Quantum Mechanics 1-3. Basic Concepts, Tools, and Applications; Angular Momentum, Spin, and Approximation; Fermions, Bosons, Photons, Correlations, and Entanglement

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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