A generation has passed since the theory of wave mechanics, or quantum mechanics, was first formulated, and it has been almost two generations since it became apparent that the atomic world is characterized by a type of discontinuous behavior not known to the macroscopic world to which our senses have most direct access. During most of this time, the theory of the mechanics of atomic-sized systems has been the concern of the research scientist, usually in physics and chemistry, and it has been taught, quite appropriately, in graduate schools. As with all great theories, however, quantum mechanics has constantly increased its domain of application, and today, for those interested in understanding basic science—even on the advanced undergraduate level—the principles of the theory have become a vital necessity. Furthermore, with the explosive growth of atomic and nuclear technology, the need for a working knowledge of quantum mechanics has been extended to many areas in engineering and applied science.
A glance at any of the modern undergraduate textbooks on atomic and solid state physics will show that quantum mechanics has "infiltrated" them. For example, there was a time when courses in atomic spectra were basically descriptions, from the experimental point of view, of energy levels, spectral lines, and selection rules. Today it is almost impossible to talk of these matters without using the only theory that adequately organizes and interprets the experiments. No one is satisfied with the relatively simple models of a generation ago. Realizing this, many authors of textbooks in modern physics undertake the Herculean task of teaching the essentials of quantum theory, as well as of describing a wide range of experiments.
The situation is clear. Quantum Mechanics should take its place earlier in the physics curriculum and should be considered to be as basic to later study as classical mechanics and electricity. When this is done, modern physics—atomic, nuclear, and solid state—can be taught more effectively.
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In classical mechanics, one does not worry about the precession of the perigee of a satellite or the nutation of a gyroscope until one has mastered Newton's Laws for the more simple cases. So in quantum mechanics, one must be concerned initially with the simple applications. Unfortunately, some of the most interesting applications involve the more advanced theory, and there is a strong temptation, for example, to wrestle in quantum mechanical terminology with "L-S coupling" when the student has only a vague idea of what a wave function is. In contrast, this textbook emphasizes simple problems, even at the expense of neglecting some favorite—and important—concepts. Since a large part of the complexity of quantum theory is due simply to geometry, we concentrate on one-dimensional systems, which clearly display a surprisingly large fraction of the key ideas and revolutionary concepts. In a first course, it is much more important to apply exact theory to simple cases than to apply approximate theory to complex cases.
The historical approach to a subject, although of great importance in demonstrating how theories are actually developed, can also be very confusing. Today, for example, one does not belabor the erroneous ideas of Newton's and Galileo's predecessors. One says rather: "Here is a theory that works. Its essential predictions can be tested fairly easily. Let us learn to use it." Later, the serious student will study the origin of the ideas in more detail.
Thus, in this book, except for a brief chapter on some of the key experimental findings that led to the quantum theory, we are content merely to postulate the theory in a page or two, and then to use it. In defense of this approach there is one excellent argument—it is efficient.
At points where our limited mastery of the theory permits comparison, we refer to the relevant experimental observations, which are, of course, the true foundation upon which the theory rests.
It must be remembered that this is a first course and in order to place it as early as possible in the student's career we have required minimum dependence on topics in advanced physics and mathematics. Thorough courses in elementary physics and in calculus are essential, however, as is some knowledge of differential equations, complex variables, and orthogonal functions. The use of numerical methods in solving the wave equation in both Cartesian and
spherical coordinates gives a maximum of insight with a minimum of mathematical technique. We avoid philosophical discussion as much as possible and concentrate on the actual use of the theory. For the sake of simplicity, we consider only bound systems and the free particle. Collision theory and matrix mechanics are left for the more advanced textbooks.
Most of the book is concerned with particles without intrinsic "spin." The subject is quantitatively treated only in the last chapter, where it is shown to follow from the postulates as a consequence of relativity.
Quantum mechanics is a discipline with which one does not easily become familiar. It is not so much because the basic ideas are difficult as because they are strange. It takes time to appreciate them, and the student of physical science should be introduced to them as early in his career as possible.
