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Elementary Quantum Mechanics
By David S. Saxon
Dover Publications, Inc.Copyright © 1996 David S. Saxon
All rights reserved.
The dual nature of matter and radiation
1. THE BREAKDOWN OF CLASSICAL PHYSICS
In the latter part of the 19th century, most physicists believed that the ultimate description of nature had already been achieved and that only the details remained to be worked out. This belief was based on the spectacular and uniform success of Newtonian mechanics, combined with Newtonian gravitation and Maxwellian electrodynamics, in describing and predicting the properties of macroscopic systems which ranged in size from the scale of the laboratory to that of the cosmos. However, as soon as experimental techniques were developed to the stage where atomic systems could be studied, difficulties appeared which could not be resolved within the laws, and even concepts, of classical physics. The necessary new laws and new concepts, developed over the first quarter of the 20th century, are those of quantum mechanics.
The difficulties encountered were of several kinds. First, there were difficulties with some of the predictions of the beautiful and general classical equipartition theorem. Straightforward applications of this theorem gave the wrong, and even a nonsensical, black-body radiation spectrum and gave wrong results for the specific heats of material systems. In both cases, the empirical result implies that only certain of the degrees of freedom participate fully in the energy exchanges leading to statistical equilibrium, while others participate little or not at all.
Second, there were difficulties in explaining the structure, and indeed the very existence, of atoms as systems of charged particles. For any such system, static equilibrium is impossible under purely electromagnetic forces, while dynamic equilibrium, for example, in the form of a miniature solar system, is equally impossible. Particles in dynamic equilibrium are accelerated and, classically, accelerated charges must radiate, thus causing rapid collapse of the orbits, whatever their precise nature might be. Accepting the fact that atoms somehow do manage to exist, there is still the problem of explaining atomic spectra, the characteristic radiation caused by the acceleration of the charged constituents of an atom when it is disturbed from its equilibrium configuration. Classically, one would expect such spectra to consist of the harmonics of a few fundamental frequencies. The observed spectra instead satisfy the Ritz combination law, which states that the frequencies are expressible as differences between a relatively few basic frequencies, or terms, and not as multiples.
A third, and more special, class of difficulties is illustrated by the photo-electric effect. Photo-emission of electrons from an illuminated surface takes place under circumstances which permit no classical explanation. The essential difficulty is this: the number of emitted electrons is proportional to the intensity of the incident light and thus to the electromagnetic energy falling on the surface, but the energy transferred to the individual photo-electrons does not depend at all upon the intensity of the illumination. Instead this energy depends upon the frequency of the light, increasing linearly with frequency above a certain threshold value, characteristic of the surface material. For frequencies below this threshold, photo-emission simply does not occur. Otherwise stated, at frequencies below threshold, no photoelectrons are emitted even if a relatively large amount of electromagnetic energy is being transmitted into the surface. On the other hand, at frequencies above threshold, no matter how weak the light source, some photo-electrons are always emitted and always with the full energy appropriate to the frequency.
The explanation of these various difficulties began in 1901, when Planck assumed the existence of energy quanta in order to obtain the desired modification of the equipartition theorem. The implication that electromagnetic radiation therefore had corpuscular aspects was emphasized, and indeed first recognized, in 1905 in Einstein's direct and simple predictions of the characteristics of photo-electric emission. It was also Einstein who first realized, two years later, that the low-temperature behavior of the specific heats of solids could be explained by quantizing the vibrational modes of internal motion of a material object according to Planck's rules. The first understanding of atomic structure and spectra came in 1913, when Bohr introduced the revolutionary idea of stationary states and gave quantum conditions for their determination. These conditions were subsequently generalized by Sommerfeld and Wilson, and the resultant theory accounted almost perfectly for the spectrum and structure of atomic hydrogen. But the Bohr theory encountered increasingly serious difficulties as attempts were made to apply it to more complex problems and to more complex systems. The helium atom, for example, proved to be completely intractable. The first indication of the ultimate solution to these problems came in 1924, when de Broglie suggested that, just as light waves exhibit particle-like behavior, so do particles exhibit wave-like behavior. Following up this suggestion, Schrödinger developed, in 1926, the famous wave equation which bears his name. Slightly earlier, and from a very different point of view, Heisenberg had arrived at a mathematically equivalent statement in terms of matrices. At about the same time, Uhlenbeck and Goudsmit introduced the idea of electron spin, Pauli enunciated the exclusion principle, and the formulation of nonrelativistic quantum mechanics was substantially completed.
