The Fundamental Principles of Quantum Mechanics With Elementary Applications

by Kemble, Edwin C.

• ISBN13: 9780486441535
• ISBN10: 0486441539
• Edition: Revised
• Format: Hardcover
• Copyright: 2005-01-27
• Publisher: Dover Publications

Summary

Written by a professor of physics at Harvard University, this volume is an inductive presentation of quantum mechanics designed for graduate students or specialists in other branches of physics. Although some acquaintance with advanced mathematics and the data of the physical sciences is preferable, this treatment is simple, clear, and easily followed. Applications of the theory are interwoven with development of the basic mathematical structure.

Table of Contents

PREFACE	vii
NOTATION	xvii
REFERENCE ABBREVIATIONS	xvii
* An asterisk before the number of a section or subsection indicates that the section or subsection so marked may be omitted or skimmed to advantage on first reading.
CHAPTER I INTRODUCTION TO DUALISTIC THEORY OF MATTER; DEVELOPMENT OF SCHRÖDINGER'S WAVE EQUATION	1	(34)
1. Historical Introduction	1	(3)
2. The Dualistic Theory of Radiation	4	(3)
3. An Analogy between Geometrical Optics and Classical Mechanics	7	(3)
4. Wave Packets and Group Velocity	10	(4)
5. The Schrödinger Wave Equation for a Single Particle	14	(5)
5a. The Time-free Equation	14	(1)
5b. The Second (General) Schrödinger Equation	15	(4)
*6. The Application of the Restricted Relativity Principle	19	(2)
7. The Wave Equation for a System of Many Particles	21	(8)
7a. Formulation of Equation	21	(2)
7b. Relation of Schrödinger Equation to the Classical Hamiltonian Function	23	(1)
*7c. The Schrödinger Equation and the Hamilton-Jacobi Equation	24	(2)
*7d. The Wave Equation for a System of Charged Particles in a Classical External Electromagnetic Field	26	(3)
8. The Physical Interpretation of the Wave Function and the Normalization Condition	29	(6)
8a. Probability and Quadratic Integrability	29	(2)
8b. Normalization and Mass Current Density	31	(2)
8c. A System Consisting of Two Independent Parts	33	(2)
CHAPTER II WAVE PACKETS AND THE RELATION BETWEEN CLASSICAL MECHANICS AND WAVE MECHANICS	35	(43)
9. Wave Packets and Group Velocity in a One-dimensional Homogeneous Medium	35	(6)
9a. The Fourier Integral Theorem	35	(2)
9b. Derivation of Group-velocity Formula	37	(4)
10. Wave Packets in Three Dimensions	41	(2)
11. Wave Surfaces and the Hamilton-Jacobi Equation of the Classical Dynamics	43	(3)
12. Wave Packets and the Motion of Particles in a Force Field; Fermat's Principle	46	(3)
13. Direct Rigorous Proof of Newton's Second Law of Motion for Wave Packets	49	(2)
14. The Statistical Interpretation of the Wave Theory of Matter	51	(7)
14a. Review of Assumptions	51	(2)
14b. Necessity of Introducing Assemblages	53	(3)
14c. Multiplicity of Energy and Momentum Values for a Definite State	56	(2)
15. The Wave Function and Measurements of Linear Momentum	58	(14)
15a. Operational Definition of Momentum for Free Particles	58	(2)
15b. Computation of Momentum Probabilities from a Wave Function.	60	(3)
*15c. Momentum of Center of Gravity	63	(3)
*15d. The Measurement of the Momenta of Particles Moving in a Force Field	66	(2)
*15e. Individual Momenta of Particles in a System	68	(1)
15f. Summary: The Determination of the Wave Function of an Assemblage	69	(3)
16. The Heisenberg Uncertainty Principle	72	(6)
CHAPTER III ONE-DIMENSIONAL ENERGY-LEVEL PROBLEMS	78	(35)
17. Boundary and Continuity Conditions; Eigenvalues and Eigenfunctions	78	(3)
18. The One-Dimensional Anharmonic Oscillator	81	(1)
19. The Qualitative Behavior of the Integral Curves: Existence of Class A Eigenfunctions	82	(5)
19a. Behavior of Integral Curves in Regions of Positive and Negative Kinetic Energy	82	(2)
19b. The Discrete Eigenfunctions	84	(1)
19c. The Continuous Spectrum of Class B Eigenfunctions	85	(1)
