Quantum Mechanics: New Quantum Theory v. 2 0720401062, 9780720401066
258 60 4MB
English Pages 486 [489] Year 1966
Table of contents :
Title
Copyright
Preface
Contents
Chapter 6. THE WAVE THEORY OF MATTER
6.36 Wave picture and particle picture
6.37 The de Broglie-Einstein relation
6.38 The Davisson-Germer experiment
6.39 The wave equation for the de Broglie wave
6.40 Matter density and energy density associated with the de Broglie wave
6.41 Simple examples
(i) The de Broglie wave in free space
(ii) The de Broglie wave contained in a box
(iii) The de Broglie wave in a field of force obeying Hooke's law
6.42 The hydrogen atom
(i) Solutions of the wave equation
(ii) The form of proper oscillations
(iii) The density distribution
(iv) The spherical wave in free space
6.43 Eigenvalues and eigenfunctions of the wave equation
(i) The eigenvalue and eigenfunction: simple examples
(ii) Eigenvalues and eigenfunctions of the de Broglie wave
(iii) The expansion theorem
(iv) Form of wave functions and the motion of a mass point
6.44 The orthogonality and expansion theorems for continuous eigenvalues
(i) General remarks
(ii) The orthogonality theorem
(iii) The expansion theorem
(iv) Introduction of the δ-function
(v) The condition to be imposed on eigenfunctions
6.45 The tunnel effect
6.46 The scattering of the de Broglie wave
(i) Incident and scattered waves
(ii) Solution of the equation for scattered wave
(iii) The asymptotic form of the scattered wave
(iv) The scattering cross section
(v) The Rutherford formula
6.47 The necessity for the quantization of a wave
Chapter 7. THE SCHRODINGER EQUATION
7.48 The preliminaries
7.49 The Schrodinger equation
(i) Matrices and vectors
(ii) A mass-point moving in one dimensional space in the field of Hooke's force
(iii) Linear operators
(iv) The correspondence between functions and vectors, and the correspondence between linear operators and matrices
(v) The Schrodinger equation
(vi) Examples
(vii) The de Broglie field
7.50 The Schrodinger equation and the de Broglie equation
7.51 Construction of Heisenberg matrices
(i) Summary of the acquired facts
(ii) Construction of matrices through Schrodinger functions
(iii) Examples
Chapter 8. THE PHYSICAL INTERPRETATION OF THE SCHRODINGER FUNCTION
8.52 The Schrodinger function and the de Broglie wave
8.53 Statistical interpretation of the Schrodinger function
(i) One-particle problems
(ii) The general case
(iii) The stationary state and the Schrodinger function
8.54 The time-dependent Schrodinger equation
(i) Case of one-particle systems
(ii) The general case
(iii) The conservation of probability
8.55 Statistical interpretation of the superposition
(i) The case of one-particle system
(ii) The general case
(iii) The case of the continuous eigenvalue
8.56 The probability for the momentum
8.57 Eigenstates
(i) A refinement of the concept "state"
(ii) Dynamical quantities and operators
(iii) Eigenstates
(iv) Example: Eigenstates of angular momentum
8.58 Probability for a physical quantity in general
8.59 Expectation value of a physical quantity
(i) Expectation value
(ii) The time variation of expectation value and the motion of the wave packet
(iii) The fluctuation
(iv) The uncertainty relation
(v) Compatibility
(vi) Probability for compatible physical quantities
(vii) The probability of a physical quantity with degeneracy
(viii) The probability of a state
8.60 The time variation of the probability and the transition probability
(i) Time variation of the probability that a physical quantity assumes a certain value
(ii) Transition of an atom by external fields
(iii) Transition due to the interaction of atom and radiation field
8.61 Diffraction and interference
Chapter 9. QUANTUM-MECHANICAL STATES
9.62 Introductory remarks
9.63 The uncertainty principle
9.64 Quantum-mechanical particle
9.65 Physical quantity and measurement
9.66 Azymuthal quantization
9.67 Interference of probabilities
9.68 Re-examination of some experiments
(i) Stern-Gerlach's experiment
(ii) The experiment of Lloyd
(iii) Compton effect and the interference of X-ray by a crystal
9.69 State and vector
Chapter 10. MANY PARTICLE SYSTEMS AND WAVE FIELDS
10.70 Preliminary remarks
10.71 Bose statistics
10.72 Symmetric states
(i) Permutation operators
(ii) The symmetric eigenfunction
(iii) Symmetric states and essentially asymmetric states
(iv) Assembly of particles without interaction
10.73 The conservation of symmetry property
10.74 Quantized wave field and assembly of Bose particles
(i) Preliminary consideration
(ii) The canonical formulation of wave field with interaction
(iii) From wave to particle
(iv) From the Bose particle system to the wave field
(v) Wave quantities and particle quantities
(vi) The commutation relation of the field quantities
(vii) The conservation of particle number
10.75 Antisymmetric states and Fermi particles
10.76 Pauli's exclusion principle and Fermi statistics
10.77 Fermi particles and wave field
Chapter 11. ANGULAR MOMENTUM AND SPIN
11.78 Introductory remark
11.79 The commutation relations among components of angular momentum
11.80 Derivation of the possible eigenvalues from the commutation relations
11.81 Combination of angular momentum vectors
(i) Preliminary remarks
(ii) Construction of eigenfunctions of the combined system
(iii) Summary
11.82 Spin of the electron
(i) One electron case
(ii) Case of two or more electrons
11.83 Coupling of the spin and the orbital angular momentum
(i) General considerations
(ii) The alkali doublet
(iii) The Zeeman effect of alkali doublets
11.84 Introduction of the exclusion principle
(i) Pauli principle in case of two electrons
(ii) Classification of symmetry properties
(iii) Symmetry character
(iv) The symmetry character and the multiplicity
(v) Pauli principle in case of three or more electrons
11.85 The orbital energy and the apparent spin-spin interaction
(i) Qualitative consideration
(ii) Semi-quantitative consideration
(iii) The field quantization and the symmetry operator
Appendix IX. Linear differential equations
Appendix X. Motion of a wave packet in the field of force obeying Hooke's law
Appendix XI. Necessary and sufficient condition for the mutual independence of y's of Eq. (43.14)
Appendix XII. The orthogonality theorem in the case of continuous eigenvalue
Appendix XIII. An example of normalization in the case of continuous eigenvalue
Appendix XIV. Green's theorem for wave equation
Appendix XV. Spreading of the general wave packet
Exercises
Index
Color plates
