relationship between the vectorial nature of S and the behavior of a spin stateupon rotation
3. Rotation of two-component spinors
Complement BIX Exercises
Chapter X. Addition of angular momenta
A. Introduction
A-1. Total angular momentum in classical mechanics
A-2. The importance of total angular momentum in quantum mechanics
B. Addition of two spin 1/2’s. Elementary method
B-1. Statement of the problem
B-2. The eigenvalues of Sz and their degrees of degeneracy
B-3. Diagonalization of S2
B-4. Results: triplet and singlet
C. Addition of two arbitrary angular momenta. General method
C-1. Review of the general theory of angular momentum
C-2. Statement of the problem
C-3. Eigenvalues of J2 and Jz
C-4. Common eigenvectors of J2 and Jz
COMPLEMENTS OF CHAPTER X, READER’S GUIDE
Complement AX Examples of addition of angular momenta
1. Addition of j1 = 1 and j2 = 1
1-a. The subspace Ԑ( J = 2)
1-b. The subspace Ԑ( J = 2)
1-c. The vector | J = 0, M = 0
2. Addition of an integral orbital angular momentum Ɩ and a spin 1/2
2-a. The subspace ɛ( J = l + 1/2)
2-b. The subspace ɛ( J = l + 1/2)
Complement BX Clebsch-Gordan coefficients
1. General properties of Clebsch-Gordan coefficients
1-a. Selection rules
1-b. Orthogonality relations
1-c. Recurrence relations
2. Phase conventions. Reality of Clebsch-Gordan coefficients
2-a. The coefficients :phase of the ket | J,J>
2-b. Other Clebsch-Gordan coefficients
3. Some useful relations
3-a. The signs of some coefficients
3-b. Changing the order of j1 and j2
3-c. Changing the sign of M ,m1 and m2
3-d. The coefficients
Complement CX Addition of spherical harmonics
1 The functions ФMJ (Ω1; Ω2)
2. The functions Fml (Ω)
3. Expansion of a product of spherical harmonics; the integral of a product ofthree spherical harmonics
Complement DX Vector operators: the Wigner-Eckart theorem
1. Definition of vector operators; examples
2. The Wigner-Eckart theorem for vector operators
2-a. Non-zero matrix elements of V in a standard basis
2-b. Proportionality between the matrix elements of J and V inside a subspace Ԑ(k, j)
2-c. Calculation of the proportionality constant; the projection theorem
3. Application: calculation of the Landé gJ factor of an atomic level
3-a. Rotational degeneracy; multiplets
3-b. Removal of the degeneracy by a magnetic field; energy diagram
Complement EX Electric multipole moments
1. Definition of multipole moments
1-a. Expansion of the potential on the spherical harmonics
1-b. Physical interpretation of multipole operators
1-c. Parity of multipole operators
1-d. Another way to introduce multipole moments
2. Matrix elements of electric multipole moments
2-a. General expression for the matrix elements
2-b. Selection rules
Complement FX Two angular momenta J1 and J2 coupled by an interaction aJ1 . J2
1. Classical review
1-a. Equations of motion
2. Quantum mechanical evolution of the average values and
2-a. Calculation of d/dƖ and d/dƖ
2-b. Discussion
3. The special case of two spin 1/2’s
3-a. Stationary states of the two-spin system
3-b. Calculation of S1 (t)
3-c. Discussion. Polarization of the magnetic dipole transitions
4. Study of a simple model for the collision of two spin 1/2 particles
4-a. Description of the model
4-b. State of the system after collision
4-c. Discussion. Correlation introduced by the collision
Complement GX Exercises
Chapter XI. Stationary perturbation theory
A. Description of the method
A-1. Statement of the problem
A-2. Approximate solution of the H (λ ) eigenvalue equation
B. Perturbation of a non-degenerate level
B-1. First-order corrections
B-2. Second-order corrections
C. Perturbation of a degenerate state
COMPLEMENTS OF CHAPTER XI, READER’S GUIDE
Complement AXI A one-dimensional harmonic oscillator subjected to a perturbing potential in x, x2, x3
1. Perturbation by a linear potential
1-a. The exact solution
1-b. The perturbation expansion
2. Perturbation by a quadratic potential
3. Perturbation by a potential in x3
3-a. The anharmonic oscillator
3-b. The perturbation expansion
3-c. Application: the anharmonicity of the vibrations of a diatomic molecule
Complement BXI Interaction between the magnetic dipoles of two spin 1/2 particles
1. The interaction Hamiltonian W
1-a. The form of the Hamiltonian W. Physical interpretation
1-b. An equivalent expression for W
1-c. Selection rules
2. Effects of the dipole-dipole interaction on the Zeeman sublevels of two fixedparticles
2-a. Case where the two particles have different magnetic moments
2-b. Case where the two particles have equal magnetic moments
2-c. Example: the magnetic resonance spectrum of gypsum
3. Effects of the interaction in a bound state
Complement CXI Van der Waals forces
1. The electrostatic interaction Hamiltonian for two hydrogen atoms
1-a. Notation
1-b. Calculation of the electrostatic interaction energy
2. Van der Waals forces between two hydrogen atoms in the 1s ground state
