MODERN PHYSICS-A SURVEY.* PART III** SAULDUSHMAN, t GENERAL ELECTRIC COMPANY, SCHENECTADY, NEWYORK Schr8dinger's Wave Equation
All this leads to the logical conclusion that the laws of ordinary mechanics cannot be applicable in the usual manner to problems in atomic mechanics. We have to devise some modification in the Newtonian method which will take into account the undulatory aspects of corpuscles when these are of atomic or electronic dimensions. Furthermore, it is an obvious deduction from the principle of indeterminism that all attempts to obtain from the solution of problems in atomic mechanics a definite answer regarding the occurrence of any individual event are foredoomed to failure. In view of the uncertainty in the definition of initial conditions we cannot expect a precise definition of subsequent occurrences. There must exist a measure of uncertainty in the prediction of events and hence all that we can determine from the solution of a problem i n atom mechanics i s the probability of any individual event. The Schrodinger wave mechanics performs exactly these two functions. Firstly, i t represents an undulatory conception grafted on the ordinary mechanical formulation of any problem in atom mechanics and secondly, once the solution is obtained (which is accomplished by well-known methods in the theory of differentialequations), the answer i s interpreted in terms of probabilities. The first stage in the solution of a problem by Schrodinger's method may therefore be described as that in which we formulate the problem in the same manner as in ordinary or classleal mechani6. That is, we assume an actual model of, say, the orbit of an electron in a hydrogen atom, or the motion of a linear harmonic oscillator. In classical mechanics the most general method for expressing the motion of a system is that which states the total energy (E) of the system under consideration as a function of the coiirdinates and the corresponding momenta. That is, the energy is expressed in terms of the kinetic and potential energies as functions of the instantaneous coordinates. Formulated in this manner the expression for E is said to be of the Hamiltonian form. Ordinarily we would proceed to solve this problem by certain standard methods and the result would enable us to describe the motions of the different parts of the system in terms of certain orbits or paths. Thus in the well-known Newtonian problem of the planetary motions, the starting *An address given before the Physical Science Section of the Tenth Ohio State Educational Conference. held in Columbus. April 3-5, 1930: and printed in the Proceedines. and also in the Gen. Elm. Rev.. . 33. 32S-35 (Tune. - . 1930):. ibid... 33. 394-400 ouly.1930). ** Part I of this series of articles appeared in J. C ~ MEDUC., . 7, 1778-87 (Aug., 1930); Part 11, ild., 7, 2077-87 (Sept., 1930). t Assistant Director, Research Laboratory, General Electric Company. 2655
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point is the postulate that the planet and sun exert an attractive force on each other which varies inversely as the square of the distance. This gives us the potential energy of the system as a function of the distance. Furthermore, we can express the kinetic energy of the planet in terms of its mass and instantaneous velocity. We can therefore write down an expression for the total energy. The problem now consists in the determination of the exact path followed by the earth in its motion under the attractive force. The "answer" deduced is that the orbit must be an ellipse with a major axis and frequency of revolution defined by the total energy of the system. However, in Schrodinger's method the use of classical coucefitions ceases after we have once formulated the total energy i n a Hamiltonian form. We forget all about the model of a corpuscle moving with definite velocity in a well-defined orbit. Instead of proceeding as in classical dynamics, we now bring in the wave concept and use the Hamiltonian expression to calculate the "wave-length of the associated undulatory motion. Thus if E represents the total energy, and V the potential energy, then E - V denotes the kinetic energy a t any instant and from the relation ( l / 2 ) m a 2= E - V
it follows that the momentum nzv = d 2 m ( ~
V)
and since the wave-length of the de Broglie wave is given by A = h/(mo) we obtain for X in the dynamical problem the result h =
h
-
d2rn(~
We now replace the motion of the particle by that of a wave-front traveling in a medium in which the velocity varies continuously from point to point. For the velocity depends upon the potential energy, V , and the latter is a function of the coordinates only. Consequently the wave-front of the associated wave-motion is continually suffering "refraction" in the field of force and the direction of the normal to the wave-front will be the same a t every point as the tangent a t that point to the orbit of the particle in the classical case. Thus the problem becomes one of determining both the direction and amplitude a t any point and instant of time of an undulatory motion for which the wave-length as a function of the coordinates is given by the above relation. In other words, the problem has now become one of undulatory mechanics, and can be formulated in terms of a differential equation which expresses the second derivative of the amplitude with respect to time in terms of the instantaneous phase velocity and of the second derivatives of the amplitude with respect to the coijrdinates; the partial diierential
