| |

| |

| |

The Formalism and Its Interpretation | |

| |

| |

| |

The Origins of the Quantum Theory | |

| |

| |

| |

Introduction | |

| |

| |

| |

The end of the Classical Period | |

| |

| |

| |

Classical Theoretical Physics | |

| |

| |

| |

Progress in the knowledge of microscopic phenomena and the appearance of quanta in physics | |

| |

| |

| |

Light Quanta or Photons | |

| |

| |

| |

The photoelectric effect | |

| |

| |

| |

The Compton effect | |

| |

| |

| |

Light quanta and interference phenomena | |

| |

| |

| |

Conclusions | |

| |

| |

| |

Quantization of Material Systems | |

| |

| |

| |

Atomic spectroscopy and difficulties of Rutherford's classical model | |

| |

| |

| |

Quantization of atomic energy levels | |

| |

| |

| |

Other examples of quantization: space quantization | |

| |

| |

| |

Correspondence Principle and the Old Quantum Theory | |

| |

| |

| |

Inadequacy of classical corpuscular theory | |

| |

| |

| |

Correspondence principle | |

| |

| |

| |

Application of the correspondence principle to the calculation of the Rydberg constant | |

| |

| |

| |

Lagrange's and Hamilton's forms of the equations of classical mechanics | |

| |

| |

| |

Bohr-Sommerfeld quantization rules | |

| |

| |

| |

Successes and limitations of the Old Quantum Theory | |

| |

| |

| |

Conclusions | |

| |

| |

| |

Matter Waves and the Schrodinger Equation | |

| |

| |

| |

Historical survey and general plan of the succeeding chapters | |

| |

| |

| |

Matter Waves | |

| |

| |

| |

Introduction | |

| |

| |

| |

Free wave packet | |

| |

| |

Phase velocity and group velocity | |

| |

| |

| |

Wave packet in a slowly varying field | |

| |

| |

| |

Quantization of atomic energy levels | |

| |

| |

| |

Diffraction of matter waves. | |

| |

| |

| |

Corpuscular structure of matter | |

| |

| |

| |

Universal character of the wave-corpuscle duality | |

| |

| |

| |

The Schrodinger Equation | |

| |

| |

| |

Conservation law of the number of particles of matter | |

| |

| |

| |

Necessity for a wave equation and conditions imposed upon this equation | |

| |

| |

| |

The operator concept | |

| |

| |

| |

Wave equation of a free particle | |

| |

| |

| |

Particle in a scalar potential | |

| |

| |

| |

Charged particle in an electromagnetic field | |

| |

| |

| |

General rule for forming the Schrodinger equation by correspondence | |

| |

| |

| |

The Time-Independent Schrodinger Equation | |

| |

| |

| |

Search for stationary solutions | |

| |

| |

| |

General properties of the equation | |

| |

| |

Nature of the energy spectrum | |

| |

| |

| |

One-Dimensional Quantized Systems | |

| |

| |

| |

Introduction | |

| |

| |

| |

Square Potentials | |

| |

| |

| |

General remarks | |

| |

| |

| |

Potential step | |

| |

| |

Reflection and transmission of waves | |

| |

| |

| |

Infinitely high potential barrier | |

| |

| |

| |

Infinitely deep square potential well | |

| |

| |

Discrete spectrum | |

| |

| |

| |

Study of a finite square well. Resonances | |

| |

| |

| |

Penetration of a square potential barrier | |

| |

| |

The "tunnel" effect | |

| |

| |

| |

General Properties of the One-Dimensional Schrodinger Equation | |

| |

| |

| |

Property of the Wronskian | |

| |

| |

| |

Asymptotic behavior of the solutions | |

| |

| |

| |

Nature of the eigenvalue spectrum | |

| |

| |

| |

Unbound states: reflection and transmission of waves | |

| |

| |

| |

Number of nodes of bound states | |

| |

| |

| |

Orthogonality relations | |

| |

| |

| |

Remark on parity | |

| |

| |

| |

Statistical Interpretation of the Wave-Corpuscle Duality and the Uncertainty Relations | |