19d. The Paradox of the Nodes	86	(1)
20. The Planck Ideal Linear Oscillator	87	(3)
20a. The Sommerfeld Polynomial Method	87	(2)
20b. Determination of the Eigenvalues	89	(1)
20c. The Eigenfunctions and Their Properties	89	(1)
21. An Approximation Method Which Correlates the Eigenvalues of Wave Mechanics with the Energy Levels of the Bohr Theory	90	(23)
21a. The B.W.K. Approximations for ψ(x)	90	(3)
21b. Application to Eigenvalue Problem	93	(2)
*21c. Zwaan's Method and the Stokes Phenomenon	95	(2)
*21d. Analysis of the Stokes Phenomenon	97	(3)
*21e. Derivation of the Connection Formulas	100	(3)
*21f. Derivation of the Sommerfeld Phase-integral Quantum Condition	103	(4)
*21g. Higher Approximations	107	(1)
*21h. Modification of Method for Radial Motion in Two-particle Problem	107	(1)
21i. Asymptotic Agreement of Wave Theory and Classical Theory Regarding Position of Particle	108	(1)
21j. The Transmission of Progressive Matter Waves through a Potential Hill	109	(4)
CHAPTER IV THE MATHEMATICAL THEORY OF COMPLETE SYSTEMS OF ORTHOGONAL FUNCTIONS	113	(27)
22. Scalar Products and Systems of Orthogonal Functions	113	(8)
22a. Expansion in a Series of Functions	113	(1)
22b. Comparison of Properties of Vectors and Functions	114	(1)
22c. Scalar Products of Vectors	115	(1)
22d. Scalar Products of Quadratically Integrable Functions of m Variables	116	(3)
22e. Spaces of Infinitely Many Dimensions	119	(1)
22f. Proof of Orthogonality of Eigenfunctions of the One-dimensional Anharmonic Oscillator Problem	120	(1)
23. Self-adjoint Operators and Equations. The Sturm-Liouville Problem.	121	(9)
23a. Self-adjoint Differential Operators in One Dimension	121	(2)
23b. Orthogonality with Respect to a Density Function p	123	(1)
23c. The Sturm-Liouville Problem	124	(1)
23d. Singular-point Boundary Conditions	125	(3)
*23e. Existence of Discrete Eigenvalues for Sturm-Liouville Problems with Singular End Points	128	(2)
24. Reduction of Eigenvalue Problems Based on Self-adjoint Differential Equations to Variational Form	130	(2)
25. Completeness of System of Discrete Eigenfunctions of a Sturm-Liouville Problem	132	(8)
*25a. The Eigenvalues as Absolute Minima	132	(3)
*25b. The Expansion of Arbitrary Functions in Terms of Eigenfunctions	135	(5)
CHAPTER V THE DISCRETE ENERGY SPECTRUM OF THE TWO-PARTICLE CENTRAL-FIELD PROBLEM	140	(22)
26. The Behavior of Solutions of an Ordinary Second-order Differential Equation near a Singular Point	140	(3)
27. The Legendre Polynomials	143	(3)
27a. General Properties of the Legendre Equation	143	(1)
27b. Explicit Determination of Eigenvalues and Eigenfunctions	144	(2)
28. The Energy Levels of the Two-particle Problem	146	(11)
28a. The Wave Equation	146	(1)
28b. Separation of the Variables	146	(1)
28c. The Azimuthal Factor of the Wave Function	147	(1)
28d. Determination of &Theta(theta) and Its Eigenvalues	148	(1)
28e. Completeness of System of Eigenfunctions	149	(1)
28f. Physical Interpretation of Quantum Numbers l and m	150	(2)
28g. Behavior of Radial Wave Functions at Boundary Points	152	(3)
28h. The Dumbbell Model of the Diatomic Molecule	155	(2)
29. The Hydrogenic Atom	157	(5)
29a. Application of the B.W.K. Method	157	(1)
29b. Application of Polynomial Method	158	(2)
29c. Generalized Laguerre Polynomials	160	(1)
29d. The Most General Eigenfunction	161	(1)
CHAPTER VI THE CONTINUOUS SPECTRUM AND THE BASIC PROPERTIES OF SOLUTIONS OF THE MANY-PARTICLE PROBLEM	162	(57)
30. The Continuous Spectrum in One-dimensional Problems	162	(16)
30a. The Nature and Use of the Eigenfunctions of the Continuous Spectrum	162	(1)
30b. The Weyl Theory	163	(2)
*30c. Formal Treatment of Continuous Spectrum as the Limit of a Discrete Spectrum	165	(3)
*30d. The Spacing of Energy Levels in Problems β and α	168	(1)
530e. The Eigendifferentials	169	(2)
*30f. Passage from the Completeness Theorem for Problem β to That for Problem α	171	(2)