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equation of the second order which is thus obtained is Schrodinger's famous equation. Solutions of this equation are then sought which will give the amplitude as a finite and continuous function of the coordinates a t any instant of time. The amplitude as a function of both coordinates and time is designated by J.. The f o m of the equationis, in general, such that i t is possible to express this function as a product of a function 4, of the coordinates only, and a cosine or sine series involving the time. Since we are also usually more interested in the equation for the "orbit," we need consider only the f o m of the function, 4. Therefore we can, in general, summarize the whole of the above process by stating that from the Hamiltonian expression for E it is possible by making use of de Broglie's equation for the wave equation to derive a certain differential equation (the Schrodinger equation) which gives the relation between the second order partial differential coefficients of a function 4 with respect to the coordinates and the quantities E and V. Whatever the physical significance of 4 (or J.) may be (this we shall discuss in a subsequent section), it is obvious that only those solutions of the differential equation can have any meaning whatever which are finite and continuous for all possible values of the coordinates. Now it is the peculiarity of Schrodinger's form of wave equation that the solutions fulfil these requirements only for certain discrete values of E, which are known as eigenwerte or characteristic values, and it is found that for the different problems in atomic mechanics these eigenwerte are identical with the values of the energy levels as calculated by the classical @ohr theory (whenever this is possible) or as determined from spectral observations. The corresponding solutions for 4 are known as eigenfunctions or characteristic functions. The whole problem of solving Schrodinger's equation is, as a matter of fact, analogous with the more familiar problems in acoustics, of detemining the positions of the nodes and the amplitudes of the loops in a vibrating string or organ pipe. The functions 4 thus resemble standing waves with
the number of nodes equal to the classical quantum number of the Bohr orbit. These functions have appreciable values only in those regions which would be occupied according to the older theory by the electronic orbits, and for distances greater than the radii of these orbits the values of 4 decrease very rapidly to zero. In other words, the amplitude of the wave motion has values greater than zero only in the neighborhood of the classical orbit. On the whole, the calculation with Schrodinger's eigenfunctions is more convenient than that involving matrices. While it was a t first thought that the Schrodinger method and that of the matrix calculus are radically different, it was shown by Eckart, Dirac, and Jordan that both methods are essentially identical from a mathematical point of view and are, in fact, both rednable to a much more general kind of mathematics, the calculus of operators.
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Significance of Schriidinger's Eigenfunction So far we have spoken of an amplitude of a wave-motion, designated by (or #) and have said nothing about a "medium." It will be remembered that when the electromagneticwave theory was first postulated it appeared necessary also to postulate a medium, the ether, in which electromagnetic waves are transmitted. As is well known, all attempts to demonstrate the existence of such a medium, as, for instance, the famous Michelson-Morley experiment, have yielded negative results, and the reason for this was provided by the theory of relativity. Furthermore, regarding the very nature of electro-magneticenergy, we have found it necessary to postulate a dualistic conception. On the one hand, in interference and diffraction we think of a wave-motion and measure wave-length (A) and frequency ( u ) . On the other hand, in phenomena involving interaction between energy and atoms or electrons (as in the photo-electric and Comptou effects) energy behaves as a corpuscular entity involving those attributes which are characteristic of corpuscle, that is, energy of magnitude, hv, and momentum of magnitude, hu/c = h/X. This association of wave and particle concepts is also seen to hold for electrons in Davisson and Germer's experiments, for when the mechanical momentum is varied, the wave-number (reciprocal of A) or wave-length effective for diffraction of the electron beam varies in just the manner given by the equation p = h/A. Now in the case of radiation, Eitistein had already suggested that the electromagnetic field in a given region determines the relative probability that the quanta go to different places: "In a set of interference fringes, the wave amplitude is strong at some places, and weak at others. The quanta go to the different places with a relative probc3ility that is given by the wave measure of intensity, viz., the s p a r e of the wave amfilit~de."~ Following this suggestion, M. Born has put forward a similar interpretation of +2 (or #2). The square of Schrodinger'sfunction for the "amplitude" of a wave-motion denotes the relative probability of occurrence of a certain event at a given point in space and at a given instant of time. Thus in the case of the hydrogen atom, we obtain a solution which gives the eigenfunctions for the different states of the atom, corresponding to the different possible orbits of the electron in the Bohr theory. These functions express the amplitude, +, as a function of the distance (7) from the nucleus and two angles which denote the longitude and colatitude, respectively. Schrijdinger originally interpreted $2 as proportional to the relative charge density of negative electricity at a given point. Since, as mentioned already, + has finite values from r = 0 to a value of r whichis slightly greater E. U. CONDON, Science, 68,193(1928). The writer is responsible for the italicized
+
portions.
