CONTENTS
Preface xi
1 The Discovery of Quantum Mechanics 1
I Introduction, 1
II Planck and Quantization, 3
III Bohr and the Hydrogen Atom, 7
IV Matrix Mechanics, 11
V The Uncertainty Relations, 13
VI Wave Mechanics, 14
VII The Final Touches of Quantum Mechanics, 20
VIII Concluding Remarks, 22
2 The Mathematics of Quantum Mechanics 23
I Introduction, 23
II Differential Equations, 24
III Kummer’s Function, 25
IV Matrices, 27
V Permutations, 30
VI Determinants, 31
vii
VII Properties of Determinants, 32
VIII Linear Equations and Eigenvalues, 35
IX Problems, 37
3 Classical Mechanics 39
I Introduction, 39
II Vectors and Vector Fields, 40
III Hamiltonian Mechanics, 43
IV The Classical Harmonic Oscillator, 44
V Angular Momentum, 45
VI Polar Coordinates, 49
VII Problems, 51
4 Wave Mechanics of a Free Particle 52
I Introduction, 52
II The Mathematics of Plane Waves, 53
III The Schro¨dinger Equation of a Free Particle, 54
IV The Interpretation of the Wave Function, 56
V Wave Packets, 58
VI Concluding Remarks, 62
VII Problems, 63
5 The Schro¨dinger Equation 64
I Introduction, 64
II Operators, 66
III The Particle in a Box, 68
IV Concluding Remarks, 71
V Problems, 72
6 Applications 73
I Introduction, 73
II A Particle in a Finite Box, 74
viii CONTENTS
III Tunneling, 78
IV The Harmonic Oscillator, 81
V Problems, 87
7 Angular Momentum 88
I Introduction, 88
II Commuting Operators, 89
III Commutation Relations of the Angular Momentum, 90
IV The Rigid Rotor, 91
V Eigenfunctions of the Angular Momentum, 93
VI Concluding Remarks, 96
VII Problems, 96
8 The Hydrogen Atom 98
I Introduction, 98
II Solving the Schro¨dinger Equation, 99
III Deriving the Energy Eigenvalues, 101
IV The Behavior of the Eigenfunctions, 103
V Problems, 106
9 Approximate Methods 108
I Introduction, 108
II The Variational Principle, 109
III Applications of the Variational Principle, 111
IV Perturbation Theory for a Nondegenerate State, 113
V The Stark Effect of the Hydrogen Atom, 116
VI Perturbation Theory for Degenerate States, 119
VII Concluding Remarks, 120
VIII Problems, 120
10 The Helium Atom 122
I Introduction, 122
CONTENTS ix
II Experimental Developments, 123
III Pauli’s Exclusion Principle, 126
IV The Discovery of the Electron Spin, 127
V The Mathematical Description of the Electron Spin, 129
VI The Exclusion Principle Revisited, 132
VII Two-Electron Systems, 133
VIII The Helium Atom, 135
IX The Helium Atom Orbitals, 138
X Concluding Remarks, 139
XI Problems, 140
11 Atomic Structure 142
I Introduction, 142
II Atomic and Molecular Wave Function, 145
III The Hartree-Fock Method, 146
IV Slater Orbitals, 152
V Multiplet Theory, 154
VI Concluding Remarks, 158
VII Problems, 158
12 Molecular Structure 160
I Introduction, 160
II The Born-Oppenheimer Approximation, 161
III Nuclear Motion of Diatomic Molecules, 164
IV The Hydrogen Molecular Ion, 169
V The Hydrogen Molecule, 173
VI The Chemical Bond, 176
VII The Structures of Some Simple Polyatomic Molecules, 179
VIII The Hu¨ckel Molecular Orbital Method, 183
IX Problems, 189
Preface xi
1 The Discovery of Quantum Mechanics 1
I Introduction, 1
II Planck and Quantization, 3
III Bohr and the Hydrogen Atom, 7
IV Matrix Mechanics, 11
V The Uncertainty Relations, 13
VI Wave Mechanics, 14
VII The Final Touches of Quantum Mechanics, 20
VIII Concluding Remarks, 22
2 The Mathematics of Quantum Mechanics 23
I Introduction, 23
II Differential Equations, 24
III Kummer’s Function, 25
IV Matrices, 27
V Permutations, 30
VI Determinants, 31
vii
VII Properties of Determinants, 32
VIII Linear Equations and Eigenvalues, 35
IX Problems, 37
3 Classical Mechanics 39
I Introduction, 39
II Vectors and Vector Fields, 40
III Hamiltonian Mechanics, 43
IV The Classical Harmonic Oscillator, 44
V Angular Momentum, 45
VI Polar Coordinates, 49
VII Problems, 51
4 Wave Mechanics of a Free Particle 52
I Introduction, 52
II The Mathematics of Plane Waves, 53
III The Schro¨dinger Equation of a Free Particle, 54
IV The Interpretation of the Wave Function, 56
V Wave Packets, 58
VI Concluding Remarks, 62
VII Problems, 63
5 The Schro¨dinger Equation 64
I Introduction, 64
II Operators, 66