| |

| |

| |

Introduction | |

| |

| |

| |

Statistical Interpretation of the Wave Functions of Wave Mechanics | |

| |

| |

| |

Probabilities of the results of measurement of the position and the momentum of a particle | |

| |

| |

| |

Conservation in time of the norm | |

| |

| |

| |

Concept of current | |

| |

| |

| |

Mean values of functions of r or of p | |

| |

| |

| |

Generalization to systems of several particles | |

| |

| |

| |

Heisenberg's Uncertainty Relations | |

| |

| |

| |

Position-momentum uncertainty relations of a quantized particle | |

| |

| |

| |

Precise statement of the position-momentum uncertainty relations | |

| |

| |

| |

Generalization: uncertainty relations between conjugate variables | |

| |

| |

| |

Time-energy uncertainty relation | |

| |

| |

| |

Uncertainty relations for photons | |

| |

| |

| |

Uncertainty Relations and the Measurement Process | |

| |

| |

| |

Uncontrollable disturbance during the operation of measurement | |

| |

| |

| |

Position measurements | |

| |

| |

| |

Momentum measurements | |

| |

| |

| |

Description of Phenomena in Quantum Theory. Complementarity and Causality | |

| |

| |

| |

Problems raised by the statistical interpretation | |

| |

| |

| |

Description of microscopic phenomena and complementarity | |

| |

| |

| |

Complementary variables | |

| |

| |

Compatible variables | |

| |

| |

| |

Wave-corpuscle duality and complementarity | |

| |

| |

| |

Complementarity and causality | |

| |

| |

| |

Development of the Formalism of Wave Mechanics and Its Interpretation | |

| |

| |

| |

Introduction | |

| |

| |

| |

Hermitean Operators and Physical Quantities | |

| |

| |

| |

Wave-function space | |

| |

| |

| |

Definition of mean values | |

| |

| |

| |

Absence of fluctuation and the eigenvalue problem | |

| |

| |

| |

Study of the Discrete Spectrum | |

| |

| |

| |

Eigenvalues and eigenfunctions of a Hermitean operator | |

| |

| |

| |

Expansion of a wave function in a series of orthonormal eigenfunctions | |

| |

| |

| |

Statistical distribution of the results of measurement of a quantity associated with an operator having a complete set of eigenfunctions with finite norm | |

| |

| |

| |

Statistics of Measurement in the General Case | |

| |

| |

| |

Difficulties of the continuous spectrum. Introduction of the Dirac [delta]-functions | |

