P 8 YS ICAL REVI EVF VOLUM E SS, NUMBER 2 JANUARY 15, 1952

A Suggested Interpretation of the Quantum Theory in Tel@as of "Hidden" Variables. IDAvID BQHM*

Palmer Physical Laboratory, Princeton University, Princeton, Sem Jersey(Received July 5, 1951)

The usual interpretation of the quantum theory is self-con-sistent, but it involves an assumption that cannot be testedexperimentally, ms. , that the most complete possible specificationof an individual system is in terms of a wave function that deter-mines only probable results of actual measurement processes.The only way of investigating the truth of this assumption is bytrying to find some other interpretation of the quantum theory interms of at present "hidden" variables, which in principle deter-mine the precise behavior of an individual system, but which arein practice averaged over in measurements of the types that cannow be carried out. In this paper and in a subsequent paper, aninterpretation of the quantum theory in terms of just such"hidden" variables is suggested. It is shown that as long as themathematical theory retains its present general form, this sug-gested interpretation leads to precisely the same results for all

physical processes as does the usual interpretation. Nevertheless,the suggested interpretation provides a broader conceptual frame-work than the usual interpretation, because it makes possible aprecise and continuous description of all processes, even at thequantum level. This broader conceptual framework allows moregeneral mathematical formulations of the theory than thoseallowed by the usual interpretation. Now, the usual mathematicalformulation seems to lead to insoluble difhculties when it is ex-trapolated into the domain of distances of the order of 10 "cmor less. It is therefore entirely possible that the interpretation sug-gested here may be needed for the resolution of these difficulties.In any case, the mere possibility of such an interpretation provesthat it is not necessary for us to give up a precise, rational, andobjective description of individual systems at a quantum level ofaccuracy.

1. INTRODUCTION

HE usual interpretation of the quantum theory isbased on an assumption having very far-reaching

implications, ~is. , that the physical state of an in-dividual system is completely specified by a wavefunction that determines only the probabilities of actualresults that can be obtained in a statistical ensemble ofsimilar experiments. This assumption has been theobject of severe criticisms, notably on the part ofEinstein, who has always believed that, even at thequantum level, there must exist precisely definableelements or dynamical variables determining (as inclassical physics) the actual behavior of each individual

system, and not merely its probable behavior. Sincethese elements or variables are not now included in thequantum theory and have not yet been detected experi-mentally, Einstein has always regarded the presentform of the quantum theory as incomplete, although headmits its internal consistency. ' ~

Most physicists have felt that objections such asthose raised by Einstein are not relevant, first, becausethe present form of the quantum theory with its usualprobability interpretation is in excellent agreementwith an extremely wide range of experiments, at leastin the domain of distances' larger than 10 " cm, and,secondly, because no consistent alternative interpreta-

* Now at Universidade de Sao Paulo, Faculdade d.e Filosofia,Ciencias, e Letras, Sao Paulo, Brasil.' Einstein, Podolsky, and Rosen, Phys. Rev. 47, 777 (1933).

2D. Bohm, Quantgm Theory (Prentice-Hall, Inc. , New York,1951), see p. 611.

3 N. Bohr, Phys. Rev. 48, 696 (1935).W. Furry, Phys. Rev, 49, 393, 476 (1936).

5 Paul Arthur Schilp, editor, Albert Einstein, Philosopher-Scientist (Library of Living Philosophers, Evanston, Illinois,1949). This book contains a thorough summary of the entirecontroversy.

6 At distances of the order of 10 "cm or smaller and for timesof the order of this distance divided by the velocity of light orsmaller, present theories become so inadequate that it is generallybelieved that they are probably not applicable, except perhaps

tions have as yet been suggested. The purpose of thispaper (and of a subsequent paper hereafter denoted byII) is, however, to suggest just such an alternativeinterpretation. In contrast to the usual interpretation,this alternative interpretation permits us to conceiveof each individual system as being in a precisely de-6nable state, whose changes with time are determinedby definite laws, analogous to (but not identical with)the classical equations of motion. Quantum-mechanicalprobabilities are regarded (like their counterparts inclassical statistical mechanics) as only a practicalnecessity and not as a manifestation of an inherentlack of complete determination in the properties ofmatter at the quantum level. As long as the presentgeneral form of Schroedinger's equation is retained, thephysical results obtained with our suggested alternativeinterpretation are precisely the same as those obtainedwith the usual interpretation. Ke shall see, however,that our alternative interpretation permits modi6ca-tions of the mathematical formulation which could noteven be described in terms of the usual interpretation.Moreover, the modifications can quite easily be for-mulated in such a way that their eGects are insigni6cantin the atomic domain, where the present quantumtheory is in such good agreement with experiment, butof crucial importance in the domain of dimensions ofthe order of 10 " cm, where, as we have seen, thepresent theory is totally inadequate. It is thus entirelypossible that some of the modifications describable interms of our suggested alternative interpretation, but

in a very crude sense. Thus, it is generally expected that in con-nection with phenomena associated with this so-called "funda-mental length, " a totally new theory will probably be needed.It is hoped that this theory could not only deal precisely with suchprocesses as meson production and scattering of elementary par-ticles, but that it would also systematically predict the masses,charges, spins, etc., of the large number of so-called "elementary"particles that have already been found, as well as those of newparticles which might be found in the future,

66

QUANTUM THEORY IN TERMS OF ''H I D DEN'' VARIABLES. I

not in terms of the usual interpretation, may be neededfor a more thorough understanding of phenomenaassociated with very small distances. We shall not,however, actually develop such modifications in anydetail in tlMsc papers.

After this article was completed, the author's atten-tion was called to similar proposals for an alternativeinterpretation of the quantum theory made by dcBroglie' in 1926, but later given up by him partly asa result of certain criticisms made by Paulis and partlybecause of additional objections raised by de Broglie7himself. f As we shall show in Appendix B of Paper II,however, all of the objections of Re Broglie and Paulicould have been met if only de Broglie had carried hisideas to their logical conclusion. The essential new stepin doing this is to apply our interpretation in the theoryof the measurement process itself as well as in thedescription of the observed system. Such a developmentof the theory of measurements is given in Paper II,'where it mill be shown in detail that our interpretationleads to precisely the same results for all experimentsas are obtained with the usual interpretation. Thefoundation for doing this is laid in Paper I, where wedevelop the basis of our interpretation, contrast itwith the usual interpretation, and apply it to a fewsimple examples, in order to illustrate the principlesinvolved.