2-a. Existence of a- C/R6 attractive potential
2-b. Approximate calculation of the constant C
3. Van der Waals forces between a hydrogen atom in the 1s state and ahydrogen atom in the 2P state
3-a. Energies of the stationary states of the two-atom system. Resonance effect
3-b. Transfer of the excitation from one atom to the other
4. Interaction of a hydrogen atom in the ground state with a conducting wall
Complement DXI The volume effect: the influence of the spatial extension of the nucleus on the atomic levels
1. First-order energy correction
1-a. Calculation of the correction
1-b. Discussion
2. Application to some hydrogen-like systems
2-a. The hydrogen atom and hydrogen-like ions
2-b. Muonic atoms
Complement EXI The variational method
1. Principle of the method
1-a. A property of the ground state of a system
1-b. Generalization: the Ritz theorem
1-c. A special case where the trial functions form a subspace
2. Application to a simple example
2-a. Exponential trial functions
2-b. Rational wave functions
3. Discussion
Complement FXI Energy bands of electrons in solids: a simple model
1. A first approach to the problem: qualitative discussion
2. A more precise study using a simple model
2-a. Calculation of the energies and stationary states
2-b. Discussion
Complement GXI A simple example of the chemical bond: the H2+ ion
1. Introduction
1-a. General method
1-b. Notation
1-c. Principle of the exact calculation
2. The variational calculation of the energies
2-a. Choice of the trial kets
2-b. The eigenvalue equation of the Hamiltonian H in the trial ket vector subspace ϝ
2-c. Overlap, Coulomb and resonance integrals
2-d. Bonding and antibonding states
3. Critique of the preceding model. Possible improvements
3-a. Results for small R
3-b. Results for R
4. Other molecular orbitals of the H+2 ion
4-a. Symmetries and quantum numbers. Spectroscopic notation
4-b. Molecular orbitals constructed from the 2P atomic orbitals
5. The origin of the chemical bond; the virial theorem
5-a. Statement of the problem
5-b. Some useful theorems
5-c. The virial theorem applied to molecules
5-d. Discussion
Complement HXI Exercises
Chapter XII. An application of perturbation theory: the fine and hyperfine structure of hydrogen
A. Introduction
B. Additional terms in the Hamiltonian
B-1. The fine-structure Hamiltonian
B-2. Magnetic interactions related to proton spin: the hyperfine Hamiltonian
C. The fine structure of the = 2 level
C-1. Statement of the problem
C-2. Matrix representation of the fine-structure Hamiltonian Wf inside the n= 2 level
C-3. Results: the fine structure of the n = 2 level
D. The hyperfine structure of the n = 1 level
D-1. Statement of the problem
D-2. Matrix representation of Whf in the 1s level
D-3. The hyperfine structure of the 1s level
E. The Zeeman effect of the 1s ground state hyperfine structure
E-1. Statement of the problem
E-2. The weak-field Zeeman effect
E-3. The strong-field Zeeman effect
E-4. The intermediate-field Zeeman effect
COMPLEMENTS OF CHAPTER XII, READER’S GUIDE
Complement AXII The magnetic hyperfine Hamiltonian
1. Interaction of the electron with the scalar and vector potentials created bythe proton
2. The detailed form of the hyperfine Hamiltonian
2-a. Coupling of the magnetic moment of the proton with the orbital angularmomentum of the electron
2-b. Coupling with the electron spin
3. Conclusion: the hyperfine-structure Hamiltonian
Complement BXII Calculation of the average values of the fine-structure Hamiltonian in the 1s, 2s and 2p states
1. Calculation of <1/R> , <1/ R2 and <1/ R3>
2. The average values
3. The average values
4. Calculation of the coefficient ξ2p associated with Wso in the 2p level
Complement CXII The hyperfine structure and the Zeeman effect for muonium and positronium
1. The hyperfine structure of the 1s ground state
2. The Zeeman effect in the 1s ground state
2-a. The Zeeman Hamiltonian
2-b. Stationary state energies
2-c. The Zeeman diagram for muonium
2-d. The Zeeman diagram for positronium
Complement DXII The influence of the electronic spin on the Zeeman effect of the hydrogen resonance line
1. Introduction
2. The Zeeman diagrams of the 1s and 2s levels
3. The Zeeman diagram of the 2p level
4. The Zeeman effect of the resonance line
4-a. Statement of the problem
4-b. The weak-field Zeeman components
4-c. The strong-field Zeeman components
Complement EXII The Stark effect for the hydrogen atom
1. The Stark effect on the n = 1 level
1-a. The shift of the 1 state is quadratic in Ԑ
1-b. Polarizability of the 1s state
2. The Stark effect on the n = 2 level
Chapter XIII. Approximation methods for time-dependent problems
A. Statement of the problem
B. Approximate solution of the Schrödinger equation
B-1. The Schrödinger equation in the {|φn>} representation
B-2. Perturbation equations
B-3. Solution to first order in λ