30g. The Fourier Integral Formulas a Special Case	173	(1)
30h. Normal Packet Functions in One Dimension	174	(1)
30i. Normal Packet Functions for the Two-particle Problem	174	(2)
*30j. Normalization of the Class B Radial Eigenfunctions for the Hydrogenic Atom	176	(2)
31. Weak Quantization. Theory of Radioactive Emission of Alpha Particles	178	(17)
31a. Weak Quantization in General	178	(1)
31b. A Model for Alpha Particle Disintegration	179	(2)
31c. Resonant Energy Intervals	181	(5)
31d. Energy Distribution in Weakly Quantized States	186	(1)
31e. The Disintegration Process	187	(5)
*31f. Complex Eigenvalues	192	(3)
32. The Existence and Properties of Solutions of the Many-particle Schrödinger Eigenvalue-eigenfunction Problem	195	(24)
32a. Introduction	195	(2)
32b. New Boundary Conditions for Physically Admissible Wave Functions	197	(4)
*32c. Approximating Arbitrary Quadratically Integrable Functions by Means of Class D Functions	201	(1)
32d. Hermitian Character of the Hamiltonian Operator	202	(4)
32e. Reduction of the Eigenvalue-eigenfunction Problem for Discrete Spectra to Variational Form	206	(1)
32f. A Lower Bound for the Energy Integral	207	(1)
32g. Behavior of Solutions of the Differential Equation at Singular Domains	208	(5)
32h. The Auger Effect	213	(1)
32i. The Discrete Eigenfunctions of the Differential Equation as Minimizing Functions	214	(1)
32j. The Continuous Spectrum and the Completeness of the System of Eigenfunctions	215	(2)
32k. Degeneracy	217	(2)
CHAPTER VII DYNAMICAL VARIABLES AND OPERATORS	219	(59)
33. The Mean Values of the Cartesian Coordinates and Conjugate Linear Momenta	219	(5)
33a. The Statistical Mean Values of the Coordinates	219	(1)
33b. The Linear Momentum Operator	220	(4)
34. The Angular-momentum Operators	224	(10)
34a. Definition of Operators	224	(1)
34b. Hermitian Character of Angular-momentum Operators	225	(1)
34c. The Expansion Theorem	226	(1)
34d. Angular Momentum of a System of Particles	227	(2)
34e. Mean Values	229	(1)
34f. The Vector Angular Momentum and Its Square; the Symmetric Top	230	(4)
35. The Energy Operators	234	(6)
35a. Calculation of Probabilities and Mean Values of Energy	234	(3)
*35b. Transformation of Hamiltonian Operator	237	(3)
36. Dynamical Variables in General	240	(38)
36a. Remarks on the Value of the General Theory	240	(2)
36b. Possibility of Defining Physical Quantities by Operators	242	(3)
36c. The Transformation of Probability Amplitudes and Dynamical Variables	245	(3)
36d. Type 1 Operators as Dynamical Variables	248	(8)
36e. Calculation of Probabilities	256	(3)
36f. Type 2 Operators as Dynamical Variables; the Method of von Neumann	259	(6)
36g. The Method of Dirac and Jordan	265	(3)
*36h. Multiplication Operators in Many Dimensions	268	(2)
36i. Transformation of Probability Amplitudes from One Arbitrary Coordinate Scheme to Another	270	(5)
36j. Dynamical Variables with Complex Eigenvalues	275	(3)
CHAPTER VIII COMMUTATION RULES AND RELATED MATTERS	278	(40)
37. Simultaneous Eigenfunctions and the Commutation of Dynamical Variables	278	(10)
37a. Operator Algebra	278	(1)
37b. Functions of a Single Operator	279	(2)
37c. Commutative Operators	281	(5)
37d. Functions of a Normal Set of Commuting Dynamical Variables.	286	(2)
38. The Conservation Laws	288	(5)
38a. Conservation of Energy	288	(1)
38b. Variation of Energy When the Hamiltonian Depends on the Time	289	(1)
38c. Conservation of an Arbitrary Dynamical Variable	290	(1)
38d. Commutation Properties of the Hamiltonian and the Angular Momentum	291	(2)
39. Conjugate Dynamical Variables and Quantum-mechanical Equations of Motion	293	(10)
39a. Conjugate Dynamical Variables	293	(7)
39b. Functions of Non-commuting Linear Operators	300	(1)
39c. An Operator Form of Hamilton's Equations of Motion	301	(2)
40. Symmetry Properties of the Wave Equation	303	(15)
