Practical Quantum Mechanics

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1 Siegfried Flügge Practical Quantum Mechanics With 78 Figures Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest

2 Contents Volume I I. General Concepts 1. Law of probability conservation 1 2. Variational principle of Schrödinger 2 3. Classical mechanics for space averages 5 4. Classical laws for angular motion 6 5. Energy conservation law 8 6. Hermitian conjugate 9 7. Construction of an hermitian operator Derivatives of an operator Time rate of an expectation value Schrödinger and Heisenberg representations Time dependent hamiltonian Repeated measurement Curvilinear coordinates Momentum space wave functions Momentum space: Periodic and aperiodic wave functions 22 П. One-Body Problems without Spin A. One-Dimensional Problems Force-free case: Basic solutions Force-free case: Wave packet Standing wave Opaque division wall Opaque wall described by Dirac Ö function Scattering at a Dirac 5 function wall Scattering at a symmetric potential barrier Reflection at a rectangular barrier Inversion of reflection Rectangular potential hole Rectangular potential hole between two walls Virtual levels 57

3 Contents Volume I IX 28. Periodic potential Dirac comb Harmonie oscillator Oscillator in Hubert space Oscillator eigenfunctions constructed by Hilbert space operators Harmonic oscillator in matrix notation Momentum space wave functions of oscillator Anharmonic oscillator Approximate wave functions Potential step Pöschl-Teller potential hole Potential hole of modified Pöschl-Teller type Free fall of a body over earth's surface Accelerating electrical field 105 B. Problems of Two or Three Degrees of Freedom without Spherical Symmetry Circular oscillator Stark effect of a two-dimensional rotator Ionized hydrogen molecule Oblique incidence of a plane wave Symmetrical top 121 С The Angular Momentum Infinitesimal rotation Components in polar coordinates Angular momentum and Laplacian Hilbert space transformations Commutators in Schrödinger representation Particles of spin Commutation with a tensor Quadrupole tensor. Spherical harmonics Transformation of spherical harmonics Construction of Hilbert space for an angular momentum component Orthogonality of spherical harmonics 143 D. Potentials of Spherical Symmetry 144 a) Bound States Angular momentum expectation values 147

4 X Contents Volume I 59. Construction of radial momentum operator Solutions neighbouring eigenfunctions Quadrupole moment Particle enclosed in a sphere Square well of finite depth Wood-Saxon potential Spherical oscillator Degeneracy of the spherical oscillator Kepler problem Hulthen potential Kratzer's molecular potential Morse potential Rotation correction of Morse formula Yukawa potential hole Isotope shift in x-rays Muonic atom ground state Central-force model of deuteron Momentum space wave functions for central force potentials Momentum space integral equation for central force potentials Momentum space wave functions for hydrogen Stark effect of a three-dimensional rotator 206 b) Problems of Elastic Scattering Interference of incident and scattered waves Partial wave expansion of plane wave Partial wave expansion of scattering amplitude Scattering at low energies Scattering by a constant repulsive potential Anomalous scattering Scattering resonances Contribution of higher angular momenta Shape-independent approximation Rectangular hole: Low-energy scattering Low-energy scattering and bound state Deuteron potential with and without hard core Low-energy cross section with and without hard core Low-energy scattering by a modified Pöschl-Teller potential hole Radial integral equation Variational principle of Schwinger 252

5 Contents Volume I XI 96. Successive approximations to partial-wave phase shift Calogero's equation Linearization of Calogero's equation Scattering length for a negative-power potential Second approximation to Calogero equation Square-well potential: Scattering length Scattering length for a Yukawa potential Improvement of convergence in a spherical harmonics series Collision-parameter integral Born scattering: Successive approximation steps Scattering by a Yukawa potential Scattering by an exponential potential Born scattering by a charge distribution of spherical symmetry Hard sphere: High energy scattering Rutherford scattering formula Partial wave expansion for the Coulomb field Anomalous scattering Sommerfeld-Watson transform Reggepole 301 E. The Wentzel-Kramers-BriUouin (WKB) Approximation Eikonal expansion Radial WKB solutions WKB boundary condition of Langer Oscillator according to WKB approach WKB eigenvalues in a homogeneous field Kepler problem in WKB approach WKB phases in the force-free case Calculation of WKB phases Coulomb phases by WKB method Quasipotential 320 F. The Magnetic Field Introduction of a magnetic field Current in presence of a magnetic field Normal Zeeman effect Paramagnetic and diamagnetic susceptibilities without spin 328

