| |
| |
| |
| 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 | |
| |
| |
| |
| Groups of transformations | |
| |
| |
| |
| Groups of transformation operators | |
| |
| |
| |
| Continuous groups and infinitesimal transformations | |
| |
| |
| Translations | |
| |
| |
| Rotations | |
| |
| |
| |
| Finite groups | |
| |
| |
| Reflections | |
| |
| |
| |
| Invariance of the equations of motion and conservation laws | |
| |
| |
| |
| Invariant observables | |
| |
| |
| |
| Symmetry of the Hamiltonian and conservation laws | |
| |
| |
| |
| Invariance properties and the evolution of dynamical states | |
| |
| |
| |
| Symmetries of the Stark and Zeeman effects | |
| |
| |
| |
| Time reversal and the principle of microreversibility | |
| |
| |
| |
| Time translation and conservation of energy | |
| |
| |
| |
| Time reversal in classical mechanics and in quantum mechanics | |
| |
| |
| |
| The time-reversal operation | |
| |
| |
| Spinless particle | |
| |
| |
| |
| General definition of time reversal | |
| |
| |
| |
| Time reversal and complex conjugation | |
| |
| |
| |
| Principle of microreversibility | |
| |
| |
| |
| Consequence: Kramers degeneracy | |
| |
| |
| |
| Real rotation-invariant Hamiltonian | |
| |
| |
| |
| Methods of Approximation | |
| |
| |
| |
| Stationary Perturbations | |
| |
| |
| |
| General introduction to Part Four | |
| |
| |
| |
| Perturbation of a non-degenerate level | |
| |
| |
| |
| Expansion in powers of the perturbation | |
| |
| |
| |
| First-order perturbations | |
| |
| |
| |
| Ground state of the helium atom | |
| |
| |
| |
| Coulomb energy of atomic nuclei | |
| |
| |
| |
| Higher-order corrections | |
| |
| |
| |
| Stark effect for a rigid rotator | |
| |
| |
| |
| Perturbation of a degenerate level | |
| |
| |
| |
| Elementary theory | |
| |
| |
| |
| Atomic levels in the absence of spin-orbit forces | |
| |
| |
| |
| Spin-orbit forces | |
| |
| |
| LS and jj coupling | |
| |
| |
| |
| The atom in LS coupling | |
| |
| |
| Splitting due to spin-orbital coupling | |
| |
| |
| |
| The Zeeman and Paschen-Back effects | |
| |
| |
| |
| Symmetry of H and removal of degeneracy | |
| |
| |
| |
| Quasi-degeneracy | |
| |
| |
| |
| Explicit forms for the perturbation expansion in all orders | |
| |
| |
| |
| The Hamiltonian H and its resolvent G(z) | |
| |
| |
| |
| Expansion of G(z), P and HP into power series in V | |
| |
| |
| |
| Calculation of eigenvalues and eigenstates | |
| |
| |
| |
| Approximate Solutions of the Time-Dependent Schrodinger Equation | |
| |
| |
| |
| Change of "representation" and perturbation treatment of a part of the Hamiltonian | |
| |
| |
| |
| Time dependent perturbation theory | |
| |
| |
| |
| Definition and perturbation calculation of transition probabilities | |
| |
| |
| |
| Semi-classical theory of Coulomb excitation of nuclei | |
| |
| |
| |
| Case when V is independent of time | |
| |
| |
| Conservation of unperturbed energy | |
| |
| |
| |
| Application to the calculation of cross-sections in the Born approximation | |
| |
| |
| |
| Periodic perturbation. Resonances | |
| |
| |
| |
| Sudden or Adiabatic Change of the Hamiltonian | |
| |
| |
| |
| The problem and the results | |
| |
| |
| |
| Rapid passage and the sudden approximation | |
| |
| |
| |
| Sudden reversal of a magnetic field | |
| |
| |
| |
| Adiabatic passage | |
| |
| |
| Generalities | |
| |
| |
| Trivial case | |
| |
| |
| |
| "Rotating axis representation" | |
| |
| |
| |
| Proof of the adiabatic theorem | |
| |
| |
| |
| Adiabatic approximation | |
| |
| |
| |
| Adiabatic reversal of a magnetic field | |
| |
| |
| |
| The Variational Method and Associated Problems | |
| |
| |
| |
| The Ritz variational method | |
| |
| |
| |
| Variational Method for Bound States | |
| |
| |
| |
| Variational form of the eigenvalue problem | |
| |
| |
| |
| Variational calculation of discrete levels | |
| |
| |
| |
| A simple example: the hydrogen atom | |
| |
| |
| |
| Discussion | |
| |
| |
| Application to the calculation of excited levels | |
| |
| |
| |
| Ground state of the helium atom | |
| |
| |
| |
| The Hartree and Fock-Dirac Atoms | |
| |
| |
| |
