4h
p -
California Association of Chemistry Teachers
Robert H. Maybury
University of Redlands Redlands, California
The Language of Quantum Mechanics
No
teacher of biology would talk to his class about insects, animals, or growing plants outside of the larger context of evolution. This great theory of Darwin provides the very mood in which an informed study of biology t,akes place today. As chemistry teachers, we have a corresponding obligation t,o tell our story of nature, the story of atoms, molecules, bonds, and electrons, in the mood appropriate to this modern day. This mood has been set by the great theories of quant,um mechanics developed in the 1920's. Them is every bit as much grandeur and sweep to these theories as t,hei-e is to the Darwinian picture of evolut,ion. Unfortunately, quantum mechanics has the appearance of a formidable mathematical language. This has prevented many science teachers from becoming familiar with it (in contrast to the theory of evolution, an essentially narrative account). Scientists are properly cautious about popularizations of their subject, particularly when they are asked to eliminate the mathematically precise statements so necessary for accuracy. Little has been done, therefore, to bring down to ordinary speech the statements of this theory. But in teaching science it is all important that the right mood prevail so that the student might be encouraged to master the intricacies of the suhject. It is the science teacher's task to create this mood, often by popularizing or simplifying the quantitative theory. A theory may have value not only as a tool, in which form it provides the scientist with knowledge inaccessible to him through experiment, but also as a language, in which form it provides a framework for better understanding of the observations and concepts of the suhject. I n the following presentation of quantum mechanics, mathematical rigor and quantitative accuracy have been sacrificed to provide the teacher of chemistry with a view of quantum mechanics which will serve him as a language for understanding the wide variety of concepts common to the work of the chemist t,oday--orhitals, resonance, hybrid bonds, and the like. The chemistry classroom abounds these days with Styrofoam models of aboms, orbitals, and molecules.
Presented as a series of lectures before the Annual Junior College Workshop in Chemistry, a program ior the improvement of instnntion in mathematics and the sciences in the Junior Colleges of California under the National Defense Education Act, Tit!e 111.
It is evident that the teacher of science strongly feels the need to rely on concrete models to produce understanding. The scientist gathers together his observations of nature into a model as a means of explaining these observations. Most models are visualizations of the subject in terms of familiar, common sense objects. I n this sense a model is closely related to an analogy, a comparison of an unfamiliar experience with a familiar ohject of experience with the hope of rendering the unfamiliar experience more recognizable or friendly, hence more understandable. So it was that Michael Faraday compared the field of a magnet to elastic ribbons or lines of force. The use of models and analogies in science must always he accompanied with warnings not to take them too far. After all, a comparison between an unfamiliar experience and something well known to us in experience would be an identity rather than an analogy if the two were identical, without the differences which eventually show up when analogy is pressed too far. The most important ideas to keep in mind in using models are that they are analogical representations of our experiences with the unknown structures of nature, that they are tentative (and will need to be altered as additional experience is gained), and that they are in no manner to be considered as duplicate pictures of the reality behind our observations. The reader may ohject to our claim that a model assists our understanding of phenomena while at the same time we deny that it is a literal picture of the reality which lies hehind the observed phenomena. Yet we must learn this lesson: we have increased our understanding, simply by putting our observations in a form with which we are more a t home! The Early Model of the Atom
A study of the atom yields a considerable array of phenomena to the experimenter. There are the sharp line series found through spectroscopy, the periodic array of the chemical elements in the periodic table, the extraordinary stability of the atom as an organized group of electrons about a nucleus, the characteristic ionization potentials of the various elements, and the magnetism manifest in a variety of effects. Early in the 20th century the atom, through study of these diverse phenomena, had become an ohject of intense curiosity to the scientist. J. J. Thomson, producing convincing evidence for the existence of the negatively charged electron as a Volume