2. THE USUAL PHYSICAL INTERPRETATIONOF THE QUANTUM THEORY

The usual physical interpretation of the quantumtheory centers around the uncertainty principle. Now,the uncertainty principle can be derived in tmo diferentways. First, we may start with the assumption alreadycriticized by Einstein, namely, that a mave functionthat determines only probabilities of actual experi-mental results nevertheless provides the most completepossible specification of the so-called "quantum state"of an individual system. With the aid of this assump-tion and with the aid of the de Broglie relation, y=kk,where k is the wave number associated with a par-tjcular fourier component of the wave function, the

7 L. de Broglie, An Introduction to the Study of W'ave M'echanics{E.P. Button and Company, Inc. , New York, 1930),see Chapters6, 9, and 10. See also Compt. rend. 183, 447 (1926); 184, 273~1927); 185, 380 (1927).

Reports on the Solvay Congress {Gauthiers-Villars et Cie.,Paris, 1928), see p. 280.

f Note added in proof.—Madelung has also proposed a similarinterpretation of the quantum theory, but like de Broglie he didnot carry this interpretation to a logical conclusion. See E. Made-lung, Z. f. Physik 40, 332 (1926), also G. Temple, Introduction toQuantum Theory (London, 1931).

~ In Paper II, Sec. 9, we also discuss von Neumann's proofLace J. von Neumann, 3Eathematisehe Gramdlagem der Qaamtem~echanik (Verlag, Julius Springer, Berlin, 1932)j that quantumtheory cannot be understood in terms of a statistical distributionof "hidden" causal parameters. %e shall show that his conclusionsdo not apply to our interpretation, because he implicitly assumesthat the hidden parameters must be associated only with theobserved system, whereas, as will become evident in these papers,our interpretation requires that the hidden parameters shall alsobe associated with the measuring apparatus.

uncertainty principle is readily deduced. '0 From thisderivation, we are led to interpret the uncertaintyprinciple as an inherent and irreducible limitation onthe precision with which it is correct for us even toconceive of momentum and position as simultaneouslydefined quantities. For if, as is done in the usual inter-pretation of the quantum theory, the wave intensityis assumed to determine only the probability of a givenposition, and if the kth Fourier component of the wavefunction is assumed to determine only the probabilityof a corresponding momentum, y= Ak, then it becomesa contradiction in terms to ask for a state in whichmomentum and position are simultaneously and pre-cisely defined.

A second possible dcl ivatlon of tlM unccl tain typrinciple is based on a theoretical analysis of theprocesses with the aid of which physically significantquantities such as momentum and position can bemeasured. In such an analysis, one finds that becausethe measuring apparatus interacts with the observedsystem by means of indivisible quanta, there will alwaysbe an irreducible disturbance of some observed prop-erty of the system. If the precise CGects of this dis-turbance could be predicted or controlled, then onecouM correct for these effects, and thus one couM stillin principle obtain simultaneous measurements ofmomentum and position, having unlimited precision.But if one could do this, then the uncertainty principlewould be violated. The uncertainty principle is, as we

have seen, however, a necessary consequence of theassumption that the wave function and its probabilityinterpretation provide the most complete possiblespecification of the state of an individual system. Inorder to avoid the possibility of a contradiction withthis assumption, Bohr"'0" and others have suggestedan additional assumption, namely, that the process oftransfer of a single quantum from observed system tomeasuring apparatus is inherently unpredictable, un-

controllable, and not subject to a detailed rationalanalysis or description. With the aid of this assumption,one can show'0 that the same uncertainty principlethat is deduced from the wave function and its proba-bility interpretation is also obtained as an inherent andunavoidable limitation on the precision of all possiblemeasurements. Thus, one is able to obtain a set ofassumptions, which permit a self-consistent formula-tion of the usual interpretation of the quantum theory.

The above point of view has been given its mostconsistent and systematic expression by Bohr, ' ' " interms of the "principle of complementarity. " In for-mulating this principle, Bohr suggests that at theatomic level we must renounce our hitherto successfulpractice of conceiving of an individual system as aunified and. precisely dehnable whole, all of whose as-pects are, in a manner of speaking, simultaneously and

'0 See reference 2, Chapter 5."N. Bohr, Atomic Theory and the DescriPtion of Eature {Cam-

bridge University Press, London, 1934}.

unambiguously accessible to our conceptual gaze. Sucha system of concepts which is sometimes called a"model, " need not be restricted to pictures, but mayalso include, for example, mathematical concepts, aslong as these are supposed to be in a precise (i.e.,one-to-one) correspondence with the objects that arebeing described. The principle of complementarityrequires us, however, to renounce even mathematica, lmodels. Thus, in Bohr's point of view, the wave func-tion is in no sense a conceptual model of an individualsystem, since it is not in a precise (one-to-one) corre-spondence with the behavior of this system, but onlyin a statistical correspondence.

In place of a precisely dered conceptual model, theprinriple of complementarity states that we are re-stricted to complementarity pairs of inherently im-prerisely defined concepts, such as position and mo-mentum, pa, rticle and wave, etc. The maximum degreeof precision of definition of either member of such apair is reriprocally related to that of the oppositemember. This need for an inherent lack of completeprecision can be understood in two ways. First, it canbe. regarded as a consequence of the fact that the ex-perimental apparatus needed for a precise measure-ment of one member of a complementary pair of vari-ables must always be such as to preclude the possibilityof a simultaneous and precise measurement of the othermember. Secondly, the assumption that an individualsystem is completely specified by the wave function andits probability interpretation implies a correspondingunavoidable lack of precision in the very conceptualstructure, with the a,id of which we can think aboutand describe the behavior of the system.