2. QUANTUM MECHANICAL CONCEPTS
The laws of quantum mechanics cannot be derived, any more than can Newton's laws or Maxwell's equations. Ideally, however, one might hope that these laws could be deduced, more or less directly, as the simplest logical consequence of some well-selected set of experiments. Unfortunately, the quantum mechanical description of nature is too abstract to make this possible; the basic constructs of quantum theory are one level removed from everyday experience. These constructs are the following:
State Functions. The description of a system proceeds through the specification of a special function, called the state function of the system, which cannot itself be directly observed. The information contained in the state function is inherently statistical or probabilistic.
Observables. Specification of a state function implies a set of observations, or measurements, of the physical properties, or attributes, of the system in question. Properties susceptible of measurement, such as energy, momentum, angular momentum, and other dynamical variables, are called observables. Observations or observables are represented by abstract mathematical objects called operators.
The process of observation requires that some interaction take place between the measuring apparatus and the system being observed. Classically, such interactions may be imagined to be as small as one pleases. Normally they are taken to be infinitesimal, in which case the system is left undisturbed by an observation. On the quantum level, however, the interaction is discrete in character, and it cannot be decreased beyond a definite limit. The act of observation thus introduces certain irreducible and uncontrollable disturbances into the system. The observation of some property A, say, will produce unpredictable changes in some other related observable B. The existence of an absolute limit to an interaction or a disturbance permits an absolute meaning to be given to the idea of size. A system may be thought of as large or small, and treated as classical or quantum mechanical, to the extent that a given irreducible interaction can be safely regarded as negligible or not.
The notion that precise observation of one property makes a second property (called complementary to the first) unobservable is a completely quantum mechanical idea with no counterpart in classical physics. The attributes of being wave-like or particle-like furnish one example of a pair of complementary properties. The wave-particle duality of quantum mechanical systems is a statement of the fact that such a system can exhibit either property, depending upon the observations to which it has been subjected. A second and more quantitative example of a pair of complementary observables is furnished by the dynamical variables, position and momentum. Observing the position of a particle, say by looking at it, which means by shining light on it, will necessarily produce a finite disturbance in its momentum. This follows because of the corpuscular nature of light; a measurement of position requires at least one photon to strike the particle, and it is this collision which produces the disturbance. One immediate consequence of this relationship between measurement and disturbance is that precise particle trajectories cannot be defined at the quantum level. The existence of a precise trajectory implies precise knowledge of both position and momentum at the same time. But simultaneous knowledge of both is not possible if measurement of one produces a significant and uncontrollable disturbance in the other, as is the case for quantum mechanical systems. We emphasize that these mutual disturbances or uncertainties are not a matter of experimental technique; they follow instead as an inevitable consequence of measurement or observation. The necessary existence of such effects in a pair of complementary variables was first enunciated by Heisenberg in his statement of the famous uncertainty principle.
We shall return to these questions later, but now we want to begin our development of the laws of quantum mechanics. Our approach, which is not the historical one, will proceed in the following way. First, in the remainder of this chapter, we shall try to make plausible some of the ideas of quantum mechanics, and particularly the ideas of complementarity and uncertainty. We shall do this by considering some experiments and observations which emphasize that matter is dual in nature and that, as one immediate consequence, the precise particle trajectories of Newtonian mechanics do not exist. This at once poses the problem of how the state of motion of a quantum mechanical system is to be characterized and how such systems are to be described. In Chapter II we answer this question by introducing the state function of a system, and we then discuss its probabilistic interpretation. In Chapter III we consider the general properties of observables and dynamical variables in quantum mechanics and give rules for obtaining their abstract operator representations. Next, in Chapters IV and V, we complete the first stage of our formulation by introducing Schrödinger's equation, which governs the time development of quantum systems. Methods of solving Schrödinger's equation for the simplest possible system, the motion of a single particle in one dimension, are discussed in Chapters VI and VII. Only in the final four chapters are we ready to treat the general problem of systems of interacting particles in three dimensions, thus making contact with the real world. Throughout our development we shall continually use the principle that the predictions of the quantum laws must correspond to the predictions of classical physics in the appropriate limit. As we shall see, this principle of correspondence plays a key role in determining the form of the quantum mechanical equations.
The emphasis throughout will be on the quantum mechanical properties of material systems. Because of its complexity, no corresponding systematic development of the quantum properties of electromagnetic fields will be presented, although relevant quantum properties will occasionally be asserted and perhaps even made plausible.
3. THE WAVE ASPECTS OF PARTICLES
The experiment which most nearly isolates the basic elements of the quantum mechanical description of nature is the scattering of a beam of electrons by a metallic crystal, first performed by Davisson and Germer in 1927. Their experiment was designed to test the prediction of de Broglie that, by analogy with the already well-established corpuscular properties of light, there is associated with a particle of momentum p a wave of wavelength λ, now called the de Broglie wavelength, given in terms of the momentum by
λ = h/p.