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than the radius of the corresponding Bohr orbit, this interpretation leads to a representation of the electron in the hydrogen atom as "smeared out" in a spherical space having approximately the same dimensions as the atom. On the basis, however, of Born's conception, the value of @ corresponding to any given point in space merely denotes the probability of finding the electron a t that point. Thus the conception of a definite orbit is re&ced by that of a certain region i n the s@ce surrounding the nucleus for which the probability of occurrence of the electran i s greater than that for any other region. This interpretation of 4 has the advantage that i t eliminates completely all speculation regarding the physical reality of a wave motion associated with a corpuscle in motion. For 4 may be a function of more than three coordmates, as, for instance, in the case of the helium atom where we are dealing with the motions of two electrons and, therefore, require six coordinates. Under these conditions 4%denotes the probability of the simultaneous occurrence of the two electrons in two specified positions, and since the probability of such an event is equal to the product of the individual probabilities there follows another important advantage of Schrodinger's method, uiz.,it permits us, theoretically, a t least ( i . e., apart from difficultiesin the mathematical technic), to solve the problem of an atom containing two, or even more, electrons. In this manner the difficulty of the three-body problem in classical dynamics does not occur in the new mechanics. For in the latter we consider @ as determined by the total field of force which is due to the interaction of all the electrons and the nucleus, whereas in the ordinary mechanics we must take into account phase relations among the electrons. As has been mentioned already, this interpretation of 4%as a probability followslogically from Heisenberg's principle of indeterminism, and, indeed, supplements it by making the whole idea more precise mathematically. We thus perceive that i t is the interplay of wave and corpuscular concepts which leads to the view of the new quantum theory as essentially a statistical theory. The problem of what occurs in any individual instance is left unanswered. All that the Schrodinger equation can yield is information about the probability for the occurrence of a certain event. In statistical mechanics this point of view has been applied since the time of Boltzmann in order to solve many problems such as those in the kinetic theory of gases, in the scattering of alpha particles by thin foils, and similar phenomena. In this mode of treatment microcoordinates were introduced and then eliminated by taking statistical averages of the results. In wave mechanics, however, we avoid this procedure, since we derive the statistical probabilities directly from the solution of the Schrodinger equation. As Born10 describes the logic involved,
" Natunuissemchaftcn, 15,238 (1927).
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We regard the particles as acted upon by Coulomb forces, but the forces are not as in classical physics proportional to the acceleration. In fact they have no direct
relation with the motion of the particles. Rather, the wave-field comes in between. The forces determine the variation of a certain magnitude, $. which depends upon the positions of all the particles simullaneously (it is therefore a function in the "configuration space") and i n this manner that .$ must saligfy a differential equation of which the coeficients depend upon the forces. The knowledge of the function 6. . oermits us to calculate the course of a . ohvsical . phenomenon, as far as it is determined by quantum mechanical laws, that is, not i n thc sense of causal determinism but i n the sense of probability.