C. An important special case: a sinusoidal or constant perturbation
C-1. Application of the general equations
C-2. Sinusoidal perturbation coupling two discrete states: the resonance phenomenon
C-3. Coupling with the states of the continuous spectrum
D. Random perturbation
D-1. Statistical properties of the perturbation
D-2. Perturbative computation of the transition probability
D-3. Validity of the perturbation treatment
E. Long-time behavior for a two-level atom
E-1. Sinusoidal perturbation
E-2. Random perturbation
E-3. Broadband optical excitation of an atom
COMPLEMENTS OF CHAPTER XIII, READER’S GUIDE
Complement AXIII Interaction of an atom with an electromagnetic wave
1. The interaction Hamiltonian. Selection rules
1-a. Fields and potentials associated with a plane electromagnetic wave
1-b. The interaction Hamiltonian at the low-intensity limit
1-c. The electric dipole Hamiltonian
1-d. The magnetic dipole and electric quadrupole Hamiltonians
2. Non-resonant excitation. Comparison with the elastically bound electronmodel
2-a. Classical model of the elastically bound electron
2-b. Quantum mechanical calculation of the induced dipole moment
2-c. Discussion. Oscillator strength
3. Resonant excitation. Absorption and induced emission
3-a. Transition probability associated with a monochromatic wave
3-b. Broad-line excitation. Transition probability per unit time
Complement BXIII Linear and non-linear responses of a two-level system subject to a sinusoidal perturbation
1. Description of the model
1-a. Bloch equations for a system of spin 1/2’s interacting with a radiofrequency field
1-b. Some exactly and approximately soluble cases
1-c. Response of the atomic system
2. The approximate solution of the Bloch equations of the system
2-a. Perturbation equations
2-b. The Fourier series expansion of the solution
2-c. The general structure of the solution
3. Discussion
3-a. Zeroth-order solution: competition between pumping and relaxation
3-b. First-order solution: the linear response
3-c. Second-order solution: absorption and induced emission
3-d. Third-order solution: saturation effects and multiple-quanta transitions
4. Exercises: applications of this complement
Complement CXIII Oscillations of a system between two discrete states under the effect of a sinusoidal resonant perturbation
1. The method: secular approximation
2. Solution of the system of equations
3. Discussion
Complement DXIII Decay of a discrete state resonantly coupled to a continuum of final states
1. Statement of the problem
2. Description of the model
2-a. Assumptions about the unperturbed Hamiltonian Hο
2-b. Assumptions about the coupling W
2-c. Results of first-order perturbation theory
2-d. Integrodifferential equation equivalent to the Schrödinger equation
3. Short-time approximation. Relation to first-order perturbation theory
4. Another approximate method for solving the Schrödinger equation
5. Discussion
5-a. Lifetime of the discrete state
5-b. Shift of the discrete state due to the coupling with the continuum
5-c. Energy distribution of the final states
Complement EXIII Time-dependent random perturbation, relaxation
1. Evolution of the density operator
1-a. Coupling Hamiltonian, correlation times
1-b. Evolution of a single system
1-c. Evolution of the ensemble of systems
1-d. General equations for the relaxation
2. Relaxation of an ensemble of spin 1/2’s
2-a. Characterization of the operators, isotropy of the perturbation
2-b. Longitudinal relaxation
2-c. Transverse relaxation
3. Conclusion
Complement FXIII Exercises
Chapter XIV. Systems of identical particles
A. Statement of the problem
A-1. Identical particles: definition
A-2. Identical particles in classical mechanics
A-3. Identical particles in quantum mechanics: the difficulties of applying the generalpostulates
B. Permutation operators
B-1. Two-particle systems
B-2. Systems containing an arbitrary number of particles
C. The symmetrization postulate
C-1. Statement of the postulate
C-2. Removal of exchange degeneracy
C-3. Construction of physical kets
C-4. Application of the other postulates
D. Discussion
D-1. Differences between bosons and fermions. Pauli’s exclusion principle
D-2. The consequences of particle indistinguishability on the calculation of physicalpredictions
COMPLEMENTS OF CHAPTER XIV, READER’S GUIDE
Complement AXIV Many-electron atoms. Electronic configurations
1. The central-field approximation
1-a. Difficulties related to electron interactions
1-b. Principle of the method
1-c. Energy levels of the atom
2. Electron configurations of various elements
Complement BXIV Energy levels of the helium atom. Configurations, terms, multiplets
1. The central-field approximation. Configurations
1-a. The electrostatic Hamiltonian
1-b. The ground state configuration and first excited configurations
1-c. Degeneracy of the configurations
2. The effect of the inter-electron electrostatic repulsion: exchange energy,spectral terms