| |

| |

| |

Expansion in a series of eigenfunctions in the general case | |

| |

| |

Closure relation | |

| |

| |

| |

Statistical distribution of the results of measurement in the general case | |

| |

| |

| |

Other ways of treating the continuous spectrum | |

| |

| |

| |

Comments and examples | |

| |

| |

| |

Determination of the Wave Function | |

| |

| |

| |

Measuring process and "filtering" of the wave packet. Ideal measurements | |

| |

| |

| |

Commuting observables and compatible variables | |

| |

| |

| |

Complete sets of commuting observables | |

| |

| |

| |

Pure states and mixtures | |

| |

| |

| |

Commutator Algebra and Its Applications | |

| |

| |

| |

Commutator algebra and properties of basic commutators | |

| |

| |

| |

Commutation relations of angular momentum | |

| |

| |

| |

Time dependence of the statistical distribution | |

| |

| |

Constants of the motion | |

| |

| |

| |

Examples of constants of the motion | |

| |

| |

Energy | |

| |

| |

Parity | |

| |

| |

| |

Classical Approximation and the WKB Method | |

| |

| |

| |

The Classical Limit of Wave Mechanics | |

| |

| |

| |

General remarks | |

| |

| |

| |

Ehrenfest's theorem | |

| |

| |

| |

Motion and spreading of wave packets | |

| |

| |

| |

Classical limit of the Schrodinger equation | |

| |

| |

| |

Application to Coulomb scattering | |

| |

| |

The Rutherford formula | |

| |

| |

| |

The WKB Method | |

| |

| |

| |

Principle of the method | |

| |

| |

| |

One-dimensional WKB solutions | |

| |

| |

| |

Conditions for the validity of the WKB approximation | |

| |

| |

| |

Turning points and connection formulae | |

| |

| |

| |

Penetration of a potential barrier | |

| |

| |

| |

Energy levels of a potential well | |

| |

| |

| |

General Formalism of the Quantum Theory (A) Mathematical Framework | |

| |

| |

| |

Superposition principle and representation of dynamical states by vectors | |

| |

| |

| |

Vectors and Operators | |

| |

| |

| |

Vector space | |

| |

| |

"Ket" vectors | |

| |

| |

| |

Dual space | |

| |

| |

"Bra" vectors | |

| |

| |

| |

Scalar product | |

| |

| |

| |

Linear operators | |

| |

| |

| |

Tensor product of two vector spaces | |

| |

| |

| |

Hermitean Operators, Projectors, and Observables | |

| |

| |

| |

Adjoint operators and conjugation relations | |

| |

| |

| |

Hermitean (or self-adjoint) operators, positive definite Hermitean operators, unitary operators | |

| |

| |

| |

Eigenvalue problem and observables | |

| |

| |

| |

Projectors (Projection operators) | |

| |

| |

| |

Projector algebra | |

| |

| |

| |

Observables possessing an entirely discrete spectrum | |

| |

| |

| |

Observables in the general case | |

| |

| |

Generalized closure relation | |

| |

| |

| |

Functions of an observable | |

| |

| |

| |

Operators which commute with an observable | |

| |

| |

Commuting observables | |

| |

| |

| |

Representation Theory | |

| |

| |

| |

General remarks on finite matrices | |

| |

| |

| |

Square matrices | |

| |

| |

| |

Extension to infinite matrices | |

| |

| |

| |

Representation of vectors and operators by matrices | |

| |

| |

| |

Matrix transformations | |

| |

| |

| |

Change of representation | |

| |

| |

| |

Unitary transformations of operators and vectors | |

| |

| |

| |

General Formalism (B) Description of Physical Phenomena | |

| |

| |

| |

Introduction | |

| |

| |

| |

Dynamical States and Physical Quantities | |

| |

| |

| |

Definition of probabilities | |

| |

| |

Postulates concerning measurement | |

| |

| |

| |

Observables of a quantized system and their commutation relations | |

| |

| |

| |

Heisenberg's uncertainty relations | |

| |

| |

| |

Definition of the dynamical states and construction of the space and | |

| |

| |

| |

One-dimensional quantum system having a classical analogue | |

| |

| |

| |

Construction of the and-space of a system by tensor product of simpler spaces | |

| |

| |

| |

The Equations of Motion | |

| |

| |

| |

Evolution operator and the Schrodinger equation | |

| |

| |

| |

Schrodinger "representation" | |

| |

| |

| |

Heisenberg "representation" | |

| |

| |

| |

Heisenberg "representation" and correspondence principle | |

| |

| |

| |

Constants of the motion | |

| |

| |

| |

Equations of motion for the mean values Time-energy uncertainty relation | |

| |

| |

| |

Intermediate representations | |

| |

| |

| |

Various Representations of the Theory | |

| |

| |

| |

Definition of a representation | |

| |

| |

| |

Wave mechanics | |

| |

| |

| |

Momentum representation ({p}-representation) | |

| |

| |

| |

An example: motion of a free wave packet | |

| |

| |

| |

Other representations. Representations in which the energy is diagonal | |

| |

| |

| |

Quantum Statistics | |

| |

| |

| |

Incompletely known systems and statistical mixtures | |

| |

| |

| |

The density operator | |

| |

| |

| |

Evolution in time of a statistical mixture | |

| |

| |

| |

Characteristic properties of the density operator | |

| |

| |

| |

Pure states | |

| |

| |

| |

Classical and quantum statistics | |

| |

| |

| |

Simple Systems | |

| |

| |

| |

Solution of the Schrodinger Equation by Separation of Variables. Central Potential | |