III The Particle in a Box, 68
IV Concluding Remarks, 71
V Problems, 72
6 Applications 73
I Introduction, 73
II A Particle in a Finite Box, 74
viii CONTENTS
III Tunneling, 78
IV The Harmonic Oscillator, 81
V Problems, 87
7 Angular Momentum 88
I Introduction, 88
II Commuting Operators, 89
III Commutation Relations of the Angular Momentum, 90
IV The Rigid Rotor, 91
V Eigenfunctions of the Angular Momentum, 93
VI Concluding Remarks, 96
VII Problems, 96
8 The Hydrogen Atom 98
I Introduction, 98
II Solving the Schro¨dinger Equation, 99
III Deriving the Energy Eigenvalues, 101
IV The Behavior of the Eigenfunctions, 103
V Problems, 106
9 Approximate Methods 108
I Introduction, 108
II The Variational Principle, 109
III Applications of the Variational Principle, 111
IV Perturbation Theory for a Nondegenerate State, 113
V The Stark Effect of the Hydrogen Atom, 116
VI Perturbation Theory for Degenerate States, 119
VII Concluding Remarks, 120
VIII Problems, 120
10 The Helium Atom 122
I Introduction, 122
CONTENTS ix
II Experimental Developments, 123
III Pauli’s Exclusion Principle, 126
IV The Discovery of the Electron Spin, 127
V The Mathematical Description of the Electron Spin, 129
VI The Exclusion Principle Revisited, 132
VII Two-Electron Systems, 133
VIII The Helium Atom, 135
IX The Helium Atom Orbitals, 138
X Concluding Remarks, 139
XI Problems, 140
11 Atomic Structure 142
I Introduction, 142
II Atomic and Molecular Wave Function, 145
III The Hartree-Fock Method, 146
IV Slater Orbitals, 152
V Multiplet Theory, 154
VI Concluding Remarks, 158
VII Problems, 158
12 Molecular Structure 160
I Introduction, 160
II The Born-Oppenheimer Approximation, 161
III Nuclear Motion of Diatomic Molecules, 164
IV The Hydrogen Molecular Ion, 169
V The Hydrogen Molecule, 173
VI The Chemical Bond, 176
VII The Structures of Some Simple Polyatomic Molecules, 179
VIII The Hu¨ckel Molecular Orbital Method, 183
IX Problems, 189
PREFACE
The physical laws and mathematical structure that constitute the basis of quantum
mechanics were derived by physicists, but subsequent applications became of interest
not just to the physicists but also to chemists, biologists, medical scientists,
engineers, and philosophers. Quantum mechanical descriptions of atomic and molecular
structure are now taught in freshman chemistry and even in some high school
chemistry courses. Sophisticated computer programs are routinely used for predicting
the structures and geometries of large organic molecules or for the indentification
and evaluation of new medicinal drugs. Engineers have incorporated the
quantum mechanical tunneling effect into the design of new electronic devices,
and philosophers have studied the consequences of some of the novel concepts
of quantum mechanics. They have also compared the relative merits of different
axiomatic approaches to the subject.
In view of the widespread applications of quantum mechanics to these areas
there are now many people who want to learn more about the subject. They may,
of course, try to read one of the many quantum textbooks that have been written,
but almost all of these textbooks assume that their readers have an extensive background
in physics and mathematics; very few of these books make an effort to
explain the subject in simple non-mathematical terms.
In this book we try to present the fundamentals and some simple applications of
quantum mechanics by emphasizing the basic concepts and by keeping the mathematics
as simple as possible. We do assume that the reader is familiar with elementary
calculus; it is after all not possible to explain the Scho¨dinger equation to
someone who does not know what a derivative or an integral is. Some of the mathematical
techniques that are essential for understanding quantum mechanics, such as
matrices and determinants, differential equations, Fourier analysis, and so on are
xi
described in a simple manner. We also present some applications to atomic and
molecular structure that constitute the basis of the various molecular structure computer
programs, but we do not attempt to describe the computation techniques in
detail.