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constituent of an atom, postulated in 1904 a "plumpudding" model of the atom consisting of a positive sphere of charge in which the negative electrons were embedded like raisins! But Rutherford's important scattering experiments followed almost immediately to indicate that, instead, the positive charge was intensely concentrated on a minute and massive nucleus. Clearly a more adequate model was called for. A big step was needed in scienceas it so often is when the number of separate pieces of data far outweigh the picture or model at hand. The big step was made by Rutherford's genius. It was the nuclear atom with which we all are familiar. Soon Niels Bohr picked up this planetary model of the atom and refined it to include the quantum idea regarding the ultimate granularity of nature which Max Planck had enunciated. Nature at base, said Planck, is not continuous or fluid-like, but grain-like: energy, light, matter, are ultimately small grains of indivisible content. The orbits of the Bohr atom provided for this quantization in the picture of the atom by limiting the electron only to certain definite and allowable orbits. For a few years matters progressed nicely; but then difficulties began to appear. New experimental facts were found through the guidance of the Bohr model which placed a strain on the model. Sornmerfeld made a valiant effort to modify the planetary atom model to allow for some of these by resorting to elliptical rather than circular orbits. But by 1920, what had seemed so beautiful in 1912 was a wilted dream among scientists throughout the world as the planetary atom model headed into deep trouble. The model had been born in imagination at one period of time but its inherent tentativeness led to its death. The time was again at hand for a new Right of the imagination to generate a new model through which understanding could be achieved. De Broglie's Wave Mechanics
A young French prince, attracted by the relativity theory Einstein had been publishing through this period of time, devoted his doctoral study to the characteristics of light. He was fascinated with the idea that light waves exhibited interference and defraction phenomena explainable only on wave theory, and yet they also showed properties attributable only to particles, the photo-electric effect, for instance. Einstein had termed these particles photons, and had associated with the wave length of the light a corresponding momentum. He expressed this: hv momentum = C
where : Planck's quantum of action (an essential constant that appears in all equations in which the quantum or granular aspect of nature is expressed) v = the frequency of the light and c = the speed of light h
=
This young prince, Louis de Broglie, nearly missed getting his degree by letting his imagination go too far (in the judgment of his professors a t the University). He made a wild suggestion that if photons with mechanical or particle-like properties were associated with 368
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light waves, why in a corresponding manner should we not associate waves, wavelengths, and all the other paraphernalia that belong to waves, with material particles such as electrons! Nothing regarding such waves associated with material objects had ever been observed and we must always credit de Broglie with an amazing imagination to have suggested such a condition. Would we be any less critical than his professors were in the face of such a daring and wild suggestion? De Broglie persisted in his efforts to convince his teachers and published his thesis finally with their less than enthusiastic permission. His startlingly novel idea was that with a material particle such as the electron there is an associated wave. The wavelength of this wave is associated with the momentum of the particle through the following relationship, in complete analogy to the photon-wave relationship for light waves: momentum (mu.)= h" V
=
h X
where: m
= =
v
c
and X
= = = =
mass of the electron velocity of the electron Planck's constant of action frequency of the wave associated with the electron the wave velocity the wavelength of the associated wave
The physics of waves says that for waves of any kind the wavelength, frequency, and wave velocity are related thus: h" = v
This allows us to work either with X (wavelength) or with v (frequency) in our equations. De Broglie knew that the photon and light wave ideas were not really clear to anyone who had tried t,o understand their association in light theory. His assertion that waves were to be associated with a material particle such as the electron did not clear up this confusion hut instead only introduced greater confusion into science. Consequently, though he did not bring any ordering into atomic science in the 1920's, his imagination provided the all-important spark of suggestion that a model of an atom need not be built on t,he particle analogy alone: pay equal attention to the wave analogy! Schrodinger Wave Model
This is precisely the kind of thinking that another European physicist, Erwin Schrodinger, began as soon as he came across de Broglie's strange idea. He proposed a model of the atom entirely in terms of waves rather than material particles as the Bohr planetary model does. I n 1925, Schrodinger startled the scientific world with the publication of his theory of wave mechanics or quantum mechanics, in which the model for the atom was taken to be a wave. And even today our chemistry with its orbitals is a direct fruit of this wave model, for the Schrodinger idea has held out for 30 or more years as a highly workable model of the atom. I n the section that follows we shall develop in some detail the framework of the Schrodinger quantum mechanics. Schrijdinger, convinced that a new model for the atom was needed, seized upon the clue to be found in de Broglie's suggestion of a wave associated with an electron. A good physicist, Schrodinrer