| |

| |

| |

Introduction | |

| |

| |

| |

Particle in a Central Potential. General Treatment | |

| |

| |

| |

Expression of the Hamiltonian in spherical polar coordinates | |

| |

| |

| |

Separation of the angular variables | |

| |

| |

Spherical harmonics | |

| |

| |

| |

The radial equation | |

| |

| |

| |

Eigensolutions of the radial equation | |

| |

| |

Nature of the spectrum | |

| |

| |

| |

Conclusions | |

| |

| |

| |

Central Square-Well Potential. Free Particle | |

| |

| |

| |

Spherical Bessel functions | |

| |

| |

| |

Free particle | |

| |

| |

Plane waves and free spherical waves | |

| |

| |

| |

Expansion of a plane wave in spherical harmonics | |

| |

| |

| |

Study of a spherical square well | |

| |

| |

| |

Two-body Problems. Separation of the Center-of-Mass Motion | |

| |

| |

| |

Separation of the center-of-mass motion in classical mechanics | |

| |

| |

| |

Separation of the center-of-mass motion of a quantized two-particle system | |

| |

| |

| |

Extension to systems of more than two particles | |

| |

| |

| |

Scattering Problems Central Potential and Phase-Shift Method | |

| |

| |

| |

Introduction | |

| |

| |

| |

Cross Sections and Scattering Amplitudes | |

| |

| |

| |

Definition of cross sections | |

| |

| |

| |

Stationary wave of scattering | |

| |

| |

| |

Representation of the scattering phenomenon by a bundle of wave packets | |

| |

| |

| |

Scattering of a wave packet by a potential | |

| |

| |

| |

Calculation of cross sections | |

| |

| |

| |

Collision of two particles | |

| |

| |

Laboratory system and center-of-mass system | |

| |

| |

| |

Scattering by a Central Potential. Phase Shifts | |

| |

| |

| |

Decomposition into partial waves | |

| |

| |

Phase-shift method | |

| |

| |

| |

Semiclassical representation of the collision | |

| |

| |

Impact parameters | |

| |

| |

| |

Potential of Finite Range | |

| |

| |

| |

Relation between phase shift and logarithmic derivative | |

| |

| |

| |

Behavior of the phase shift at low energies | |

| |

| |

| |

Partial waves of higher order | |

| |

| |

Convergence of the series | |

| |

| |

| |

Scattering by a hard sphere | |

| |

| |

| |

Scattering Resonances | |

| |

| |

| |

Scattering by a deep square well | |

| |

| |

| |

Study of a scattering resonance | |

| |

| |

Metastable states | |

| |

| |

| |

Observation of the lifetime of metastable states | |

| |

| |

| |

Various Formulae and Properties | |

| |

| |

| |

Integral representations of phase shifts | |

| |

| |

| |

Dependence upon the potential | |

| |

| |

Sign of the phase shifts | |

