Quantum Mechanics

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  • Copyright: 2003-03-28
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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
Introductionp. 3
The end of the Classical Periodp. 4
Classical Theoretical Physics
Progress in the knowledge of microscopic phenomena and the appearance of quanta in physics
Light Quanta or Photonsp. 11
The photoelectric effect
The Compton effect
Light quanta and interference phenomena
Quantization of Material Systemsp. 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 Theoryp. 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
Matter Waves and the Schrodinger Equation
Historical survey and general plan of the succeeding chaptersp. 45
Matter Wavesp. 49
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 Equationp. 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 Equationp. 71
Search for stationary solutions
General properties of the equation
Nature of the energy spectrum
One-Dimensional Quantized Systems
Introductionp. 77
Square Potentialsp. 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 Equationp. 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
Introductionp. 115
Statistical Interpretation of the Wave Functions of Wave Mechanicsp. 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 Relationsp. 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 Processp. 139
Uncontrollable disturbance during the operation of measurement
Position measurements
Momentum measurements
Description of Phenomena in Quantum Theory. Complementarity and Causalityp. 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
Introductionp. 162
Hermitean Operators and Physical Quantitiesp. 163
Wave-function space
Definition of mean values
Absence of fluctuation and the eigenvalue problem
Study of the Discrete Spectrump. 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 Casep. 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 Functionp. 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 Applicationsp. 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
Classical Approximation and the WKB Method
The Classical Limit of Wave Mechanicsp. 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 Methodp. 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 vectorsp. 243
Vectors and Operatorsp. 245
Vector space
"Ket" vectors
Dual space
"Bra" vectors
Scalar product
Linear operators
Tensor product of two vector spaces
Hermitean Operators, Projectors, and Observablesp. 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 Theoryp. 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
Introductionp. 294
Dynamical States and Physical Quantitiesp. 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 Motionp. 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 Theoryp. 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 Statisticsp. 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
Introductionp. 343
Particle in a Central Potential. General Treatmentp. 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
Central Square-Well Potential. Free Particlep. 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 Motionp. 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
Introductionp. 369
Cross Sections and Scattering Amplitudesp. 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 Shiftsp. 385
Decomposition into partial waves
Phase-shift method
Semiclassical representation of the collision
Impact parameters
Potential of Finite Rangep. 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 Resonancesp. 396
Scattering by a deep square well
Study of a scattering resonance
Metastable states
Observation of the lifetime of metastable states
Various Formulae and Propertiesp. 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
Introductionp. 411
The Hydrogen Atomp. 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 Scatteringp. 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
Introductionp. 432
Eigenstates and Eigenvectors of the Hamiltonianp. 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 Propertiesp. 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 Dimensionsp. 451
General treatment of the isotropic oscillator in p dimensions
Two-dimensional isotropic oscillator
Three-dimensional isotropic oscillator
Distributions, [delta]-"Function" and Fourier Transformationp. 462
Special Functions and Associated Formulaep. 479
Symmetries and Invariance
Angular Momentum in Quantum Mechanics
Introductionp. 507
Eigenvalues and eigenfunctions of angular momentump. 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]}
Orbital angular momentum and the spherical harmonicsp. 519
The spectrum of l[superscript 2] and l[subscript z]
Definition and construction of the spherical harmonics
Angular momentum and rotationsp. 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
Spinp. 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 momentap. 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 operatorsp. 569
Representation of scalar operators
Irreducible tensor operators
Representation of irreducible tensor operators
Wigner-Eckhart theorem
Systems of Identical Particles. Pauli Exclusion Principle
Identical particles in quantum theoryp. 582
Symmetrization postulatep. 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
Applicationsp. 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
Introductionp. 632
Mathematical complements. Antilinear operatorsp. 633
Three useful theorems
Antilinear operators in Hilbert space
Antilinear transformations
Antilinear operators and representations
Transformations and groups of transformationsp. 643
Transformations of the dynamical variables and dynamical states of a system
Groups of transformations
Groups of transformation operators
Continuous groups and infinitesimal transformations
Finite groups
Invariance of the equations of motion and conservation lawsp. 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 microreversibilityp. 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 Fourp. 685
Perturbation of a non-degenerate levelp. 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 levelp. 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
Explicit forms for the perturbation expansion in all ordersp. 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 Hamiltonianp. 722
Time dependent perturbation theoryp. 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 Hamiltonianp. 739
The problem and the results
Rapid passage and the sudden approximation
Sudden reversal of a magnetic field
Adiabatic passage
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 methodp. 762
Variational Method for Bound Statesp. 763
Variational form of the eigenvalue problem
Variational calculation of discrete levels
A simple example: the hydrogen atom
Application to the calculation of excited levels
Ground state of the helium atom
The Hartree and Fock-Dirac Atomsp. 773
The self-consistent field method
Calculation of E[Phi]
The Fock-Dirac equations
The Hartree equations
The Structure of Moleculesp. 781
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
Introductionp. 801
Free Wave Green's Function and the Born Approximationp. 802
Integral representations of the scattering amplitude
Cross sections and the T matrix
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 Wavesp. 822
Generalized Born approximation
Generalization of the Born expansion
Green's functions for distorted waves
Definition and formal properties of T
Note on the 1/4 potentials
Complex Collisions and the Born Approximationp. 832
Cross sections
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
Multiple scattering
Variational Calculations of Transition Amplitudesp. 856
Stationary expressions for the phase shifts
The variational calculation of phase shifts
Extension to complex collisions
General Properties of the Transition Matrixp. 863
Conservation of flux
Unitarity of the S matrix
The Bohr-Peierls-Placzek relation (optical theorem)
Invariance properties of the T matrix
Elements of Relativistic Quantum Mechanics
The Dirac Equation
General Introductionp. 875
Relativistic quantum mechanics
Notation, various conventions and definitions
The Lorentz group
Classical relativistic dynamics
The Dirac and Klein-Gordon Equationsp. 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 Equationp. 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 Solutionsp. 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 Equationp. 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 Theoryp. 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
Introductionp. 959
Quantization of a Real Scalar Fieldp. 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 Systemp. 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 Radiationp. 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 Radiationp. 1029
Quantization of free radiation
Plane waves
Radiation momentum
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 Matricesp. 1053
Elements of Group Theoryp. 1079
General Indexp. 1125
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