Many authors present quantum mechanics by means of the axiomatic approach,
which leads to a rigorous mathematical representation of the subject. However, in
some instances it is not easy for an average reader to even understand the axioms,
let alone the theorems that are derived from them. I have always looked upon quantum
mechanics as a conglomerate of revolutionary new concepts rather than as a
rigid mathematical discipline. I also feel that the reader might get a better understanding
and appreciation of these concepts if the reader is familiar with the background
and the personalities of the scientists who conceived them and with the
reasoning and arguments that led to their conception. Our approach to the presentation
of quantum mechanics may then be called historic or conceptual but is perhaps
best described as pragmatic. Also, the inclusion of some historical background
makes the book more readable.
I did not give a detailed description of the various sources I used in writing the
historical sections of the book because many of the facts that are presented were
derived from multiple sources. Some of the material was derived from personal
conversations with many scientists and from articles in various journals. The
most reliable sources are the original publications where the new quantum mechanical
ideas were first proposed. These are readily available in the scientific literature,
and I was intrigued in reading some of the original papers. I also read various
biographies and autobiographies. I found Moore’s biography of Schro¨edinger, Constance
Reid’s biographies of Hilbert and Courant, Abraham Pais’ reminiscences,
and the autobiographies of Elsasser and Casimir particularly interesting. I should
mention that Kramers was the professor of theoretical physics when I was a student
at Leiden University. He died before I finished my studies and I never worked under
his supervision, but I did learn quantum mechanics by reading his book and by
attending his lectures.
Finally I wish to express my thanks to Mrs. Alice Chen for her valuable help in
typing and preparing the manuscript.
HENDRIK F. HAMEKA
The physical laws and mathematical structure that constitute the basis of quantum
mechanics were derived by physicists, but subsequent applications became of interest
not just to the physicists but also to chemists, biologists, medical scientists,
engineers, and philosophers. Quantum mechanical descriptions of atomic and molecular
structure are now taught in freshman chemistry and even in some high school
chemistry courses. Sophisticated computer programs are routinely used for predicting
the structures and geometries of large organic molecules or for the indentification
and evaluation of new medicinal drugs. Engineers have incorporated the
quantum mechanical tunneling effect into the design of new electronic devices,
and philosophers have studied the consequences of some of the novel concepts
of quantum mechanics. They have also compared the relative merits of different
axiomatic approaches to the subject.
In view of the widespread applications of quantum mechanics to these areas
there are now many people who want to learn more about the subject. They may,
of course, try to read one of the many quantum textbooks that have been written,
but almost all of these textbooks assume that their readers have an extensive background
in physics and mathematics; very few of these books make an effort to
explain the subject in simple non-mathematical terms.
In this book we try to present the fundamentals and some simple applications of
quantum mechanics by emphasizing the basic concepts and by keeping the mathematics
as simple as possible. We do assume that the reader is familiar with elementary
calculus; it is after all not possible to explain the Scho¨dinger equation to
someone who does not know what a derivative or an integral is. Some of the mathematical
techniques that are essential for understanding quantum mechanics, such as
matrices and determinants, differential equations, Fourier analysis, and so on are
xi
described in a simple manner. We also present some applications to atomic and
molecular structure that constitute the basis of the various molecular structure computer
programs, but we do not attempt to describe the computation techniques in
detail.
Many authors present quantum mechanics by means of the axiomatic approach,
which leads to a rigorous mathematical representation of the subject. However, in
some instances it is not easy for an average reader to even understand the axioms,
let alone the theorems that are derived from them. I have always looked upon quantum
mechanics as a conglomerate of revolutionary new concepts rather than as a
rigid mathematical discipline. I also feel that the reader might get a better understanding
and appreciation of these concepts if the reader is familiar with the background
and the personalities of the scientists who conceived them and with the
reasoning and arguments that led to their conception. Our approach to the presentation
of quantum mechanics may then be called historic or conceptual but is perhaps
best described as pragmatic. Also, the inclusion of some historical background
makes the book more readable.
I did not give a detailed description of the various sources I used in writing the
historical sections of the book because many of the facts that are presented were
derived from multiple sources. Some of the material was derived from personal
conversations with many scientists and from articles in various journals. The
most reliable sources are the original publications where the new quantum mechanical
ideas were first proposed. These are readily available in the scientific literature,
and I was intrigued in reading some of the original papers. I also read various
biographies and autobiographies. I found Moore’s biography of Schro¨edinger, Constance
Reid’s biographies of Hilbert and Courant, Abraham Pais’ reminiscences,
and the autobiographies of Elsasser and Casimir particularly interesting. I should
mention that Kramers was the professor of theoretical physics when I was a student
at Leiden University. He died before I finished my studies and I never worked under
his supervision, but I did learn quantum mechanics by reading his book and by
attending his lectures.
Finally I wish to express my thanks to Mrs. Alice Chen for her valuable help in
typing and preparing the manuscript.
HENDRIK F. HAMEKA
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