| |

| |

| |

The Born approximation | |

| |

| |

| |

Effective range theory | |

| |

| |

The Bethe formula | |

| |

| |

| |

The Coulomb Interaction | |

| |

| |

| |

Introduction | |

| |

| |

| |

The Hydrogen Atom | |

| |

| |

| |

Schrodinger equation of the hydrogen atom | |

| |

| |

| |

Order of magnitude of the binding energy of the ground state | |

| |

| |

| |

Solution of the Schrodinger equation in spherical coordinates | |

| |

| |

| |

Energy spectrum. Degeneracy | |

| |

| |

| |

The eigenfunctions of the bound states | |

| |

| |

| |

Coulomb Scattering | |

| |

| |

| |

The Coulomb scattering wave | |

| |

| |

| |

The Rutherford formula | |

| |

| |

| |

Decomposition into partial waves | |

| |

| |

| |

Expansion of the wave [psi subscript c] in spherical harmonics | |

| |

| |

| |

Modifications of the Coulomb potential by a short-range interaction | |

| |

| |

| |

The Harmonic Oscillator | |

| |

| |

| |

Introduction | |

| |

| |

| |

Eigenstates and Eigenvectors of the Hamiltonian | |

| |

| |

| |

The eigenvalue problem | |

| |

| |

| |

Introduction of the operators a, a and N | |

| |

| |

| |

Spectrum and basis of N | |

| |

| |

| |

The {N} representation | |

| |

| |

| |

Creation and destruction operators | |

| |

| |

| |

{Q} representation. Hermite polynomials | |

| |

| |

| |

Applications and Various Properties | |

| |

| |

| |

Generating function for the eigenfunctions u[subscript n](Q) | |

| |

| |

| |

Integration of the Heisenberg equations | |

| |

| |

| |

Classical and quantized oscillator | |

| |

| |

| |

Motion of the minimum wave packet and classical limit | |

| |

| |

| |

Harmonic oscillators in thermodynamic equilibrium | |

| |

| |

| |

Isotropic Harmonic Oscillators in Several Dimensions | |

| |

| |

| |

General treatment of the isotropic oscillator in p dimensions | |

| |

| |

| |

Two-dimensional isotropic oscillator | |

| |

| |

| |

Three-dimensional isotropic oscillator | |

| |

| |

| |

Distributions, [delta]-"Function" and Fourier Transformation | |

| |

| |

| |

Special Functions and Associated Formulae | |

| |

| |

| |

Symmetries and Invariance | |

| |

| |

| |

Angular Momentum in Quantum Mechanics | |

| |

| |

| |

Introduction | |

| |

| |

| |

Eigenvalues and eigenfunctions of angular momentum | |

| |

| |

| |

Definition of angular momentum | |

| |

| |

| |

Characteristic algebraic relations | |

| |

| |

| |

Spectrum of J[superscript 2] and J[subscript z] | |

| |

| |

| |

Eigenvectors of J[superscript 2] and J[subscript z]. Construction of the invariant subspaces E(j) | |