was of course familiar with the available experimental information then pertaining to atoms. The spectroscopists had published their work showing that distinct lines characterized atomic spectra. B o b and others had emphasized that any adequate model of the atom must incorporate the quantum ideas of Planck, i.e., the model must provide for a series of discrete energy levels in the atom. Bohr had shown how this requirement could be met by arbitrarily introducing quantwn numbers, a series of integers, into his theory. As Schrodinger reflected upon these facts, the insight came to him that the standing waves associated with vibrating objects, such as strings or drum heads and the like, were mathematically expressible by equations in which a set of integers necessady and naturally appeared. That was the point he sensed to be crucial. He needed only to find the right vihrating ohject as a model and the standing waves for this object would have equations which contain this set of integers; these equations would then represent the electrons in an atom. Amazingly enough, he found the answer to his search for the right vibrating object in some mathematical work by William Hamilton, the mathematical genius of 100 years earlier. The vihrating ohject was a flooded planet-a uniformly deep ocean over a spherical planet. Hamilton, evidently working out some mathematics connected with problems of the tides in which the moon perturbs the earth's ocean in a periodic manner, had idealized the problem by writing out the complete mathematics of the vibrations, that is, the fundamental wave and the higher overtones, for a uniformly deep ocean over the surface of the earth. Schrodinger recognized in an instant that the solutions to the wave equation for this model were the equations for these standing waves, and that these could represent the electrons in an atom. Before taking a closer look a t this matter of waves, wave equations, and solutions to wave equations, let us note that the familiar orbitals that we use in our chemistry teaching, the s, the p, and the d orbitals with their characteristic shapes, are close to being threedimensional plots of the mathematical equations for these standing waves in this vihrating, flooded-planet model worked out by Hamilton and applied to the atom by Schr~dinger. To carry our story beyond this point, especially to appreciate the way these integers arise in any mathematical treatment of waves, some familiarity with the most elementary aspects of wave motion is required. Suppose, then, that a string is stretched tautly between two posts. Waves can be set up in this string simply by plucking it with the finger. Because the string is not infinite in length hut instead is bounded on each end, the vibration of the string when plucked leads to a set of standing waves. The standing wave shown in
Figure 1 is termed the fundamental mode of vibration of the string, and corresponds to a wave in which the entire length of the string from one end to the other undergoes a displacement from the rest position. Other, higher modes of vibration can be induced in this string; the first ouertone occurs (Fig. 2 ) when the string vibrates in such a way that its center point remains motionless and unmoved from its rest position. This fixed point on the string between the two tied-down ends is termed a node. Higher overtones occur in this string when additional nodes are located in the string as shown in Figure 2. These wave forms shown in Figures 1 and 2 may he described more succinctly by simple trigonometric expressionsin which y, the height above the rest position of any given point x along the string, is related to this point x. Thus, y = a sin s(x/Z), in which y is the amplitude, x is the position at a given point along the string, 1 is the length of the bounded string, and a, a constant, describes the fundamental mode of vibration. Each of the overtones is expressed similarly:
v.
=
a sin n s ( s / l )
where n = 2 for the first overtone, n = 3 for the second overtone, and so on. Clearly this elementary consideration of a vibrating string has revealed one striking and highly significant feature regarding wave motion, namely, the perfectly natural association with the trigonometric expressions for the modes of vibration of a set of integers, the n = 1, 2, 3, 4 . . ., in the above equations. Schrodiuger, a competent physicist thoroughly familiar with these mathematical aspects of vihrating objects, knew that a similar set of integers would emerge from a mathematical treatment of the vihrating flooded-planet model. Again in more mathematical terms, what he recognized was that imposing boundary conditions upon the equation for wave motion of any vibrating ohject generates a set of expressions for the corresponding standing waves. For a string, the equation for wave motion is:
First Overtone
%
A
Figure 1.
x-
Second Overtone
O x-
Third Overtone
figure 2.
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I n this equation, u is the velocity of the wave in the string and indicates that this wave equation refers to a vibration in a long, long string in which a wave pulse travels along the string toward one end or the other. An interesting and useful thing happens to this wave equation when we tie the string down at both ends (as in the case of the string that we considered above, tied to two posts). Mathematically, we say that we are imposing houndary conditions on the wave equation. These conditions for the string tied on each end can be written mathematically as: at x
=
0,y = 0; and at z = 2, y
=
0,again
When these boundary conditions are taken into account, the wave equation can be solved by a mathematical treatment referred to as "turning a differential equation crank." Out fall the solutions to the wave equation, the set of standing waves existing in this particular string tied down a t each end:
.