40a. Symmetry Properties in General	303	(2)
40b. The Reflection Operators	305	(1)
40c. The Rotation Operator	306	(2)
40d. The Permutation Operators	308	(2)
40e. Degeneracy and the Integrals of the Schrödinger Equation	310	(3)
40f. The Normal Degeneracy of the Energy Levels of Free Atomic Systems	313	(5)
CHAPTER IX THE MEASUREMENT OF DYNAMICAL VARIABLES	318	(30)
41. General Theory of Measurement	318	(16)
41a. Fundamental Characteristics of Measurements	318	(2)
41b. Pure States and Mixtures	320	(2)
41c. Postulates Regarding Retrospective and Predictive Measurements	322	(4)
41d. The Reduction of the Wave Packet	326	(5)
41e. Classical Orbits and Wave Packets	331	(3)
42. More About Measurements	334	(14)
42a. Conjugate Variables and Measurements	334	(1)
42b. Impossibility of Measurements Which Imply Distinction between Particles of Same Species	335	(6)
42c. A Classification of Observations	341	(1)
42d. Measurements as Correlations	342	(1)
42e. The Observing Mechanism Not Entirely Classical	343	(5)
CHAPTER X MATRIX THEORY	348	(32)
43. Matrix Algebra	348	(4)
44. Matrices and Operators	352	(14)
44a. The Derivation of Matrices from Operators	352	(3)
44b. Canonical Matrix Transformations	355	(4)
44c. Matrix Form of the Eigenvalue-eigenfunction Problem	359	(4)
*44d. Matrices with Continuous Elements	363	(3)
45. The Matrix Theory of Heisenberg, Born, and Jordan	366	(8)
45a. Fundamental Postulates	366	(2)
45b. Correlation of the Heisenberg and Schrödinger Theories	368	(2)
45c. Solution of Matrix Equations of Motion for an Ideal Linear Oscillator	370	(2)
45d. Reduction of the Fundamental Problem of Matrix Mechanics to a Principal-axis Transformation	372	(2)
46. The Bohr Correspondence Principle and Its Relationship to Matrix Theory	374	(6)
46a. The Bohr Postulates	374	(1)
46b. The Bohr Correspondence Principle and the Heisenberg Matrix Theory	375	(5)
CHAPTER XI THEORY OF PERTURBATIONS WHICH Do NOT INVOLVE THE TIME	380	(47)
47. The Perturbation Theory for Nondegenerate Problems	380	(8)
47a. First-order Perturbations	380	(4)
47b. Second-order Perturbations	384	(2)
47c. An Example: The Diatomic Molecule	386	(2)
48. The Perturbation Theory for Degenerate Problems	388	(10)
48a. First-order Energy Perturbations	388	(3)
48b. Second-order Energy Perturbations	391	(3)
*48c. Van Vleck's Method for Second-order Perturbations	394	(2)
48d. Simplification of Perturbation Calculations by Means of Integrals of the Perturbed Hamiltonian	396	(2)
49. The Energy Levels of an Hydrogenic Atom in a Uniform Magnetic Field (Spin Neglected)	398	(5)
49a. Derivation of Hamiltonian Operator	398	(1)
49b. Legitimacy of the Perturbation Method	399	(1)
49c. First-order Energy Correction; Relation to Magnetic Moment and Larmor Precession	400	(2)
49d. The Second-order Energy Correction	402	(1)
50. The Energy Levels of an Hydrogenic Atom in a Uniform Electric Field.	403	(5)
51. The Variational Method	408	(11)
51a. Reduction of the Variational Problem to Algebraic Form	408	(2)
51b. The Ritz Method	410	(5)
*51c. Higher Roots of the Secular Equation	415	(1)
51d. General Observations Regarding the Use of the Variational Method	416	(2)
51e. Modifications of the Method; Construction of Eigenfunctions from Non-orthogonal System	418	(1)
52. The Problem of the Hydrogen Molecule	419	(8)
52a. The Fixed-nuclei Problem	419	(1)
52b. The Heitler and London Calculation	420	(5)
52c. The Method of James and Coolidge	425	(2)
CHAPTER XII QUANTUM STATISTICAL MECHANICS AND THE EINSTEIN TRANSITION PROBABILITIES	427	(47)
53. Quantum Statistical Mechanics	427	(21)
53a. The General Theory of Perturbations Which May Involve the Time	427	(4)
53b. The Adiabatic Theorem	431	(1)
53c. The Fundamental Problems of Quantum Statistical Mechanics	432	(2)
53d. The Conventional Characterization of a Chaotic Assemblage	434	(5)