In classical dynamics the problem is always stated thus: Given the initial conditions in all their details, also the forces acting on the particles and the total energy, what is the condition at any given subsequent instant of time? On the other hand. we find that in the case of oroblems in atomic physics it is impossible to formulate the problem in this manner. Because of the indeterminacy of the initial conditions all we can state is the probability of a given condition a t any subsequent instant of time. There exists a certain probability that the system shall be in a given state and also another assignable probability that it shall be in another given state. I t is in this sense that we must abandon the causality principle as ap plied to atomic physics. This principle can no longer be regarded as valid for individual occurrences, but rather for the probability function, 4%. All we can predict is the manner in which this probability must vary with time. I t is, therefore, in this sense only that we can still regard the causality principle as applicable to quantum phenomepa. The interpretation of 4%as a probability and the significance of this from the point of view of the classical principle of causality are still more strikingly emphasized by the method used in the new mechanics for calculating the intensity of spectral lines. On the basis of the Correspondence Principle, Bohr derived the conclusion that transitions between certain types of energy levels cannot occur, and that, therefore, the intensities of the line corresponding to these transitions must be zero. But according to both the matrix theory and Schrodinger's wave mechanics the intensity of a spectral line is given by the probability of the coexistence of the functions & and &for the initial and final states of the atomic system. That is, the intensity of any line depends upon the magnitude of the product of two functions, one for the initial and the other for the final state. In this deduction it is readily evident how far we have diverged from any classical notions of causality. For the most treasured feature of this principle is that the end state cannot have any effect on the phenomena which lead up to it. How, then, are we to interpret the wave-mechanics calculation of intensities of spectral lines? Perhaps the best way is to accept the result because it leads to the correct answer, and forget the causality principle, at least in atomic problems.
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This means that our criterion for applying the principle of causality fails in the case of atomic systems. Can we ever he sure that two atoms are in the same initial state? The answer is necessarily in the negative, for in attempting to observe the state of the atom we must bombard it with atoms, photons, or electrons, and in each case the act of observing changes the state in an indeterminate manner. Hence, according to Bohr and Heisenberg, we must conclude that in the atom space and time are in some manner complementary, and we therefore can no longer apply to such cases the conceptions which we have found so useful for ordinary or large-scale phenomena. Some Applications of Quantum Mechanics It would take us much beyond the scope of the present.paper to discuss the applications of the new mechanics to various prohlems of atomic and molecular structure, to line and band spectra, and still more recently to prohlems of chemical valency. By means of the new quantum theory, it has been found possible not only to derive all the results obtained by the older Bohr theory, but also to account for observations which could not he made to fit in with the latter. Furthermore, the theory represents, in many ways, a more logical approach toward the ultimate goal of any branch of science--a quantitative correlation of observable magnitudes. I t is, however, of special interest to mention briefly one consequence of the wave mechanics conception which is of greatest importance and yet represents a totally unexpected depart& from the usual point of view. In ordinary mechanics, we accept as a necessary consequence the statement that it is impossible for a particle ever to occur a t a point in space for which the potential energy (V) would exceed the total energy (E). For instance, in the case of a vibrating pendulum, the maximum amplitude of oscillation is determined by the condition that at this point the potential energy is just equal to the total energy. But in quantum mechanics, the path of the particle is no longer confined to just that region in which V is less than or equal to E. The function 4, which defines the amplitude, and c$~, which determines the probability of occurrence of the particle at any point are found to havefinite values for regions which extend beyond those in which the motion of the particle would be conlined according to classical mechanics. In other words, there exists a certain measurable probability (which is usually very small) that the particle will be found in those regions which, according to ordinary mechanics, are forbidden. The reason for this surprising result is evidently to be found in the Uncertainty Principle, for in atomic mechanics there is no longer any precise correlation between simultaneous value of position and momentum such as is demanded by Newton's laws. Now let us consider the case of two regions of low potential energy (designated as regions I and 11) separated by a central one of relatively high