where n is an integer having values 1, 2, 3 . . . I n going from the general form of the wave equation to the solutions given above, several steps have been omitted; these should be hinted a t here. F i s t of all, in the general wave equation it is clear that y is really a function of both x and t. The first step in solving that equation is to get rid of the time dependency, so that an equation for standing waves with y as a function of x only remains. The mathematician who solves this differential wave equation gets rid of the time by separating the wave equation into two parts, one part depending only on x, the other part depending only on the time. I n eliminating the part that depends on t h e finds it necessary to introduce a term that imparts to the remaining portion of the equation, dependent only on x, a periodic or wave motion form. This term he writes into the equation either as a frequency or as a wavelength. That this wave property of wavelength actually is embodied in the expression y = A sin n a (xll) can he demonstrated thus: in the fundamental mode of vibration, the wave length, XI, is equal to twice the length of the string; in the f i s t overtone, Xz is equal to the length of the string; and so forth. For all the harmonic vibrations in this string, the relationship between I and X, can be stated in this way:
wave motion in three dimensions:
E"+-a=+ + -a = - - I a=* ax"y2 asp u* at* In this equation the Greek letter \I. is used to represent the wave amplitude which in our string equation was represented by y; u stands for the velocity of the wave impulse through space; x, y, and z are the usual Cartesian coordinates. Alternatively this equation may be written in r, 8, and p, the spherical coordinates. Refering to one of the books cited in the bibliography will reveal the form this equation takes in t,hese coordinates. A procedure similar to that used in obtaining the expressions for standing waves in the string is now followed. First, the time is eliminated from the wave equation to yield an equation in which \I. is a function only of the space coordinates x, y, and z or r, 8, and p. This is accomplished by dividing the equation into two parts, one containing time and one only the space coordinates, and introducing into the part which depends on space coordinates a term that indicates the periodic or wave-like aspect of this part. The result of these operations is an eqnation of this form:
or its equivalent in r, 8, and p. Finally, the boundary conditions appropriate to the flooded-planet model are imposed upon this equation for wave motion to generate a set of expressions for the standing waves. The genius of Schrodinger is most readily appreciated a t just this point, for he was the first to recognize the fruitfulness of placing de Broglie's electronic wave length, X =h/mv, into this equation for wave motion in three dimensions. Thus, placing h/mv into the equation above where X appears completes the analogy between an ordinary wave of experience and the electron considered as a wave. After this suhstitution, the wave eqnation now appears as:
A few changes using algebra alter this equation into the form of the well-known Schrodinger wave equation. First of all, classical mechanics states that: E = kinetic energy E = muZ V E = (mu)* -+V 2m
+
Our solutions may Ly written, then, in this form: = A , sin 2r(xjX,). When we consider the vibrations in objects more complicated than the string-the drum-head or a solid object, for instance-we find that acoustical theory in each case provides a wave eqnation similar to that for the string. Then in each case boundary conditions suggested by the actual geometry of the vibrating object can be imposed on the wave equation. These cause the wave equation to yield solutions describing standing waves made up of afundamental, overtone and higher overtones. I n every case these equations are rclated through a series of integers just as we found with the simple string example. I n setting up his wave equation for an atom, SchrG dinger first had to write out the general eqnation for
v.
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+ potential energy
or rearranging: Substituting this relationship for momentum into our wave equation, the wave equation now takes on the form:
or rearranged:
which is the well-known Schrodinger wave equation. When this eqnation describes electrons in an atom, V represents the electrostatic potential energy of the
electrons in the field of the nucleus. Understandably, this field depends upon the nuclear charge. The part of the equation:
is commonly abbreviated Vz$. V may just as weU be expressed in terms of the spherical coordinates r, 8, and + rather than the Cartesian coordinates x, y, and z. The wave equation now may be written more tersely:
This probability interpretation of J. is such a revolutionary idea that some special attention to the proper interpretation of this amplitude function, $2, needs to he given. Let us ask the question: Where is the electron in an atom, say the hydrogen atom, an atom with a nucleus with one positive charge and a single electron? Where is the electron on the basis of this quantum mechanical interpretation Born has given to J.2? When the Schrodinger equation is solved for the hydrogen atom, the solutions are equations of the form:
*
The symbol H is given to
=
Ae-),
where A and k are constants and r is the distance from the nucleus. A graph of J.2, which is said to he proportional to the probability of finding the electron at a given point T from the nucleus plotted vs r has the form shown in Figure 3.