It is only at the classical level that we can correctlyneglect the inherent lack of precision in all of our con-ceptual models; for here, the incomplete determinationof physical properties implied by the uncertainty prin-ciple produces effects that are too small to be of prac-tical significance. Our ability to describe classicalsystems in terms of precisely definable models is, how-ever, an integral part of the usual interpretation of thetheory. For without such models, we would have noway to describe, or even to think of, the result of anobservation, which is of course always 6nally carriedout at a classical level of accuracy. If the relationshipsof a given set of classically describa, ble phenomenadepend signihcantly on the essentially quantum-me-chanical properties of matter, however, then the prin-ciple of complementarity states that no single model ispossible which could provide a precise and rationalanalysis of the connections between these phenomena.In such a case, we are not supposed, for example, toattempt to describe in detail how future phenomenaarise out of past phenomena. Instead, we shouM simplyaccept without further analysis the fact that futurephenomena do in fact somehow manage to be produced,in a way that is, however, necessarily beyond the possi-bility of a detailed description. The only aim of a

mathematical theory is then to predict the statisticalrelations, if any, connecting these phenomena.

3. CRITICISM OF THE USUAL INTERPRETATION OFTHE QUANTUM THEORY

The usual interpretation of the quantum theory canbe criticized on many grounds. ' In this paper, however,we shall stress only the fact that it requires us to giveup the possibility of even conceiving precisely whatmight determine the behavior of an individual systemat the quantum level, without providing adequateproof that such a renunciation is necessary. The usualinterpretation is admittedly consistent; but the meredemonstration of such consistency does not exclude thepossibility of other equally consistent interpretations,which would involve additional elements or parameterspermitting a detailed causal and continuous descriptionof all processes, and not requiring us to forego thepossibility of conceiving the quantum level in preciseterms. From the point of view of the usual interpreta-tion, these additional elements or parameters could becalled "hidden" variables. As a matter of fact, when-ever we have previously had recourse to statisticaltheories, we have always ultimately found that thelaws governing the individual members of a statisticalensemble could be expressed in terms of just suchhidden variables. For example, from the point of viewof macroscopic physics, the coordinates and momentaof individual atoms are hidden variables, which in alarge scale system manifest themselves only as sta-tistical averages. Perhaps then, our present quantum-mechanical averages are similarly a manifestation ofhidden variables, which have not, however, yet beendetected directly.

Now it may be asked why these hidden variablesshould have so long remained undetected. To answerthis question, it is helpful to consider as an analogy theearly forms of the atomic theory, in which the existenceof atoms was postulated in order to explain certainlarge-scale effects, such as the laws of chemical com-bination, the gas laws, etc. On the other hand, thesesame effects could also be described directly in termsof existing macrophysical concepts (such as pressure,volume, temperature, mass, etc.); and a correct de-scription in these terms did not require any reference toatoms. Ultimately, however, effects were found whichcontradicted the predictions obtained by extrapolatingcertain purely macrophysical theories to the domain ofthe very small, and which could be understood cor-rectly in terms of the assumption that matter is com-posed of atoms. Similarly, we suggest that if there arehidden variables underlying the present quantumtheory, it is quite likely that in the atomic domain, theywill lead to effects that can also be described adequatelyin the terms of the usual quantum-mechanical concepts;while in a domain assoriated with much smaller dimen-sions, such as the level associated with the "fundamentallength" of the order of 10 " cm, the hidden variables

QUANTUM THEORY IN TERMS OF ''HIDDEN'' VARIABLES. I 169

may lead to completely new sects not consistent withthe extrapolation of the present quantum theory downto this level.

If, as is certainly entirely possible, these hidden vari-ables are actually needed for a correct description atsmall distances, we could easily be kept on the wrongtrack for a long time by restricting ourselves to theusual interpretation of the quantum theory, which ex-cludes such hidden variables as a matter of principle.It is therefore very important for us to investigate ourreasons for supposing that the usual physical inter-pretation is likely to be the correct one. To this end, weshall begin by repeating the two mutually consistentassumptions on which the usual interpretation is based(see Sec. 2):

(1) The wave function with its probability inter-pretation determines the most complete possible speci-fication of the state of an individual system.

(2) The process of transfer of a single quantum fromobserved system to measuring apparatus is inherentlyunpredictable, uncontrollable, and unanalyzable.

Let us now inquire into the question of whether thereare any experiments that could conceivably provide atest for these assumptions. It is often stated in con-nection with this problem that the mathematical ap-paratus of the quantum theory and its physical in-terpretation form a consistent whole and that thiscombined system of mathematical apparatus andphysical interpretation is tested adequately by theextremely wide range of experiments that are in agree-ment with predictions obtained by using this system.If assumptions (1) and (2) implied a unique mathe-matical formulation, then such a conclusion would bevalid, because experimental predictions could then befound which, if contradicted, would clearly indicatethat these assumptions were wrong. Although assump-tions (1) and (2) do limit the possible forms of themathematical theory, they do not limit these formssuKciently to make possible a unique set of predictionsthat could in principle permit such an experimentaltest. Thus, one can contemplate practically arbitrarychanges in the Hamiltonian operator, including, forexample, the postulation of an unlimited range of newkinds of meson fields each having almost any conceiv-able rest mass, charge, . spin, magnetic moment, etc.And if such postulates should prove to be inadequate,it is conceivable that we may have to introduce non-local operators, nonlinear fields, S-matrices, etc. Thismeans that when the theory is found to be inadequate(as now happens, for example, at distances of the orderof 10 "cm), it is always possible, and, in fact, usuallyquite natural, to assume that the theory can be madeto agree with experiment by some as yet unknownchange in the mathematical formulation alone, notrequiring any fundamental changes in the physical in-terpretation. This means that as long as we accept theusual physical interpretation of the quantum theory,we cannot be led by any conceivable experiment to

4. NEW PHYSICAL INTERPRETATION OFSCHROEDINGER'S EQUATION

We shall now give a general description of our sug-gested physical interpretation of the present mathe-matical formulation of the quantum theory. We shallcarry out a more detailed description in subsequentsections of this paper.