C. W. S.
Urbana, Illinois May, 1959
CONTENTS
1. The Experimental Basis of Quantum Mechanics 1
Problems, 7
2. Basic Postulates 12
2.1. Matter waves, 12
2.2. The basic postulates of quantum mechanics, 13
2.3. Probability, 19
2.4. The wave equation for *F*, 25 Problems, 26
3. The Solution of the Wave Equation 29
3.1. The separation of the time-dependent wave equation, 29
3.2. The solution of the amplitude equation for the harmonic
oscillator, using numerical methods, 30
3.3. The particle in a one-dimensional box, finite walls, 41
3.4. The box with infinite walls, 48
3.5. Mathematical description of the eigenfunctions of the
harmonic oscillator, 50
3.6. The correspondence principle, 52 Problems, 54
4. The Wave Equation in Three Dimensions 62
4.1. The basic postulates for three dimensions and two
particles, 62
4.2. The particle in a rectangular box, 66
4.3. The particle in a central field, 72
4.4. The ^-dependent equation, 76
4.5. The ^-dependent equation, 78
4.6. The /--dependent equation, 87
4.7. The energy levels of the hydrogen atom, 94
4.8. The complete hydrogen atom eigenfunctions, 94
4.9. The energy levels of a physical system, 97
4.10. Conclusion, 98
4.11. Summary of Chapters 3 and 4, 99 Problems, 101
5. The Superposition of States, and Some Calculations Using the Wave 108
Function
5.1. The superposition of states, 109
5.2. The calculation of system energy, 118
5.3. The calculation of position, 123
5.4. The calculation of momentum, 126
5.5. Limitations on measurement in quantum mechanics, 128
5.6. Wave packets and the scattering of particles, 130 Problems, 143
6. Angular Momentum 148
6.1. The angular momentum operators, 148
6.2. The expectation value of the z-component of the angular
momentum, 152
6.3. The expectation value of the magnitude of the angular
momentum, 156 Problems, 159
7. Steady-State Perturbation Theory. Nondegenerate Case 162
7.1. Perturbation theory, nondegenerate level, 164
7.2. A sample calculation for a nondegenerate level, 174
7.3. Summary, 180 Problems, 181
8. Steady-State Perturbation Theory. Degenerate Case 184
8.1. Analysis of a twofold degenerate level, 184
8.2. Example: Analysis of a twofold degenerate level for a
single particle in a rectangular box, 191
8.3. Multiple degeneracy, 197
8.4. The unique relationship between H' and the zero-order
eigenfunctions, 198
8.5. Summary: First-order perturbation theory for a twofold
degenerate state, 199 Problems, 201
9. Identical Particles 205
9.1. Two identical particles in a one-dimensional box, 206
9.2. The symmetry properties of the first-order wave functions,
212
9.3. Some consequences of the symmetry properties of wave
functions, 214
9.4. Particles with nonoverlapping wave functions, 221
9.5. The Pauli exclusion principle, 224
9.6. Summary, 229 Problems, 233
10. Time-Dependent Perturbation Theory 239
10.1. Time-dependent perturbation theory, 241
10.2. Constant perturbation, 248
10.3. Harmonic perturbation, 251
10.4. The harmonic oscillator in a periodic electric field, 256
10.5. An example: The vibration spectrum of the diatomic
molecule, 265
10.6. The importance of time-dependent perturbations, 269
10.7. Summary, 271 Problems, 273
11. The Relativistic Wave Equation and the Origin of Electron Spin 278
11.1. The relationship between energy, momentum, and mass in the special theory of relativity, 278
1.2. The relativistic Hamiltonian in linear form, 281
1.3. Matrix operators, 283
1.4. The Dirac matrices, 287
1.5. The Dirac wave equation for a free particle, 289
1.6. Particles with negative total energy, 300
11.7. The Dirac particle in the one-dimensional well, 301
11.8. Identical Dirac particles, and the exclusion principle for
electrons, 308
11.9. Singlet and triplet states, 314
11.10. The nonrelativistic spin wave functions, 322
11.11. Summary, 324 Problems, 329
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