| The self-consistent field method | |
| |
| |
| |
| Calculation of E[Phi] | |
| |
| |
| |
| The Fock-Dirac equations | |
| |
| |
| |
| Discussion | |
| |
| |
| |
| The Hartree equations | |
| |
| |
| |
| The Structure of Molecules | |
| |
| |
| |
| Generalities | |
| |
| |
| Separation of the electronic and nuclear motions | |
| |
| |
| |
| Motion of the electrons in the presence of fixed nuclei | |
| |
| |
| |
| The adiabatic approximation | |
| |
| |
| |
| Hamiltonian for the nuclei in the adiabatic approximation | |
| |
| |
| |
| The Born-Oppenheimer method | |
| |
| |
| |
| Notions on diatomic molecules | |
| |
| |
| |
| Collision Theory | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Free Wave Green's Function and the Born Approximation | |
| |
| |
| |
| Integral representations of the scattering amplitude | |
| |
| |
| |
| Cross sections and the T matrix | |
| |
| |
| Microreversibility | |
| |
| |
| |
| The Born approximation | |
| |
| |
| |
| Integral equation for scattering | |
| |
| |
| |
| The Born expansion | |
| |
| |
| |
| Validity criterion for the Born approximation | |
| |
| |
| |
| Elastic scattering of electrons by an atom | |
| |
| |
| |
| Central potential | |
| |
| |
| Calculation of phase shifts | |
| |
| |
| |
| Green's function as an operator | |
| |
| |
| Relation to the resolvent of H[subscript 0] | |
| |
| |
| |
| Generalization to Distorted Waves | |
| |
| |
| |
| Generalized Born approximation | |
| |
| |
| |
| Generalization of the Born expansion | |
| |
| |
| |
| Green's functions for distorted waves | |
| |
| |
| |
| Applications | |
| |
| |
| Definition and formal properties of T | |
| |
| |
| |
| Note on the 1/4 potentials | |
| |
| |
| |
| Complex Collisions and the Born Approximation | |
| |
| |
| |
| Generalities | |
| |
| |
| Cross sections | |
| |
| |
| |
| Channels | |
| |
| |
| |
| Calculation of cross sections | |
| |
| |
| T matrices | |
| |
| |
| |
| Integral representations of the transition amplitude | |
| |
| |
| |
| The Born approximation and its generalizations | |
| |
| |
| |
| Scattering of fast electrons by an atom | |
| |
| |
| |
| Coulomb excitation of nuclei | |
| |
| |
| |
| Green's functions and integral equations for stationary scattering waves | |
| |
| |
| |
| Scattering of a particle by two scattering centers | |
| |
| |
| |
| Simple scattering | |
| |
| |
| Interference | |
| |
| |
| |
| Multiple scattering | |
| |
| |
| |
| Variational Calculations of Transition Amplitudes | |
| |
| |
| |
| Stationary expressions for the phase shifts | |
| |
| |
| |
| The variational calculation of phase shifts | |
| |
| |
| Discussion | |
| |
| |
| |
| Extension to complex collisions | |
| |
| |
| |
| General Properties of the Transition Matrix | |
| |
| |
| |
| Conservation of flux | |
| |
| |
| Unitarity of the S matrix | |
| |
| |
| |
| The Bohr-Peierls-Placzek relation (optical theorem) | |
| |
| |
| |
| Microreversibility | |
| |
| |
| |
| Invariance properties of the T matrix | |
| |
| |
| |
| Elements of Relativistic Quantum Mechanics | |
| |
| |
| |
| The Dirac Equation | |
| |
| |
| |
| General Introduction | |
| |
| |
| |
| Relativistic quantum mechanics | |
| |
| |
| |
| Notation, various conventions and definitions | |
| |
| |
| |
| The Lorentz group | |
| |
| |
| |
| Classical relativistic dynamics | |
| |
| |
| |
| The Dirac and Klein-Gordon Equations | |
| |
| |
| |
| The Klein-Gordon equation | |
| |
| |
| |
| The Dirac equation | |
| |
| |
| |
| Construction of the space E[superscript (s)] | |
| |
| |
| Dirac representation | |
| |
| |
| |
| Covariant form of the Dirac equation | |
| |
| |
| |
| Adjoint equation | |
| |
| |
| Definition of the current | |
| |
| |
| |
| Invariance Properties of the Dirac Equation | |
| |
| |
| |
| Properties of the Dirac matrices | |
| |
| |
| |
| Invariance of the form of the Dirac equation in an orthochronous change of referential | |
| |
| |
| |
| Transformation of the proper group | |
| |
| |
| |
| Spatial reflection and the orthochronous group | |
| |
| |
| |
| Construction of covariant quantities | |
| |
| |
| |
| A second formulation of the invariance of form: transformation of states | |