We begin with the one-particle Schroedinger equa-tion, and shall later generalize to an arbitrary numberof particles. This wave equation is

i7iBQ/Bt = —(t'i'/2m) VQ+ V(x)P.

Now P is a complex function, which can be expressed as

P =E e p(ixS/5),

where R and S are real. We readily verify that the equa-tions for E. and S are

BRfRV'S+2VR VS],

Bt 2m

(VS)'+V(x)—

2m R

h2 V2R-

2m

give up this interpretation, even if it should happen tobe wrong. The usual physical interpretation thereforepresents us with a considerable danger of falling intoa trap, consisting of a self-closing chain of circularhypotheses, which are in principle unverifiable if true.The only way of avoiding the possibility of such a trapis to study the consequences of postulates that con-tradict assumptions (1) and (2) at the outset. Thus,we could, for example, postulate that the precise out-come of each individual measurement process is inprinciple determined by some at present "hidden"elements or variables; and we could then try to findexperiments that depended in a unique and reproducibleway on the assumed state of these hidden elements orvariables. If such predictions are verified, we shouldthen obtain experimental evidence favoring the hy-pothesis that hidden variables exist. If they are notverified, however, 'the correctness of the usual in-terpretation of the quantum theory is not necessarilyproved, since it may be necessary instead to alter thespecific character of the theory that is supposed todescribe the behavior of the assumed hidden variables.

We conclude then that a choice of the present in-terpretation of the quantum theory involves a realphysical limitation on the kinds of theories that we wishto take into consideration. From the arguments givenhere, however, it would seem that there are no secureexperimental or theoretical grounds on which we canbase such a choice because this choice follows fromhypotheses that cannot conceivably be subjected to anexperimental test and because we now have an al-ternative interpretation.

470 DA VI D BOB M

It is convenient to write P(x)=R'(x), or R=P'*where P(x) is the probability density. We then obtain

8P t' VSq—+V}P }=0,at E m)BS (VS)' 1't' ~PP 1 (VP)'—+ +V(x) — —— =0. (6)Bt 2m 4m P 2 P'

Now, in the classical limit (t~—'&0) the above equationsare subject to a very simple interpretation. The func-tion S(x) is a solution of the Hamilton-Jacobi equation.If we consider an ensemble of particle trajectories whichare solutions of the equations of motion, then it is awell-known theorem of mechanics that if all of thesetrajectories are normal to any given surface of constantS, then they are normal to all surfaces of constant S,and VS(x)/m will be equal to the velocity vector, v(x),for any particle passing the point x. Equation (5) cantherefore be re-expressed as

M'/Bt+V (Pv) =0. (7)

This equation indicates that it is consistent to regardP(x) as the probability density for particles in ourensemble. For in that case, we can regard Pv as themean current of particles in this ensemble, and Eq. (7)then simply expresses the conservation of probability.

I.et us now see to what extent this interpretation canbe given a meaning even when kAO. To do this, let usassume that each particle is acted on, not only by a"classical" potential, V(x) but also by a "quantum-mechanical" potential,

—O' V'P 1 (VP)' —O' V'RU(x) =

4m P 2 P' 2m E

Then Eq. (6) can still be regarded as the Hamilton-Jacobi equation for our ensemble of particles, VS(x)/mcan still be regarded as the particle velocity, and Eq. (5)can still be regarded as describing conservation ofprobability in our ensemble. Thus, it would seem thatwe have here the nucleus of an alternative interpreta-tion for Schroedinger's equation.

The first step in developing this interpretation in amore explicit way is to associate with each electron aparticle having precisely definable and continuouslyvarying values of position and momentum. The solu-tion of the modified Hamilton-Jacobi equation (4)defines an ensemble of possible trajectories for thisparticle, which can be obtained from the Hamilton-Jacobi function, S(x), by integrating the velocity,v(x) = VS(x)/m. The equation for S implies, however,that the particles moves under the action of a forcewhich is not entirely derivable from the classical po-tential, V(x), but which also obtains a contribution fromthe "quantum-mechanical" potential, U(x) = (—h'/2m)XV2R/R. The function, R(x), is not completely arbi-trary, but is partially determined in terms of S(x) by

the differential Eq. (3). Thus R and S can be said tocodetermine each other. The most convenient way ofobtaining R and S is, in fact, usually to solve Eq. (1)for the Schroedinger wave function, P, and then to usethe relations,

f= U+iW=RI cos(S/ti)+i sin(S/8)],R'= U'+ V' S=k tan '(W/U).

Since the force on a particle now depends on a func-tion of the absolute value, R(x), of the wave function,P(x), evaluated at the actual location of the particle,we have electively been led to regard the wave func-tion of an individual electron as a mathematical repre-sentation of an objectively real field. This field exertsa force on the particle in a way that is analogous to,but not identical with, the way in which an electro-magnetic field exerts a force on a charge, and a mesonfield exerts a force on a nucleon. In the last analysis,there is, of course, no reason why a particle should notbe acted on by a f-field, as well as by an electromagneticfield, a gravitational field, a set of meson fields, andperhaps by still other fields that have not yet beendiscovered.

The analogy with the electromagnetic (and other)field goes quite far. For just as the electromagneticfield obeys Maxwell's equations, the f-field obeysSchroedinger's equation. In both cases, a completespecification of the fields at a given instant over everypoint in space determines the values of the fields forall times. In both cases, once we know the field func-tions, we can calculate force on a particle, so that,if we also know the initial position and momentum ofthe particle, we can calculate its entire trajectory.