allowing the equation to appear as: Here is a revolutionary new wave equation. Its solutions will correspond to a fundamental and the higher harmonics of a wave for a vibrating substance. This vibrating substance is the model for the electron of wave length A. Schrodinger reached this final stage by recognizing that the boundary condition to be imposed on this wave equation had, of course, to correspond to something l i e the geometry of an atom. This he set as follows: the amplitude $ should become zero at a distance of infinity away from the center of the atom. Imposing this novel boundary condition on this wave equation and "turning the crank" of the differential equation, he obtained solutions which were the mathematical equations of a series of the standing waves of this peculiar object he had chosen as a model for the atom. These parts of these solutions which depend only on the angles 8 and 6 match exactly solutions for the flooded planet wave equation. The orbitals we use so commonly are, very closely, three dimensional plots of these solutions. The Probability Interpretation of $
Now that a set of waves has been found which represents the electrons in an atom, we face (as Schrodinger faced) the puzzling question of the physical meaning of these waves. Schrodinger thought these waves actually were the electrons and as a result he pictured an electron spread out over space in the shape of these waves. But other physicists showed that this was a highly untenable position. Many ideas were put forth concerning the interpretation of this amplitude of the wave, J.,until finally Max Born proposed an answer of a most revolutionary nature. He startled the scientific world with his assertion that this wave with an amplitude of J. was not the electron at all, but was a wave of probability or something very close to this. Actually, he asserted that the square of J. (fi2) described the probability that the electron would he found a t each particular point in space. I n giving the wave this probability interpretation he left, of course, as an unanswered question the problem of the nature of the electron itself.
0
I -
Figure 3.
How shall we interpret this graph in our attempt to reach an answer to the question, "where is the electron?" The best way is for us to play a End of game. Pretend that the nucleus of the hydrogen atom were a golf ball placed on a chair. We, as some sort of atomicsized observers with pencil, paper, and a stop watch in hand, stand at a distance one foot away from the golf hall. We are now to look for the electron as it passes our observation point in its flight in the vicinity of the nucleus. The electron may be imagined to be a bee flying in this space. Occasionally the bee will fly in front of our nose while we stand a t this observation point one foot away from the golf ball on the chair and when it does, we place a stroke on our notepad with our pencil. Keeping a record of the number of times we see it a t this spot in a five-minute interval, we then take up a new observation position out 2 ft from the golf ball where we repeat this experiment of tabulating appearances of the electron "bee" for another five-minute interval. Then we move out 3 ft from the golf ball, make counts for five minutes, and so on until we have moved out 10 or 20 ft from the golf ball. Now we plot the results of these scores at each station to see what the record may show us. Of course, our distances would be Angstrom units, not feet, if we were to conduct this imaginary game in the realm of .the real atom. As we can see (Fig. 4), the summary of our experiences of
Figure 4
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"seeing" the electron "bee" resembles the plot of the $2 function above. For the purposes of a rough analogy this is a perfectly good way to interpret what we mean about the answer to the question, "where is the electron?" The wave equation gives a solution, $, which, when squared and plotted against distance from the nucleus, yields a probability plot. This plot tells us a t any given point r the number of times per unit time that we would experience the electron if we should look for it at that point. We must hasten to clean up this analogy a little bit in one respect, and that is to destroy the notion that the electron can be thought of as literally running around inside the atom making appearances a t different points. I n reality it is absolutely impossible to pin the electron down as a particle with a continuous, unbroken path, running around like a bee, making appearances here and there. The very act of attempting to "see" the electron leads to the electron showing itself as a point a t a given position. It is also incorrect to think of the electron as being like a bee flying around having intrinsic but unrevealed position in between these acts of ours to measure its position. Our only knowledge of this electron is that given by this $2 plot which is a statistical knowledge-a brand new way of knowing an object, indeed! The $2 plot is a summary of a large number of our experiences of the position of the electron. It is meaningless to ask just what the electron is other than this view. This is the most honest presentation possible of the new spirit in which we must talk about electrons in atomic systems and shows the radical sense in which wave mechanics alters our view of fundamental matter. The Orbital Model