2-a. Choice of a basis of Ԑ(n,l,n,l) adapted to the symmetries of W
2-b. Spectral terms. Spectroscopic notation
2-c. Discussion
3. Fine-structure levels; multiplets
Complement CXIV Physical properties of an electron gas. Application to solids
1. Free electrons enclosed in a box
1-a. Ground state of an electron gas; Fermi energy EF
1-b. Importance of the electrons with energies close to EF
1-c. Periodic boundary conditions
2. Electrons in solids
2-a. Allowed bands
2-b. Position of the Fermi level and electric conductivity
Complement DXIV Exercises
Appendix I: Fourier series and Fourier transforms
1. Fourier series
1-a. Periodic functions
1-b. Expansion of a periodic function in a Fourier series
1-c. The Bessel-Parseval relation
2. Fourier transforms
2-a. Definitions
2-b. Simple properties
2-c. The Parseval-Plancherel formula
2-d. Examples
2-e. Fourier transforms in three-dimensional space
Appendix II: The Dirac δ -“function”
1. Introduction; principal properties
1-a. Introduction of the δ-“function”
1-b. Functions that approach δ
1-c. Properties of δ
2. The δ-''function” and the Fourier transform
2-a. The Fourier transform of δ
2-b. Applications
3. Integral and derivatives of the δ -“function”
3-a. δ is the derivative of the “unit step-function”
3-b. Derivatives of δ
4. The δ-“function” in three-dimensional space
Appendix III: Lagrangian and Hamiltonian in classical mechanics
1. Review of Newton’s laws
1-a. Dynamics of a point particle
1-b. Systems of point particles
1-c. Fundamental theorems
2. The Lagrangian and Lagrange’s equations
3. The classical Hamiltonian and the canonical equations
3-a. The conjugate momenta of the coordinates
3-b. The Hamilton-Jacobi canonical equations
4. Applications of the Hamiltonian formalism
4-a. A particle in a central potential
4-b. A charged particle placed in an electromagnetic field
5. The principle of least action
5-a. Geometrical representation of the motion of a system
5-b. The principle of least action
5-c. Lagrange’s equations as a consequence of the principle of least action
BIBLIOGRAPHY OF VOLUMES I AND II
INDEX
EULA
Claude Cohen-Tannoudji, Bernard Diu , Frank Laloë - Quantum Mechanics Volume 3. 2nd Edition. Wiley (2019)
Cover
Title Page
Copyright Page
Directions for Use
Foreword
VOLUME III
Table of contents
Chapter XV. Creation and annihilation operators for identical particles
A. General formalism
A-1. Fock states and Fock space
A-2. Creation operators
A-3. Annihilation operators
A-4. Occupation number operators (bosons and fermions)
A-5. Commutation and anticommutation relations
A-6. Change of basis
B. One-particle symmetric operators
B-1. Definition
B-2. Expression in terms of the operators and
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
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
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
3-c. Common expression
4. Total number of particles
4-a. Fermions
4-b. 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
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
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
3-c. Critical velocity, metastable flow
3-d. Generalization; topological aspects
Complement EXV Fermion system, Hartree-Fock approximation
1. Foundation of the method
1-a. Trial family and Hamiltonian
1-b. Energy average value
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
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
2-c. Variational grand potential
2-d. Optimization
3. Temperature dependent mean field equations
3-a. Form of the equations
3-b. Properties and limits of the equations
3-c. Differences with the zero-temperature Hartree-Fock equations (fermions)
3-d. Zero-temperature limit (fermions)
3-e. Wave function equations
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
3-b. Resolution of the equations by graphical iteration
3-c. Physical discussion
4. Bosons: equation of state, attractive instability
4-a. Repulsive bosons
4-b. Attractive bosons
Chapter XVI. Field operator
A. Definition of the field operator
A-1. Definition
A-2. Commutation and anticommutation relations
B. Symmetric operators
B-1. General expression
B-2. Simple examples
B-3. Field spatial 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. Twoand 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
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-2. Undetermined particle number
B-3. Pairs of particles and pairs of individual states
C. Properties of the kets characterizing the paired states
C-1. Normalization
C-2. Average value and root mean square deviation of particle number
C-3. “Anomalous” average values
D. Correlations between particles, pair wave function
D-1. Particles in the same spin state
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-3. Basis of excited states, quasi-particles
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