In this connection, it is worth while to recall that theuse of the Hamilton-Jacobi equation in solving for themotion of a particle is only a matter of convenienceand that, in principle, we can always solve directly byusing Newton's laws of motion and the correct boundaryconditions. The equation of motion of a particle actedon by the classical potential, V(x), and the "quantum-mechanical" potential, Eq. (8), is

md'x/dt'= —V I V(x) —(5'/2m) V'R/R I . (Sa)

It is in connection with the boundary conditionsappearing in the equations of motion that we find theonly fundamental difference between the f-field andother fields, such as the electromagnetic Geld. For inorder to obtain results that are equivalent to those ofthe usual interpretation of the quantum theory, we arerequired to restrict the value of the initial particlemomentum to y= VS(x). From the application ofHamilton-Jacobi theory to Eq. (6), it follows that thisrestriction is consistent, in the sense that if it holdsinitially, it will hold for all time. Our suggested newinterpretation of the quantum theory implies, however,that this restriction is not inherent in the conceptualstructure. We shall see in Sec. 9, for example, that it is

OF ''HIDDEN'' VARIABI ES. I

quite consistent in our interpretation to contemplatemodihcations in the theory, which permit an arbitraryrelation between y and V'S(x). The law of force on, theparticle can, however, be so chosen that in the atomicdomain, p turns out to be very nearly equal to VS(x)/m,while in processes involving very small distances, thesetwo quantities may be very diGerent. In this way, wecan improve the analogy between the f-Geld and theelectromagnetic field (as well as between quantummechanics and classical mechanics).

Another important difference between the P-Geldand the electromagnetic held is that, whereas Schroed-inger's equation is homogeneous in P, Maxwell's equa-tions are inhomogeneous in the electric and magnetic6elds. Since inhomogeneities are needed to give rise toradiation, this means that, our present equations imp1ythat the f-Geld is not radiated or absorbed, but simplychanges its form while its integrated intensity remainsconstRnt. This rcstllctlon to R holTlogcncous cquRtlonis, however, like the restriction to a homogeneous equa-tion is, however, like the restriction to y=&S(x), notinherent ln thc conccptuRl stI'uctuI'c of our ncw in-

terpretation. Thus, in Sec. 9, we shall show that onecan consistently postulate inhomogeneities in the equa-tion governing P, which produce important effects onlyat very small distances, and negligible effects in theatomic domain. If such inhomogeneities are actuallypl'cscIlt, then tllc 'lp-Geld will bc sub]cct 'to bciIlg cIIlit tedand absorbed, but only in connection with px'ocesses

associated with very small distances. Once the P-fieldhas been emitted, however, it will in all atomic processessimply obey Schroedinger's equation as a very goodapproximation. Nevertheless, at very small distances,the value of the f-Geld would, as in the case of the elec-tromagnetic Geld, depend to some extent on the actuallocation of the particle.

Let us now consider the meaning of the assumptionof a statistical ensemble of particles with a probabilitydensity equal to E(x)=R'(x) =

~p(x) ~'. From Eq. (5),

it follows that this assumption is consistent, providedthat P satisfies Schroedinger's equation, and v = V'S(x)/m. This probability density is numerically equal to theprobability density of particles obtained in the usualinterpretation. In the usual interpretation, however,the need for a probability description is regarded asinherent in the very structure of matter (see Sec. 2),whereas in our interpretation, it arises, a,s we shall secin Paper II, because from one measurement to thenext, we cannot in practice predict or control the pre-cise location of a particle, as a result of correspondingunpredictable and uncontrollable disturbances intro-duced by the measuring apparatus. Thus, ln oUl 1Q-

terpretation, the use of a statistical ensemble is (as inthe case of classical statistical mechanics) only a prac-tical necessity, and not a reAection of an inherentimitation on the precision with which it is correct forus to conceive of the variables defining the state of thesystem. Moreover, it is clear that if in connection with

very small distances we are ultimately required to giveup the special assumptions that f satisGes Schroed-inger's equation and that v= VS(x)/III, then ~it ~' willcea,se to satisfy a conservation equation and will there-fore also cease to be able to represent the probabilitydensity of particles. Nevertheless, there wouM still be atrue probability density of particles which is conserved.Thus, it would become possible in principle to And ex-periments in which ~f~' could be distinguished fromthe probability density, and therefore to prove that theusual interpretation, which gives ~p~s only a proba-bility interpretation must be inadequate. Moreover,we shall see in Paper II that with the aid of suchmodidcations in the theory, we couM in principlemeasure the particle positions Rnd momenta precisely,Rnd thus vlolRtc thc unccI'tMnty pI'lnclplc. As long Rs

we restrict ourselves to conditions in which Schroed-lilgcr s cquatloI1 is satlsflcd, alld Ill wlllcll v= VS(x)/tÃhowever, the uncertainty principle will remain anCAective practical limitation on the possible precisionof measurements. This means that at present, theparticle positions and momenta should be regarded as"hidden" variables, since as wc shall sec in Paper II,we are not now able to obtain experiments that localizethem to a region smaller than that in which the intensityof the f-Geld is appreciable. Thus, we cannot yet findclear-cut experimental proof that the assumption ofthese variables is necessary, although it is entirelyposslblc that ln thc domalQ of very small dlstaQccsnew modi6cations in the theory may have to be intro-duced, which would permit a proof of the existence ofthe definite particle position and momentum to beobtained.

We conclude that our suggested interpretation of thequantum theory provides a much broader conceptualframework than that provided by the usual interpreta-tion, for all of the results of the usual interpretation areobtained from our interpretation if we make the follow-ing three special assumptions which are mutuallyconsistent:

(1) That the P-6eld satis6es Schroedinger's equation.(2) That the particle momentum is restricted to y= V'S(x).(3) That ere do not predict or control the precise location of the

particle, but have, in practice, a statistical ensemble with proba-bility density P(x)= rP(x) )'. The nse of statistics is, however,not inherent in the conceptual structure, but merely a conse-quence of our ignorance of the precise initial conditions of theparticle.

As we shall see in Sec. 9, it is entirely possible that abetter theory of phenomena involving distances of theorder of 10 "cm or less would require us to go beyondthe limitations of these special assumptions. Our prin-cipal purpose in this paper (and in Paper II) is to show,however, that if one makes these special assumptions,our interpretation leads in all possible experiments tothe same predictions as are obtained from the usual1QtcrPI'ctRtloQ.