6 Contents Volume II Ш. Particles with Spin A. One-Body Problems Construction of Pauli matrices Eigenstates of Pauli matrices Spin algebra Spinor transformation properties Spin electron in a central field Quadrupole moment of a spin state Expectation values of magnetic moments Fine structure Plane wave of spin i particles Free electron spin resonance 20 B. Two- and Three-Body Problems Spin functions for two particles Spin-dependent central force between nucleons Powers of spin operators Angular momentum eigenfunctions of two spin particles Tensor force operator Deuteron with tensor interaction Electrical quadrupole and magnetic dipole moments ofdeuteron Spin functions of three particles Neutron scattering by molecular hydrogen 40 IV. Many-Body Problems A. Few Particles Two repulsive particles on a circle Three-atomic linear molecule Centre-of-mass motion 51

7 Contents Volume II XIII 151. Virial theorem Slater determinant Exchange in interaction terms with Slater determinant Two electrons in the atomic ground state Excited states of the helium atom Excited S states of the helium atom Lithium ground state Exchange correction to lithium ground state Dielectric susceptibility Diamagnetic susceptibility of neon Van der Waals attraction Excitation degeneracy Neutral hydrogen molecule Scattering of equal particles Anomalous proton-proton scattering Inelastic scattering 95 B. Very Many Particles: Quantum Statistics Electron gas in a metal Paramagnetic susceptibility of a metal Field emission, uncorrected for image force Field emission, corrected for image force White dwarf Thomas-Fermi approximation Amaldi correction for a neutral atom Energy of a Thomas-Fermi atom Virial theorem for the Thomas-Fermi atom Tietz approximation of a Thomas-Fermi field Variational approximation of Thomas-Fermi field Screening of К electrons 131 V. Non-Stationary Problems 179. Two-level system with time-independent perturbation Periodic perturbation of two-level system Dirac perturbation method Periodic perturbation: Resonance Golden Rule for scattering Born scattering in momentum space Coulomb excitation of an atom PhotoefTect 153

8 XIV Contents Volume II 187. Dispersion of light. Oscillator strengths Spin flip in a magnetic resonance device 160 VI. The Relativ istic Dirac Equation 189. Iteration of the Dirac equation Plane Dirac waves of positive energy Transformation properties of a spinor Lorentz covariants Parity transformation Charge conjugation Mixed helicity states Spin expectation value Algebraic properties of a Dirac wave spinor Current in algebraic formulation Conduction current and polarization current Splitting up of Dirac equations into two pairs Central forces in Dirac theory Kepler problem in Dirac theory Hydrogen atom fine structure Radial Kepler solutions at positive kinetic energies Angular momentum expansion of plane Dirac wave Scattering by a central force potential Continuous potential step Plane wave at a potential jump Reflected intensity at a potential jump 222 VII. Radiation Theory 210. Quantization of Schrödinger field Scattering in Born approximation Quantization of classical radiation field Emission probability of a photon Angular distribution of radiation Transition probability Selection rules for dipole radiation Intensities of Lyman lines Compton effect Bremsstrahlung 249

9 Contents Volume II XV Mathematical Appendix Coordinate systems 257 Г function 258 Bessel functions 260 Legendre functions 264 Spherical harmonics 267 The hypergeometric series 271 The confluent series 274 Some functions defined by integrals 276 Index for Volumes I and II 279

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