The universal constant h is Planck's constant or the quantum of action. Motivating de Broglie was the desire to provide a basis for understanding, in terms of fitting an integral number of half-wavelengths into a Bohr orbit, Bohr's apparently arbitrary quantization condition. In any case, Davisson and Germer observed that the electrons of momentum p scattered by the crystal were indeed distributed in a diffraction pattern, exactly as would be x-rays of the same wavelength scattered by the same crystal; and thus they directly, conclusively and quantitatively verified de Broglie's hypothesis.
The quantum of action is seen to have the dimensions of momentum-length or, equivalently, of energy-time, and its numerical value is
h = 6.625 × 10-27 erg-sec.
In most quantum mechanical applications it turns out to be more convenient to use the quantity h/2π, which is abbreviated as [??] and is called "h bar." It has the numerical value
[??] [equivalent to] h/2π = 1.054 × 10-27 erg-sec.
In terms of [??], the de Broglie relation can be rewritten in the form
[??] = λ/2π = [??]/p.
where we have introduced the reduced wavelength [??] (called "lambda bar"), which is physically a more significant length characterizing the wave than is the wavelength itself. It is also convenient to define the wave number k (strictly speaking, the reduced wave number) as the reciprocal of [??]. Thus we can also write the de Broglie relation in the form
p = [??]k.
To collect these relations in a single expression let us write, finally,
p = h/λ = 2π[??]/λ = [??]/[??] = [??]k. (1)
The de Broglie hypothesis, and the Davisson-Germer experiment, are in sharp conflict with classical physics in that both particle and wave properties are assigned to the same entity. The nature and implications of the conflict can be made much clearer by imagining the experiment to be performed with a beam of electrons so limited in intensity that only a single electron is scattered by the crystal and recorded at a time. In that event, no diffraction pattern at all would be observed at first; a given electron would be scattered in some direction or other in an apparently random way. However, as time went on and the slowly accumulating number of scattered electrons mounted into the thousands and millions, it would become increasingly clear that more electrons are scattered in some directions than in others, and thus the diffraction pattern would gradually emerge.
The following conclusions can be drawn from the results of the Davis-son-Germer experiment:
(a) Electrons exhibit both particle and wave properties. The quantitative connection between these is expressed by the de Broglie relation, equation (1).
(b) The exact behavior of a given electron cannot be predicted, only its probable behavior.
(c) Precisely defined trajectories do not exist at the quantum level.
(d) The probability that an electron is observed to be in a given region is proportional to the intensity of its associated wave field.
(e) The superposition principle applies to de Broglie waves, just as it does to electromagnetic waves.
Conclusions (a) and (b) require no further comment. Conclusion (c) follows from (b), because classically a particle moves along a unique trajectory under the influence of specified forces for given initial conditions. Conclusion (d) is inferred from the parallelism between the x-ray and electron diffraction patterns from a given crystal. Finally, conclusion (e) follows from the fact that the diffraction pattern is produced by interference of secondary waves generated at each atomic site in the crystal, that is, by a linear combination or superposition of these scattered waves.
These conclusions are the starting point for our whole development of quantum mechanics. They have been reached without reference to the specific character of the interaction between electrons (or x-rays either, for that matter) with the atoms in the crystal and without reference to the details of the diffraction pattern formed as a result of that interaction. This is no oversight, however, for our argument is based entirely on the behavior of a crystal as a three-dimensional diffraction grating, calibrated by observation of its effects upon x-rays of known properties. Nonetheless, it is a little unsatisfying, pedagogically speaking, to have reached such significant conclusions without exploring all the details. Unfortunately, these details require an understanding of the interaction of an electron with the atoms in a crystalline solid, and this interaction cannot be understood before we understand quantum mechanics itself. For that reason we shall now consider two highly idealized "crucial" experiments which will force us to essentially the same conclusions in a more or less transparent way. These experiments are one-dimensional versions of scattering and diffraction, and they involve nothing but the simplest kinds of systems. However, as will shortly become apparent, our experiments are actually performable only in principle and not in practice.
In the first experiment, as shown in Figure 1(a), a particle of positive charge e and mass m is sent with momentum p down the axis of a long drift tube, the walls of which are at ground potential. Aligned with the first drift tube, and infinitesimally separated from it is a second drift tube at a higher potential V0.
Excerpted from Elementary Quantum Mechanics by David S. Saxon. Copyright © 1996 David S. Saxon. Excerpted by permission of Dover Publications, Inc..
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