It is now easy to understand why the adoption of the

DAVID BOHM

usual interpretation of the quantum theory would tendto lead us away from the direction of our suggestedalternative interpretation. For in a theory involvinghidden variables, one would normally expect that thebchavlor of aQ lndlvldual systcn1 should not dcpcQdon the statistical ensemble of which it is a member,because this ensemble refers to a series of similar butdisconnected experiments carried out under equivalentinitial conditions. In our interpretation, however, the"quantum-mechanical" potential, U(x), acting on anindividual particle depends on a wave intensity, P(x),that is also numerically equal to a probability densityin our ensemble. In the terminology of the usual in-terpretation of the quantum theory, in which onetacitly assumes that the wave function has only oneinterpretation; namely, in terms of a probability, oursuggested ncw lntcrprctatloQ would look like a mysteri-ous dependence of the individual on the statisticalensemble of which it is a member. In our interpretation,such a dependence is perfectly rational, because thewave function can consistently be interpreted both asa force and as a probability density. "

It is instructive to carry our analogy between theSchroedinger field and other kinds of fields a bit further.To do this, we can derive the wave Eqs. (5) and (6)from a Hamiltonian functional. Ke begin by writingdown the expression for the mean energy as it is ex-pressed in the usual quantum theory:

The mean particle energy is found by averaging E(x)with the weighting function, P(x). We obtain

{@ensemble= Jt P(X)E(X)dXaverage

(vs)'~P(x) +V(x) dx — I RVsRdx.

2m&

A little integration by parts yields

(vs)'(E)ensemb1e= P(X) +V(X)

2tlZaverage

5' (VP)'+ dx=H. (11)

8m I"

These are, however, the same as the correct waveEqs. (5) and (6).

Kc can now show that the mean particle energyaveraged over our ensemble is equal to the usual quan-tum mechanical mean value of the Hamiltonian, H. Todo this, we note that according to Eqs. (3) and (6), theenergy of a particle is

BS(x) (V'S)' 5' PEE(x)= — = +V(x)—2' 2m E

II= P*( — Vs+ V(x) (Pdx2m )

A2

I~~I'+V( )I&I' d .2'Writing P= P& e px(iS/5), we obtain

(V'S)' iP (VP)'P= P(x) +U(x)+ — dx.

Sm I"We shall now reinterpret P(x) as a Geld coordinate,

defined at each point, x, and we shall tentative]y assumethat S(x) is the momentum, canonically conjugate toP(x). That such an assumption is appropriate can beverified by finding the Hamiltonian equations of motionfor P(x}and S(x},under the assumption that the Hamil-tonian functional is equal to H (See Eq. (9)). Theseequations of motion are

8HP= =—7 (PVS)

b5

-(~s)' ass pPP 1 (VP)'q+V(x)—2'

"This consistency is guaranteed by the conservation Eq. {7).The questions of why an arbitrary statistical ensemble tends todecay into an ensemble with a probability density equal to P*Pwill be discussed in Paper II, Sec. 7.

S. THE STATIONARY STATE

YVe shall now show how the problem of stationarystates is to be treated in our interpretation of thequantum theory.

The following seem to be reasonable requirements inour interpretation for a stationary state:

(1) The particle energy should be a constant of themotion,

(2) The quantum-mechanical potential should beindependent of time.

(3) The probability density in our statistical en-semble should be independent of time.

It is easily verified that these requirements can besatisfied with the assumption that

P(x, t) =Pp(x) exp( —sEt/is)

=Re(x)expt i(4 (x)—Er)/hg. (12)

From the above, we obtain S=C (x)—Et. According tothe generalized Hamilton-Jacobi Eq. (4), the particleenergy is given by

BS/83= —E.Thus, we verify that the particle energy is a constantof the motion. Moreover, since P=R'= ~P~', it followsthat P (and R) are independent of time. This meansthat both the probability density in our ensemble andthe quantum-mechanical potential are also timeindependent.

QUANTUM THEORY IN TERMS OF ''HI D DEN'' VAR lABLES. I

The reader will readily verify that no other form ofsolution of Schroedinger's equation will satisfy all threeof our criteria for a stationary state.

Since P is now being regarded as a mathematicalrepresentation of an objectively real force field, itfollows that (like the electromagnetic field) it shouldbe everywhere 6nite, continuous, and single valued.These requirements will guarantee in all cases that occurin practice that the allowed values of the energy in astationary state, and the corresponding eigenfunctionsare the same as are obtained from the usual interpreta-tion of the theory.

In order to show in more detail what a stationarystate means in our interpretation, we shall now considerthree examples of stationary states.

Case 1:"s"State

The first case that we shall consider is an "s" state.In an "s"state, the wave function is

f=f(r) exp[i(n —Et)/It], (13)

where 0. is an arbitrary constant and r is the radiustaken from the center of the atom. We conclude thatthe Hamilton-Jacobi function is

S=n —8].The particle velocity is

v= VS=0.

The particle is therefore simply standing still, whereverit may happen to be. How can it do this? The absenceof motion is possible because the applied force, —V'V(x),is balanced by the "quantum-mechanical" force, (ft'/2m)V(PR/R), produced by the Schroedinger f-fieldacting on its own particle. There is, however, a sta-tistical ensemble of possible positions of the particle,with a probability density, P(x) = (f(r))'.

Case 2: State with Nonzero AngularMomentum

In a typical state of nonzero angular momentum,we have

1t =f„'(r)Pi (cos8)exp[i(P —Et+Am&)/kj, (14)

where 8 and it are the colatitude and azimuthal polarangles, respectively, Pp is the associated Legendrepolynomial, and P is a constant. The Hamilton-Jacobifunction is S=P Zt+hnnf& From—this resul. t it followsthat the s component of the angular momentum isequal to km. To prove this, we write

I.,=xP„ yp. =xBS/By —yBS/Bx=BS/B—y=hm (15).Thus, we obtain a statistical ensemble of trajectorieswhich can have diferent forms, but all have the same"quantized" value of the s component of the angularmomentum.

Case 3:A Scattering Problem

Let us now consider a scattering problem. Becauseit is comparatively easy to analyze, we shall discuss ahypothetical experiment, in which an electron is in-cident in the s direction with an initial momentum, Po,on a system consisting of two slits. "After the electronpasses through the slit system, its position is measuredand recorded, for example, on a photographic plate.