| |

| |

| |

Standard representation {J[superscript 2] J[subscript z]} | |

| |

| |

| |

Conclusion | |

| |

| |

| |

Orbital angular momentum and the spherical harmonics | |

| |

| |

| |

The spectrum of l[superscript 2] and l[subscript z] | |

| |

| |

| |

Definition and construction of the spherical harmonics | |

| |

| |

| |

Angular momentum and rotations | |

| |

| |

| |

Definition of rotation | |

| |

| |

Euler angles | |

| |

| |

| |

Rotation of a physical system | |

| |

| |

Rotation operator | |

| |

| |

| |

Rotation of observables | |

| |

| |

| |

Angular momentum and infinitesimal rotations | |

| |

| |

| |

Construction of the operator R ([alpha] [beta] [gamma]) | |

| |

| |

| |

Rotation through an angle 2[pi] and half-integral angular momenta | |

| |

| |

| |

Irreducible invariant subspaces | |

| |

| |

Rotation matrices R[superscript (j)] | |

| |

| |

| |

Rotational invariance and conservation of angular momentum | |

| |

| |

Rotational degeneracy | |

| |

| |

| |

Spin | |

| |

| |

| |

The hypothesis of electron spin | |

| |

| |

| |

Spin 1/2 and the Pauli matrices | |

| |

| |

| |

Observables and wave functions of a spin 1/2 particle. Spinor fields | |

| |

| |

| |

Vector fields and particles of spin 1 | |

| |

| |

| |

Spindependent interactions in atoms | |

| |

| |

| |

Spin-dependent nucleon-nucleon interactions | |

| |

| |

| |

Addition of angular momenta | |

| |

| |

| |

The addition problem | |

| |

| |

| |

Addition theorem for two angular momenta | |

| |

| |

| |

Applications and examples | |

| |

| |

| |

Eigenvectors of the total angular momentum | |

| |

| |

Clebsch-Gordon coefficients | |

| |

| |

| |

Application: two-nucleon system | |

| |

| |

| |

Addition of three or more angular momenta | |

| |

| |

Racah coefficients. "3sj" symbols | |

| |

| |

| |

Irreducible tensor operators | |

| |

| |

| |

Representation of scalar operators | |

| |

| |

| |

Irreducible tensor operators | |

| |

| |

Definition | |

| |

| |

| |

Representation of irreducible tensor operators | |

| |

| |

Wigner-Eckhart theorem | |

| |

| |

| |

Applications | |

| |

| |

| |

Systems of Identical Particles. Pauli Exclusion Principle | |

| |

| |

| |

Identical particles in quantum theory | |

| |

| |

| |

Symmetrization postulate | |

| |

| |

| |

Similar particles and the symmetrical representation | |

| |

| |

| |

Permutation operators | |

| |

| |

| |

Algebra of permutation operators | |

| |

| |

Symmetrizers and antisymmetrizers | |

| |

| |

| |

Identical particles and the symmetrization postulate | |

| |

| |

| |

Bosons and Bose-Einstein statistics | |

| |

| |

| |

Fermions and Fermi-Dirac statistics | |

| |

| |

Exclusion principle | |

| |

| |

| |

It is always necessary to symmetrize the wave-function | |

| |

| |

| |

Applications | |

| |

| |

| |

Collision of two spinless identical particles | |

| |

| |

| |

Collision of two protons | |

| |

| |

| |

Statistics of atomic nuclei | |

| |

| |

| |

Complex atoms | |

| |

| |

Central field approximation | |

| |

| |

| |

The Thomas-Fermi model of the atom | |

| |

| |

| |

Nucleon systems and isotopic spin | |

| |

| |

| |

Utility of isotopic spin | |

| |

| |

Charge independence | |

| |

| |

| |

Invariance and Conservation Theorems. Time Reversal | |

| |

| |

| |

Introduction | |

| |

| |

| |

Mathematical complements. Antilinear operators | |

| |

| |

| |

Three useful theorems | |

| |

| |

| |

Antilinear operators in Hilbert space | |

| |

| |

| |

Antilinear transformations | |

| |

| |

| |

Antilinear operators and representations | |

| |

| |

| |

Transformations and groups of transformations | |

| |

| |

| |

Transformations of the dynamical variables and dynamical states of a system | |

| |

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Groups of transformations | |

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Groups of transformation operators | |

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Continuous groups and infinitesimal transformations | |

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Translations | |

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Rotations | |

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Finite groups | |

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Reflections | |

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Invariance of the equations of motion and conservation laws | |

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Invariant observables | |

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Symmetry of the Hamiltonian and conservation laws | |

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Invariance properties and the evolution of dynamical states | |

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Symmetries of the Stark and Zeeman effects | |

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Time reversal and the principle of microreversibility | |

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Time translation and conservation of energy | |

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Time reversal in classical mechanics and in quantum mechanics | |

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The time-reversal operation | |

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Spinless particle | |

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General definition of time reversal | |

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Time reversal and complex conjugation | |

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Principle of microreversibility | |

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Consequence: Kramers degeneracy | |

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Real rotation-invariant Hamiltonian | |

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Methods of Approximation | |

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Stationary Perturbations | |

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General introduction to Part Four | |

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Perturbation of a non-degenerate level | |

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Expansion in powers of the perturbation | |

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First-order perturbations | |

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Ground state of the helium atom | |

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Coulomb energy of atomic nuclei | |

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Higher-order corrections | |

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Stark effect for a rigid rotator | |

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Perturbation of a degenerate level | |

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Elementary theory | |

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Atomic levels in the absence of spin-orbit forces | |

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Spin-orbit forces | |

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LS and jj coupling | |

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The atom in LS coupling | |

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Splitting due to spin-orbital coupling | |