| |
| |
| |
| Invariance of the law of motion | |
| |
| |
| |
| Transformation operators | |
| |
| |
| Momentum, angular momentum, parity | |
| |
| |
| |
| Conservation laws and constants of the motion | |
| |
| |
| |
| Time reversal and charge conjugation. | |
| |
| |
| |
| Gauge invariance | |
| |
| |
| |
| Interpretation of the Operators and Simple Solutions | |
| |
| |
| |
| The Dirac equation and the correspondence principle | |
| |
| |
| |
| Dynamical variables of a Dirac particle | |
| |
| |
| |
| The free electron | |
| |
| |
| Plane waves | |
| |
| |
| |
| Construction of the plane waves by a Lorentz transformation | |
| |
| |
| |
| Central potential | |
| |
| |
| |
| Free spherical waves | |
| |
| |
| |
| The hydrogen atom | |
| |
| |
| |
| Non-Relativistic Limit of the Dirac Equation | |
| |
| |
| |
| Large and small components | |
| |
| |
| |
| The Pauli theory as the non-relativistic limit of the Dirac theory | |
| |
| |
| |
| Application: hyperfine structure and dipole-dipole coupling | |
| |
| |
| |
| Higher-order corrections and the Foldy-Wouthuysen transformation | |
| |
| |
| |
| FW transformation for a free particle | |
| |
| |
| |
| FW transformation for a particle in a field | |
| |
| |
| |
| Electron in a central electrostatic potential | |
| |
| |
| |
| Discussions and conclusions | |
| |
| |
| |
| Negative Energy Solutions and Positron Theory | |
| |
| |
| |
| Properties of charge conjugate solutions | |
| |
| |
| |
| Abnormal behavior of the negative energy solutions | |
| |
| |
| |
| Reinterpretation of the negative energy states | |
| |
| |
| Theory of "holes" and positrons | |
| |
| |
| |
| Difficulties with the "hole" theory | |
| |
| |
| |
| Field Quantization. Radiation Theory | |
| |
| |
| |
| Introduction | |
| |
| |
| |
| Quantization of a Real Scalar Field | |
| |
| |
| |
| Classical free field | |
| |
| |
| Normal vibrations | |
| |
| |
| |
| Quantization of the free field | |
| |
| |
| |
| Lagrangian of the field | |
| |
| |
| Momentum conjugate to [Phi](r) | |
| |
| |
| |
| Complex basis functions | |
| |
| |
| |
| Plane waves | |
| |
| |
| Definition of the momentum | |
| |
| |
| |
| Spherical waves | |
| |
| |
| Definition of the angular momentum | |
| |
| |
| |
| Space and time reflections | |
| |
| |
| |
| Coupling With an Atomic System | |
| |
| |
| |
| Coupling to a system of particles | |
| |
| |
| |
| Weak coupling and perturbation treatment | |
| |
| |
| |
| Level shifts | |
| |
| |
| |
| Emission of a corpuscle | |
| |
| |
| |
| Quantum theory of decaying states | |
| |
| |
| Line width | |
| |
| |
| |
| Elastic scattering | |
| |
| |
| Dispersion formula | |
| |
| |
| |
| Resonance scattering | |
| |
| |
| Formation of a metastable state | |
| |
| |
| |
| Absorption of a corpuscle (photo-electric effect) | |
| |
| |
| Radiative capture | |
| |
| |
| |
| Classical Theory of Electromagnetic Radiation | |
| |
| |
| |
| The equations of the classical Maxwell-Lorentz theory | |
| |
| |
| |
| Symmetries and conservation laws of the classical theory | |
| |
| |
| |
| Self-energy and classical radius of the electron. | |
| |
| |
| |
| Electromagnetic potential. | |
| |
| |
| Choice of the gauge | |
| |
| |
| |
| Longitudinal and transverse parts of a vector field | |
| |
| |
| |
| Elimination of the lopgitudinal field | |
| |
| |
| |
| Energy, momentum, angular momentum | |
| |
| |
| |
| Hamiltonian for free radiation | |
| |
| |
| |
| Hamiltonian for radiation coupled to a set of particles | |
| |
| |
| |
| Quantum Theory of Radiation | |
| |
| |
| |
| Quantization of free radiation | |
| |
| |
| Photons | |
| |
| |
| |
| Plane waves | |
| |
| |
| Radiation momentum | |
| |
| |
| |
| Polarization | |
| |
| |
| |
| Multipole expansion | |
| |
| |
| Photons of determined angular momentum and parity | |
| |
| |
| |
| Coupling with an atomic system | |
| |
| |
| |
| Emission of a photon by an atom | |
| |
| |
| Dipole emission | |
| |
| |
| |
| Low energy Compton scattering | |
| |
| |
| The Thomson formula | |
| |
| |
| |
| Vector Addition Coefficients and Rotation Matrices | |
| |
| |
| |
| Elements of Group Theory | |
| |
| |
| General Index | |