Now, in the usual interpretation of the quantumtheory, the electron is described by a wave function.The incident part of the wave function is fo exp(iPos/fi); but when the wave passes through the slit sy' stem,it is modified by interference and diffraction effects,so that it will develop a characteristic intensity patternby the time it reaches the position measuring instru-ment. The probability that the electron will be detectedbetween x and x+dx is

~lt (x) ~'dx. If the experiment is

repeated many times under equivalent initial condi-tions, one eventually obtains a pattern of hits on thephotographic plate that is very reminiscent of theinterference patterns of optics.

In the usual interpretation of the quantum theory,the origin of this interference pattern is very dif6cultto understand. For there may be certain points wherethe wave function is zero when both slits are open, butnot zero when only one slit is open. How can theopening of a second slit prevent the electron from reach-ing certain points that it could reach if this slit wereclosed? If the electron acted completely like a classicalparticle, this phenomenon could not be explained at all.Clearly, then the wave aspects of the electron musthave something to do with the production of the inter-ference pattern. Yet, the electron cannot be identicalwith its associated wave, because the latter spreadsout over a wide region. On the other hand, when theelectron's position is measured, it always appears atthe detector as if it were a localized particle.

The usual interpretation of the quantum theory notonly makes no attempt to provide a single preciselydefined conceptual model for the production of thephenomena described above, but it asserts that nosuch model is even conceivable. Instead of a singleprecisely defined conceptual model, it provides, aspointed out in Sec. 2, a pair of complementary models,vis. , particle and wave, each of which can be mademore precise only under conditions which necessitatea reciprocal decrease in the degree of precision of theother. Thus, while the electron goes through the slitsystem, its position is said to be inherently ambiguous,so that if we wish to obtain an interference pattern, itis meaningless to ask through which slit an individualelectron actually passed. Within the domain of spacewithin which the position of the electron has no mean-ing we can use the wave model and thus describe thesubsequent production of interference. If, however, we

"This experiment is discussed in some detail in reference 2,Chapter 6, Sec. 2.

DAVrD BOH M

tried to define the position of the electron as it passedthe slit system more accurately by means of a measure-ment, the resulting disturbance of its motion producedby the measuring apparatus would destroy the inter-ference pattern. Thus, conditions would be created inwhich the particle model becomes more precisely de-fined at the expense of a corresponding decrease in thedegree of definition of the wave model. When the posi-tion of the electron is measured at the photographicplate, a similar sharpening of the degree of definition ofthe particle model occurs at the expense of that of thewave model.

In our interpretation of the quantum theory, thisexperiment is described causally and continuously interms of a single precisely definable conceptual model.As we have already shown, we must use the same wavefunction as is used in the usual interpretation; butinstead we regard it as a mathematical representationof an objectively real Geld that determines part of theforce acting on the particle. The initial momentum ofthe particle is obtained from the incident wave func-tion, exp(epos/k), as p=8s/Os= po We .do not in prac-tice, however, control the initial location of the par-ticle, so that although it goes through a definite slit,we cannot predict which slit this will be. The particleis at all times acted on by the "quantum-mechanical"potential, U=(—A'/2m)PR/R. While the particle isincid. ent, this potential vanishes because R is then aconstant; but after it passes through the slit system,the particle encounters a quantum-mechanical po-tential that changes rapidly with position. The subse-quent motion of the particle may therefore becomequite complicated. Nevertheless, the probability thata particle shall enter a given region, dx, is as in the usualmterpretation, equal to

~P(x) ~

'dx. We therefore deducethat the particle can never reach a point where thewave function vanishes. The reason is that the "quan-tum-mechanical" poteQtial, U, becomes iQGQlte wlienR becomes zero. If the approach to infinity happens tobe through positive values of U, there will be an in-finite force repelling the particle away from the origin.If the approach is through negative values of U, theparticle will go through this point with infinite speed,and thus spend no time there. In either case, we obtaina simple and precisely definable conceptual model ex-plaining why particles can never be found at pointswhere the wave function vanishes.

If one of the slits is closed, the "quantum-mechanical"potential is correspondingly altered, because the P-Geldis changed, and the particle may then be able to reachcertain points which it was unable to reach when bothslits were open. The slit is therefore able to a6ect themotion of the particle only indirectly, through itseffect on the Schroedinger P-Geld. Moreover, as we shallsee in Paper II, if the position of the electron is meas-ured while it is passing through the slit system, themeasuring apparatus will, as in the usual interpretation,create a disturbance that destroys the interference

pattern. In our interpretation, however, the necessityfor this destruction is not inherent in the conceptualstructure; and as we shall see, the destruction of theinterference pattern could in principle be avoided bymeans of other ways of making measurements, wayswhich are conceivable but not now actually possible.

Writing &=R(x„x,)expLi5(x„x,)/h] and R'=P, weobtain

BP—+—[V'i PV'i5+W2 P~72Sj=0,8$ m

(16)

as (v,s)'+(v,s)'—+- +V(xi, x2)2m

jz2

[Vi2R+722R] =0. (17)2mR

The above equations are simply a six-dimensionalgeneralization of the similar three-dimensional Eqs. (5)and (6) associated with the one-body problem. In thetwo-body problem, the system is described thereforeby a six-dimensional Schroedinger wave and by a six-dimensional trajectory, specifying the actual locationof each of the two particles. The velocity of this tra-jectory has components, V'iS/nz and V'g/m, respec-tively, in each of the three-dimensional surfaces associ-ated with a given particle. P(xi, x2) then has a dualinterpretation. First, it defines a "quantum-mechanical"potential, acting on each particle

V(xi, x2) = —(jr'/2mR) [V'i2R+ V'22R].

This potential introduces an additional effective inter-action between particles over and above that due to theclassically inferrable potential V(x). Secondly, the func-tion P(xi, x2) can consistently be regarded as theprobability density of representative points (xi, x&) inour six-dimensional ensemble.