Now we are prepared to discuss the orbitals in somewhat the same manner. The s orbital, as has always been noticed, is represented as a sphere. This spherical orbital, a mathematical surface of spherical symmetry, is actually a plot of J'2 (in terms only of its angular dependency). The significance of this surface is that within it the electron would be found, if it were being looked for, about 90% of the time. Were we to go looking for the electron by making position measure ments, there is a nine-out-of-ten chance that the electron would register a position signal with us inside of the surface of this sphere. We can consider the p orbitals in a similar manner. There are three of these, which are mathematical surfaces inside of which 90% of the time we are likely to find an electron if we go looking for it. The same thing can be said for the five d orbitals. Time must be spent studying this interpretation until its full significance is grasped. I n review, these surfaces are mathematical surfaces representing the fiZfunction (at least the part of that depends on angle). They give the region inside of which the electron makes an appearance 90% of the time in experiments performed to show up electron position. We must remember that Schrodinger had recognized the natural occurrence of a set of integers every time a wave equation is solved with boundary conditions applying. This was one of the most rewarding outcomes of solving the Schrodinger equation for the atom, because the quantnm numbers n, I, and m (the principal quantum number, the quantum number for angular momentum, and the magnetic quantum number) all 372 / journal o f Chemical Education
appear automatically aspart of the solutions for the wave equation. The existence of energy levels in this wave picture of the atom is of interest. Just as a vibrating string possesses an energy that is characteristic for each harmonic mode of its vibration, so the energy of the electron in an atom (or the energy level of the atom) is related to the orbital mode of the electron in that atom. Quantum mechanics says all this very succinctly. The Schrodinger wave equation, written as: H*
=
E*
yields a value for E, the energy of the atom, which is a function of the quantum number n primarily. The magnitude of E is calculated by the above equation. $ represents a given state of the electron and, of course, a particular solution to the wave equation. There is a definite value of E that depends upon the J' in question. If $ is the 1s $, for instance, meaning that the electron is in the 1s orbital, then E has a value characteristic of the 1s state. If J' is the 2s orbital, that is, the electron is in the 2s state, E will have a correspondingly different value. For the hydrogen atom these values of E can be calculated this way for each fi corresponding to the various harmonics, and these energy levels can be plotted on a graph.' Summary
This treatment of the atom has admittedly been a very qualitative handling of the Schrodinger wave mechanics and a more precise account of this story should be read in a standard text. But the important point established here is that these orbitals and the corresponding ideas they lead to in chemistry-bonding, hybrid orbitals, etc.-are consequences of a radically different model of the atom, a wave model, initiated by de Broglie and completed by Schrodinger in his wave mechanics. We must establish this change in essential viewpoint with our students to set the proper mood for their study. They will fill in with the more accurate details in their advanced study of quantum mechanics. Today the knowledge of the snb-atomic world of electrons in atoms has become considerably more complicated and mysterious as a result of the investigations of the early part of this century. It is clear that the electron exhibits behavior a t one time like that of a particle, for instance, in the photoelectric experiment, whereas at other times its behavior is that of a wave, for instance, in diffraction experiments with crystals. This peculiar behavior of the electron has been tidily summarized in the complementarity theory of Niels Bohr in which he says that our knowledge of the electron is a lot like that of the proverbial blind man observing the elephant. Our knowledge depends upon the particular kind of observation we make. When we set up experiments to observe the electron as a wave, the electron obliges us by behaving as a wave. When we set up experiments to observe the particle character of the electron, the electron obliges us by behaving as a particle. What the electron really is, whether
' For example, see Figure 2-19 in PAULINQ,LINUS,"Nature of the Chemical Bond," 3rd ed., Cornell University Press, Ithaca, N. Y., 1960, p. 56. See also Figure 4 on page 290 of the June. 1962, issue of mrs JOURNAL.
particle or wave or some combination of particle and wave, is not a question we can answer. The electron has the capacity for exhibiting both these behaviors and probably has the capacity to exhibit other behaviors which we may not yet have set about to observe. Our position is at least more tenable if we keep clear in our minds the role of our models in science. To urge that the electron must he a particle or a wave is to mistake the model for reality. We remarked a t the outset that our models are, a t best, analogies which attempt simply to erect a comparison between something familiar and something strange to us. Waves and particles are familiar everyday objects to us. The electron is still a mystery. Thus, we try to make it friendlier to us by a t one time comparing it to a wave, a t another time comparing it to a particle. Let us remember that we come to terms with our experience of nature's mysteries by turning to models, but these
models are not the ultimate realities behind the mysteries. Fortunately, models are useful to us. I n science we must learn to be content with usefulness as the best we can have with our finite understanding in a world of infinite mystery. Bibliography D'Asao, A,, "The Rise of the New Phyaies," 2 Vol., Dover Publications, New York, 1951. R. B., "Scientific Explanation," Harper TorchBRAITHWAITE, book 515, New York, 1953. COULSON, C. A,, "Valence," Oxford, London, 1952. HOFBT~AN, BANESH,"The Strange Story of The Quantum," Harper & Brothers, New York, 1947. HEISENBEER, W., " P h r ~ i and ~ s Philosophy," Harper & Brothera, New York, 1958. J. W., "Wave Mechanics and Valency," John Wiley & LINNETT, Sons, Inc., New York, 1960. C. W., "Introduction to Quantum Mechmics," Henry SHERWIN, Holt & Co., New York, 1959.
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