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The Zeeman and Paschen-Back effects | |

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Symmetry of H and removal of degeneracy | |

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Quasi-degeneracy | |

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Explicit forms for the perturbation expansion in all orders | |

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The Hamiltonian H and its resolvent G(z) | |

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Expansion of G(z), P and HP into power series in V | |

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Calculation of eigenvalues and eigenstates | |

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Approximate Solutions of the Time-Dependent Schrodinger Equation | |

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Change of "representation" and perturbation treatment of a part of the Hamiltonian | |

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Time dependent perturbation theory | |

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Definition and perturbation calculation of transition probabilities | |

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Semi-classical theory of Coulomb excitation of nuclei | |

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Case when V is independent of time | |

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Conservation of unperturbed energy | |

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Application to the calculation of cross-sections in the Born approximation | |

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Periodic perturbation. Resonances | |

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Sudden or Adiabatic Change of the Hamiltonian | |

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The problem and the results | |

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Rapid passage and the sudden approximation | |

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Sudden reversal of a magnetic field | |

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Adiabatic passage | |

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Generalities | |

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Trivial case | |

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"Rotating axis representation" | |

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Proof of the adiabatic theorem | |

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Adiabatic approximation | |

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Adiabatic reversal of a magnetic field | |

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The Variational Method and Associated Problems | |

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The Ritz variational method | |

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Variational Method for Bound States | |

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Variational form of the eigenvalue problem | |

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Variational calculation of discrete levels | |

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A simple example: the hydrogen atom | |

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Discussion | |

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Application to the calculation of excited levels | |

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Ground state of the helium atom | |

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The Hartree and Fock-Dirac Atoms | |

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The self-consistent field method | |

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Calculation of E[Phi] | |

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The Fock-Dirac equations | |

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Discussion | |

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The Hartree equations | |

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The Structure of Molecules | |

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Generalities | |

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Separation of the electronic and nuclear motions | |

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Motion of the electrons in the presence of fixed nuclei | |

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The adiabatic approximation | |

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Hamiltonian for the nuclei in the adiabatic approximation | |

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The Born-Oppenheimer method | |

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Notions on diatomic molecules | |

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Collision Theory | |

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Introduction | |

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Free Wave Green's Function and the Born Approximation | |

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Integral representations of the scattering amplitude | |

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Cross sections and the T matrix | |

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Microreversibility | |

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The Born approximation | |

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Integral equation for scattering | |

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The Born expansion | |

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Validity criterion for the Born approximation | |

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Elastic scattering of electrons by an atom | |

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Central potential | |

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Calculation of phase shifts | |

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Green's function as an operator | |

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Relation to the resolvent of H[subscript 0] | |

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Generalization to Distorted Waves | |

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Generalized Born approximation | |

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Generalization of the Born expansion | |

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Green's functions for distorted waves | |

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Applications | |

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Definition and formal properties of T | |

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Note on the 1/4 potentials | |

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Complex Collisions and the Born Approximation | |

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Generalities | |

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Cross sections | |

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Channels | |

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Calculation of cross sections | |

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T matrices | |

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Integral representations of the transition amplitude | |

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The Born approximation and its generalizations | |

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Scattering of fast electrons by an atom | |

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Coulomb excitation of nuclei | |

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Green's functions and integral equations for stationary scattering waves | |

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Scattering of a particle by two scattering centers | |

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Simple scattering | |

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Interference | |

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Multiple scattering | |

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Variational Calculations of Transition Amplitudes | |

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Stationary expressions for the phase shifts | |

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The variational calculation of phase shifts | |

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Discussion | |

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Extension to complex collisions | |

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General Properties of the Transition Matrix | |

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Conservation of flux | |

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Unitarity of the S matrix | |

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The Bohr-Peierls-Placzek relation (optical theorem) | |

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Microreversibility | |

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Invariance properties of the T matrix | |

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Elements of Relativistic Quantum Mechanics | |

