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The Formalism and Its Interpretation | |
The Origins of the Quantum Theory | |
Introduction | p. 3 |
The end of the Classical Period | p. 4 |
Classical Theoretical Physics | |
Progress in the knowledge of microscopic phenomena and the appearance of quanta in physics | |
Light Quanta or Photons | p. 11 |
The photoelectric effect | |
The Compton effect | |
Light quanta and interference phenomena | |
Conclusions | |
Quantization of Material Systems | p. 21 |
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 | p. 27 |
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 | p. 45 |
Matter Waves | p. 49 |
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 | p. 59 |
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 | p. 71 |
Search for stationary solutions | |
General properties of the equation | |
Nature of the energy spectrum | |
One-Dimensional Quantized Systems | |
Introduction | p. 77 |
Square Potentials | p. 78 |
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 | p. 98 |
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 | p. 115 |
Statistical Interpretation of the Wave Functions of Wave Mechanics | p. 116 |
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 | p. 129 |
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 | p. 139 |
Uncontrollable disturbance during the operation of measurement | |
Position measurements | |
Momentum measurements | |
Description of Phenomena in Quantum Theory. Complementarity and Causality | p. 149 |
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 | p. 162 |
Hermitean Operators and Physical Quantities | p. 163 |
Wave-function space | |
Definition of mean values | |
Absence of fluctuation and the eigenvalue problem | |
Study of the Discrete Spectrum | p. 171 |
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 | p. 179 |
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 | p. 196 |
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 | p. 206 |
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 | p. 214 |
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 | p. 231 |
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 | p. 243 |
Vectors and Operators | p. 245 |
Vector space | |
"Ket" vectors | |
Dual space | |
"Bra" vectors | |
Scalar product | |
Linear operators | |
Tensor product of two vector spaces | |
Hermitean Operators, Projectors, and Observables | p. 254 |
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 | p. 273 |
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 | p. 294 |
Dynamical States and Physical Quantities | p. 296 |
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 | p. 310 |
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 | p. 323 |
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 | p. 331 |
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 | p. 343 |
Particle in a Central Potential. General Treatment | p. 344 |
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 | p. 355 |
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 | p. 361 |
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 | p. 369 |
Cross Sections and Scattering Amplitudes | p. 369 |
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 | p. 385 |
Decomposition into partial waves | |
Phase-shift method | |
Semiclassical representation of the collision | |
Impact parameters | |
Potential of Finite Range | p. 389 |
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 | p. 396 |
Scattering by a deep square well | |
Study of a scattering resonance | |
Metastable states | |
Observation of the lifetime of metastable states | |
Various Formulae and Properties | p. 404 |
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 | p. 411 |
The Hydrogen Atom | p. 412 |
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 | p. 421 |
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 | p. 432 |
Eigenstates and Eigenvectors of the Hamiltonian | p. 433 |
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 | p. 441 |
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 | p. 451 |
General treatment of the isotropic oscillator in p dimensions | |
Two-dimensional isotropic oscillator | |
Three-dimensional isotropic oscillator | |
Distributions, [delta]-"Function" and Fourier Transformation | p. 462 |
Special Functions and Associated Formulae | p. 479 |
Symmetries and Invariance | |
Angular Momentum in Quantum Mechanics | |
Introduction | p. 507 |
Eigenvalues and eigenfunctions of angular momentum | p. 508 |
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 | p. 519 |
The spectrum of l[superscript 2] and l[subscript z] | |
Definition and construction of the spherical harmonics | |
Angular momentum and rotations | p. 523 |
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 | p. 540 |
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 | p. 555 |
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 | p. 569 |
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 | p. 582 |
Symmetrization postulate | p. 586 |
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 | p. 603 |
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 | p. 632 |
Mathematical complements. Antilinear operators | p. 633 |
Three useful theorems | |
Antilinear operators in Hilbert space | |
Antilinear transformations | |
Antilinear operators and representations | |
Transformations and groups of transformations | p. 643 |
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 | p. 655 |
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 | p. 664 |
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 | p. 685 |
Perturbation of a non-degenerate level | p. 686 |
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 | p. 698 |
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 | p. 712 |
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 | p. 722 |
Time dependent perturbation theory | p. 724 |
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 | p. 739 |
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 | p. 762 |
Variational Method for Bound States | p. 763 |
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 | p. 773 |
The self-consistent field method | |
Calculation of E[Phi] | |
The Fock-Dirac equations | |
Discussion | |
The Hartree equations | |
The Structure of Molecules | p. 781 |
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 | p. 801 |
Free Wave Green's Function and the Born Approximation | p. 802 |
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 | p. 822 |
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 | p. 832 |
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 | p. 856 |
Stationary expressions for the phase shifts | |
The variational calculation of phase shifts | |
Discussion | |
Extension to complex collisions | |
General Properties of the Transition Matrix | p. 863 |
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 | p. 875 |
Relativistic quantum mechanics | |
Notation, various conventions and definitions | |
The Lorentz group | |
Classical relativistic dynamics | |
The Dirac and Klein-Gordon Equations | p. 884 |
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 | p. 896 |
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 | p. 919 |
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 | p. 933 |
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 | p. 949 |
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 | p. 959 |
Quantization of a Real Scalar Field | p. 960 |
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 | p. 979 |
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 | p. 1009 |
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 | p. 1029 |
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 | p. 1053 |
Elements of Group Theory | p. 1079 |
General Index | p. 1125 |
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