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potential energy (region 111). According to ordinary mechanics, if the particle is moving in either region I or 11, then it must certainly remain there for all time, unless it is supplied with additional energy from an external source sufficient to bring it over the "hill" which comprises region 111. But on the basis of the fact that @ varies continuously from one region of low potential energy through region 111,into the other region of low potential energy, it follows that we can no longer regard a transition from region I to 11, or vice versa, as impossible. Rather, for the certainty of classical mechanics regarding the impossibility of such an occurrence we must substitute, according t o the new point of view, the existence of a finite probability that in the course of time such a transition may O C G U ~ . ' ~ It is by applying this consequence of the new theory that it has been found possible, for the first time, to obtain an interpretation of such a periodic phenomena as the disintegration of radioactive elements, electron emission from cold metals under the action of strong fields, and the photoelectric effect. Concluding Remarks To a large number it has seemed that the new point of view necessarily leads to an abandonment of Bohr's model of the atom as constituted of a nucleus with one or more electrons revolving about this nucleus in elliptic orbits. That, however, this opinion is not quite justified must be evident when we consider that in the very development of Schrodinger's equation in any problem the first step involves the formulation of the energy in terms of a model. It is only the mathematical method of solving the problem which has been modified, but the electronic orbits must still remain. Perhaps they emerge in not quite as concrete a form as the older theory represented them. The simple elliptic orbit is replaced by an annular shape, or some other more complicated spacial figure at all points of which we may regard the electron as located at any instant with a certain measure of probability. The picture has, as it seems, become more "fuzzy" on the edges; this must be so because of the essential indeterminism in our knowledge of phenomena within the atom itself. But for the interpretation of many of the observations in atomic physics the older view need not necessarily be any less "true" than it ever was. For, after all, a scientific theory is a framework wherein we may fit together the broken parts of a picture puzzle, and, oftentimes, if we are content to select a small number of parts, the framework may be made to appear a little less complicated. Yet, granted all this, the fact remains that we must recognize the same dualism in regard to both matter and radiation. Whether we shall ultiThis conception has been very thoroughly discussed by R. W. GURNEY and E. V. CONDON, Phys. %.,33, 127 (1929).
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mately find a monism behind this dualism, only the future can reveal. If this event does occur, it will be becanse of additional experimental observations which cannot be adapted to conceptions held valid at present. Philosophically, the doctrine of indeterminism and its consequent dednction that atomic phenomena are governed by laws of probability, will appear to many as an extremely unsatisfactory denouement for a branch of saence which hitherto has always prided itself on the preciseness of its deductions. P. W. Bridgman, in his extremely interesting book on "The Logic of Modem Physics," has expressed this feeling in the following remarks: In general, we cannot admit for a minute that a statistical method, unless used to smooth out irrelevant details, can ever mark more than a temporary stage in our progress, because the assumption of events taking place according to pure chance constitutes the complete negation of our fundamental assumption of connectivity;such statistical methods always indicate the presence of physical complications which it must be our aim to disentangle eventually.
And from a pure philosopher comes the complaint that physics has now become metaphysic~!'~Perhaps the philosopher may be excused for not understanding Schrodinger's equation. What does it all mean? Are there structures beyond the electron and the quantum, as Bridgman would suggest? Must we conceive nature to be ultimately simple, just becanse we have an esthetic predilection for such a conception? On the contrary, the hisiory of science teaches us that as we search deeper into the mysteries of nature: the more complex does it appear, and hence there is no reason, except that of mental inertia, for expecting nature to remain always s i m p l e b y which we mean that we can make mechanical models which will show us how the "wheels go round." It would, of course, be extremely convenient if nature could always have arranged things to suit our standardized modes of thinking at any period. But the history of human progress has been one in which vision and iaterpretation have grown with actual knowledge, and therefore we must not be shocked too severely if the present ideas of quantum mechanics appear on the one hand as a negation of our fondest expectations of the past, and on the other, a mass of abstract and almost transcendental mathematical notions. Time will undoubtedly breed familiarity and, perhaps ultimately, disdain for the "simple" ideas of the present. But that the future will reveal to us still more complex phenomena and that these in turn will necessitate still more radical ideas-these things none dare deny.
(Concluded) W. DUMT, "The Mansions of Philosophy," p. 64. It is worthwhile for any scientifically trained person to read the remarks of Dr. Durant, for it is always of interest to see ourselves as others see us, even if we believe them to be mistaken.