1-b. Pairs in a singlet spin state
2. Average values in a paired state
2-a. Average value of a field operator; pair wave function, and order parameter
2-b. Average value of a product of two field operators; factorization of the order parameter
2-c. Application to the computation of the correlation function (singlet pairs)
3. Commutation relations of field operators
3-a. Particles in the same spin state
3-b. Singlet pairs
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
3-b. Total energy
4. Spinless bosons
4-a. Choice of the variational state
4-b. Different contributions to the 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
2. Distribution functions, correlations
2-a. One-particle distribution
2-b. Two-particle distribution, internal pair wave function
2-c. Properties of the pair wave function, coherence length
3. Physical discussion
3-a. Modification of the Fermi surface and phase locking
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
4-d. Excitation energies
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.
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-2. Description in the reciprocal space
A-3. Elimination of the longitudinal fields from the expression of the physical quantities
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-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
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
A. Quantization of the radiation in the Coulomb gauge
A-1. Quantization rules
A-2. Radiation contained in a box
A-3. Heisenberg equations
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
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-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
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
Complement CXIX Angular momentum exchange between atoms and photons
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
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-2. Non-monochromatic 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-2. Resonant scattering
E-3. Inelastic scattering Raman scattering
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
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
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
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
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
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
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
Complement EXX Detection of one- or two-photon wave packets, interference
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. Oneor 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
2-c. Interference signals for a product of two one-photon wave packets
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
5. Example of wave packets with two entangled photons
5-a. Parametric down-conversion
5-b. Temporal correlations between the two photons generated in parametric down-conversion
Chapter XXI. Quantum entanglement, measurements, Bell’s inequalities
A. Introducing entanglement, goals of this chapter
B. Entangled states of two spin- 1/2systems
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-2. Coupling with the environment, decoherence; “pointer states”
D-3. Uniqueness of the measurement result
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-2. Bohr’s reply, non-separability
F-3. Bell’s inequality
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
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
Complement CXXI Measurement induced relative phase between two condensates
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 P (x1)
2-b. Entanglement between the two modes after the first detection
2-c. Measuring the double density P (x2, x1)
2-d. Discussion
3. Detection of a large number Q of particles
3-a. Probability of a multiple detection sequence
3-b. Discussion; emergence of a relative phase
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
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
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
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
4-b. Several operators
Appendix V: Lagrange multipliers
1. Function of two variables
2. Function of variables
Appendix VI: Brief review of Quantum Statistical Mechanics
1. Statistical ensembles
1-a. Microcanonical ensemble
1-b. Canonical ensemble
1-c. Grand canonical ensemble
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
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
2-c. Reality, normalization, operator form
2-d. Gaussian wave packet
2-e. Semiclassical situations
2-f. Quantum situations where the Wigner distribution is not a probability distribution
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
3-d. Evolution of the density operator
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
Bibliography of volume III
Index
EULA