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# Quantum Mechanics

**by**Messiah, Albert

### 9780486409245

0486409244

Paperback

3/28/2003

Dover Publications

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## Summary

Simple enough for students yet sufficiently comprehensive to serve as a reference for working physicists, this classic text is celebrated for its clarity and coherence of presentation as well as the author's fluid and literate style. Subjects include a detailed treatment of formalism and its interpretation, an analysis of simple systems, symmetries and invariance, methods of approximation, and a review of the elements of relativistic quantum mechanics. "Strongly recommended"-American Journal of Physics.

## Table of Contents

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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