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The Dirac Equation | |

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General Introduction | |

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Relativistic quantum mechanics | |

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Notation, various conventions and definitions | |

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The Lorentz group | |

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Classical relativistic dynamics | |

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The Dirac and Klein-Gordon Equations | |

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The Klein-Gordon equation | |

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The Dirac equation | |

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Construction of the space E[superscript (s)] | |

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Dirac representation | |

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Covariant form of the Dirac equation | |

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Adjoint equation | |

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Definition of the current | |

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Invariance Properties of the Dirac Equation | |

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Properties of the Dirac matrices | |

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Invariance of the form of the Dirac equation in an orthochronous change of referential | |

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Transformation of the proper group | |

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Spatial reflection and the orthochronous group | |

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Construction of covariant quantities | |

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A second formulation of the invariance of form: transformation of states | |

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Invariance of the law of motion | |

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Transformation operators | |

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Momentum, angular momentum, parity | |

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Conservation laws and constants of the motion | |

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Time reversal and charge conjugation. | |

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Gauge invariance | |

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Interpretation of the Operators and Simple Solutions | |

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The Dirac equation and the correspondence principle | |

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Dynamical variables of a Dirac particle | |

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The free electron | |

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Plane waves | |

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Construction of the plane waves by a Lorentz transformation | |

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Central potential | |

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Free spherical waves | |

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The hydrogen atom | |

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Non-Relativistic Limit of the Dirac Equation | |

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Large and small components | |

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The Pauli theory as the non-relativistic limit of the Dirac theory | |

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Application: hyperfine structure and dipole-dipole coupling | |

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Higher-order corrections and the Foldy-Wouthuysen transformation | |

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FW transformation for a free particle | |

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FW transformation for a particle in a field | |

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Electron in a central electrostatic potential | |

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Discussions and conclusions | |

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Negative Energy Solutions and Positron Theory | |

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Properties of charge conjugate solutions | |

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Abnormal behavior of the negative energy solutions | |

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Reinterpretation of the negative energy states | |

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Theory of "holes" and positrons | |

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Difficulties with the "hole" theory | |

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Field Quantization. Radiation Theory | |

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Introduction | |

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Quantization of a Real Scalar Field | |

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Classical free field | |

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Normal vibrations | |

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Quantization of the free field | |

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Lagrangian of the field | |

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Momentum conjugate to [Phi](r) | |

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Complex basis functions | |

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Plane waves | |

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Definition of the momentum | |

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Spherical waves | |

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Definition of the angular momentum | |

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Space and time reflections | |

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Coupling With an Atomic System | |

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Coupling to a system of particles | |

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Weak coupling and perturbation treatment | |

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Level shifts | |

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Emission of a corpuscle | |

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Quantum theory of decaying states | |

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Line width | |

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Elastic scattering | |

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Dispersion formula | |

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Resonance scattering | |

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Formation of a metastable state | |

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Absorption of a corpuscle (photo-electric effect) | |

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Radiative capture | |

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Classical Theory of Electromagnetic Radiation | |

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The equations of the classical Maxwell-Lorentz theory | |

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Symmetries and conservation laws of the classical theory | |

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Self-energy and classical radius of the electron. | |

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Electromagnetic potential. | |

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Choice of the gauge | |

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Longitudinal and transverse parts of a vector field | |

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Elimination of the lopgitudinal field | |

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Energy, momentum, angular momentum | |

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Hamiltonian for free radiation | |

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Hamiltonian for radiation coupled to a set of particles | |

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Quantum Theory of Radiation | |

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Quantization of free radiation | |

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Photons | |

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Plane waves | |

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Radiation momentum | |

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Polarization | |

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Multipole expansion | |

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Photons of determined angular momentum and parity | |

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Coupling with an atomic system | |

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Emission of a photon by an atom | |

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Dipole emission | |

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Low energy Compton scattering | |

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The Thomson formula | |

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Vector Addition Coefficients and Rotation Matrices | |

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Elements of Group Theory | |

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General Index | |