53e. Transition Probabilities and Statistical Equilibrium for Chaotic Assemblages	439	(7)
53f. The Gibbs Canonical Assemblage for Systems of the Most General Type	446	(2)
54. The Absorption and Emission of Radiation: Perturbation of an Atomic System by a Classical Radiation Field	448	(21)
54a. The Einstein Derivation of the Planck Radiation Formula	448	(2)
54b. Elementary Approaches to the Quantum Theory of the Einstein Transition Probabilities	450	(2)
54c. The Perturbing Hamiltonian for a Classical Radiation Field	452	(2)
54d. The Born Transition Probability	454	(4)
54e. The Einstein Transition Probabilities	458	(4)
54f. Spectroscopic Stability	462	(1)
*54g. Magnetic Dipole and Electric Quadrupole Radiation	462	(7)
55. Some Elementary Selection Rules for Electric Dipole Radiation	469	(5)
55a. The Harmonic Oscillator	469	(1)
55b. Selection Rules for the Two-particle Problem	470	(1)
55c. Fine Structure and Polarization of Spectrum Lines in Simple Zeeman Effect	471	(3)
CHAPTER XIII INTRODUCTION TO THE PROBLEM OF ATOMIC STRUCTURE: ELECTRON SPIN.	474	(49)
56. The Atomic Problem as a Two-particle Problem	474	(7)
56a. The Empirical Basis for the Idealized Bohr Atom Model	474	(4)
56b. Derivation of the Ritz Formula	478	(3)
57. The Bohr Assignment of Electronic Quantum Numbers	481	(10)
57a. The Quantum Numbers of the Valence Electrons in the Spectra of the Alkalies and Alkaline Earths	481	(3)
57b. Perturbation Theory and the Significance of an Assignment of Quantum Numbers to Inner Electrons	484	(7)
58. The Electron-spin Hypothesis	491	(12)
58a. The Empirical Fine Structure of Spectrum Lines	491	(4)
58b. The Combination of Angular Momenta	495	(3)
58c. The Lande Magnetic Core Theory	498	(2)
58d. Solution of the Fine-structure Problem by the Electron-spin Hypothesis	500	(3)
59. The Fine Structure of the Spectra of Atomic Systems with a Single Valence Electron	503	(4)
60. The Approximate Relativistic Theory of the Hydrogen Atom	507	(3)
61. The Pauli Wave-mechanical Formulation of the Theory of Electron Spin	510	(13)
61a. Nature of the Configuration Space and Wave Functions	510	(2)
61b. Preliminary Discussion of Spin Operators and Spin Matrices	512	(7)
61c. Application of the Pauli Theory to the Alkali Doublets	519	(4)
CHAPTER XIV THE THEORY OF THE STRUCTURE OF MANY-ELECTRON ATOMS	523	(34)
62. General Formulation of the Problem	523	(5)
62a. The Configuration Space	523	(1)
62b. The Hamiltonian Operator	524	(2)
62c. The Perturbation Form of the General Atomic Problem	526	(2)
63. Problem B: The Spin-orbit Energy Neglected	528	(12)
63a. Integrals of the Motion	528	(5)
63b. Antisymmetric Functions and the Empirical Pauli Exclusion Rule	533	(3)
63c. Closed Shells	536	(1)
63d. Terms Originating in a Given Configuration	537	(3)
64. Selection Rules for Electric Dipole Radiation	540	(7)
64a. The Laporte Rule	540	(1)
64b. Selection Rules for the Central-field Problem	541	(2)
64c. Selection Rules for Problems B and C	543	(4)
65. The Helium Atom and Exchange Energy	547	(8)
65a. Two-electron Atoms	547	(6)
65b. The Exchange Phenomenon	553	(2)
66. Diagonal Sums and the Problem B Energy Levels	555	(2)
APPENDICES	557	(42)
A. The Calculus of Variations and the Principle of Least Action	557	(7)
B. Derivation of Equation (15.7)	564	(4)
C. Theorems Regarding the Linear Oscillator Problem	568	(4)
D. Mathematical Notes on the B.W.K. Method	572	(7)
E. The Reduction of Certain Boundary-value Problems Based on Self-adjoint Differential Equations to Variational Form	579	(4)
F. The Legendre Polynomials and Associated Legendre Functions	583	(2)
G. The Generalized Laguerre Orthogonal Functions	585	(3)
H. Two Theorems Relating to the Continuous Spectrum	588	(4)
I. Concerning the Expansion of Hf in Spherical Harmonics	592	(2)
J. The Jacobi Polynomials	594	(2)
K. Schlapp's Method	596	(3)
NAME INDEX	599	(4)
SUBJECT INDEX	603