The extension to an arbitrary number of particlesis straightforward, and we shaH quote only the resultshere. We introduce the wave function, Q=R(x„x2,

x„)expLiS(xi, x2 x„)/hj and deGne a 3n-dimen-sional trajectory, where e is the number of particles,which describes the behavior of every particle in thesystem. The velocity of the ith particle is v, = V;S(xi,x2 ' 'x„)/fs. Tile flilictloli P(xi, x2 x )=R llas two

6. THE MANY-BODY PROBLEM

We shall now extend our interpretation of the quan-tum theory to the problem of many bodies. Vfe beginwith the Schroedinger equation for two particles. (Forsimplicity, we assume that they have equal masses,but the extension of our treatment to arbitrary masseswill be obvious. )

8$ h'ih =—— (Vip+ V'2Q)+ V(xi, x~)P.

Bt 2m

OF ' 'H I D DEN'' VARIABLES. I

interpretations. First, it defines a "quantum-mechani-cal" potential

pp

U(xg, x2. x )= — Q V,2E(xg, x2 x„). (18)2fgg @=1

Secondly, P(xg) x2' ' ' xn) 1s equal 'to the density ofrepresentative points (x~, xq x„) in our 3n-dimen-sional ensemble.

We see here that the "effective potential, " U(x~, x2,~ x„), acting on a particle is equivalent to that pro-duced by a "many-body" force, since the force betvreen

any tvro particles may depend significantly on thelocation of every other particle in the system. An ex-ample of the effects of such a force is given by the ex-clusion principle. Thus, if the wave function is anti-symmetric, we deduce that the "quantum-mechanical"forces vrill be such as to prevent two particles from everreaching the same point in space, for in this case, vre

must have I'=0.

/. TRANSITIONS BETWEEN STATIONARY STATES-THE FRANCE-HERTZ EXPERIMENT

Our interpretation of the quantum theory describesall processes as basically causal and continuous. Howthen can it lead to a correct description of processessuch as the Franck-Hertz experiment, the photoelectriceffect, and the Compton effect, which seem to callmost strikingly for an interpretation in terms of dis-continuous and incompletely determined transfers ofenergy and momentums In this section, we shall ansvrerthis question by applying our suggested interpretationof the quantum theory in the analysis of the Franck-Hertz experiment. Here, we shall see that the ap-parently discontinuous nature of the process of transferof energy from the bombarding particle to the atomicelectron is brought about by the "quantum-mechanica1"potential, U= (—h'/2m)P'E/R, which does not neces-sarily become small when the wave intensity becomessmall. Thus, even if the force of interaction betweentwo particles is very weak, so that a correspondinglysmall disturbance of the Schroedinger wave function isproduced by the interaction of these particles, thisdisturbance is capable of bringing about very largetransfers of energy and momentum between the par-ticles in a very short time. This means that if we viewonly the end results, this process presents the aspectof being discontinuous. Moveover, vre shall see thatthe precise value of the energy transfer is in principledetermined by the initial position of each particle andby the initial form of the wave function. Since we cannotin practice predict or control the initial particle posi-tions with complete precision, vre are also unable topredict or control the final outcome of such an experi-ment, and can, in practice, predict only the probabilityof a given outcome. Because the probability that theparticles will enter a region with coordinates, xI, x2, isproportional to E'(x~, x2), we conclude that although

a Schroedinger wave of lovr intensity can bring aboutlarge transfers of energy, such a process is (as in theusual interpretation) highly improbable.

In Appendix A of Paper II, we shall see that similarpossibilities arise in connection with the interaction ofthe electromagnetic fmld vrith charged matter, so thatelectromagnetic vraves can very rapidly transfer a fullquantum of energy (and momentum) to an electron,even after they have spread out and fallen to a verylow intensity. In this way, we shall explain the photo-electric eGect and the Compton eGect. Thus„we aleable in our interpretation to understand by means of acausal and continuous model just those properties ofmatter and light vrhich seem most convincingly to re-quire the assumption of discontinuity and incompletedeterminism.

Before we discuss the process of interaction betweentvro particles, vre shall 6nd it convenient to analyze theproblem of an isolated, single particle that happens tobe in a nonstationary state. Because the 6eld function

P is a solution of Schroedinger s equation, we can line-arly suppose stationary-state solutions of this equationand in this way obtain new solutions. As an illustration,let us consider a superposition of two solutions

P- Cygne(x) exp( —iEit/5)+ C2$2(x) exp( —iE2i/k),

where Cq, C~, g&, and f2 are real. Thus we write fq=E&,1/2=22, and

P=expL —i(E&+E2)t/2h]IC&R& expt i(E)—E2)t/—2k]+C2E2 exp/i(Eg —E2)t/2k] I.

Writing P= E exp(i5/h), we obtain

E2—C12+12(x)+C22+22 (x)+2CgCpEg(x)Eg(x)cosL(Eg —E2)t/2A], (19)

S+(Eg—E2)t/2

C2E, (x)—Cog(x) (Eg—E2)ttan

C&g(x)+ CgEg(x)

We see immediately that the particle experiences a"quan«m-mecha»cai" potential, U(x) = (—&/2~) &'&/E, which fluctuates with angular frequency, m=(E~—E2)/5, and that the energy of this particle, E= —BS/BI,, and its momentum y= V'S, fluctuate with the sameangular frequency. If the particle happens to enter aregion of space vrhere R is small, these Quctuations canbecome quite violent. We see then that, in general, theorbit of a particle in a nonstationary state is very ir-regular and complicated, resembling Brovrnian motionmore closely than it resembles the smooth track of aplanet around the sun.

If the system is isolated, these Quctuations will con-tinue forever. The result is quite reasonable, since as iswell knovrn, a system can make a transition from onestationary state to another only if it can exchange en-

DA V I D

ergy with some other system. In order to treat theproblem of transition between stationary states, wemust therefore introduce another system capable ofexchanging energy with the system of interest. In thissection, we shall discuss the Franck-Hertz experiment,in which t'his other system consists of a bombardingparticle. For the sake of illustration, let us suppose thatwe have hydrogen atoms of energy Eo and wave func-tion, fe(x), which are bombarded by particles thatcRn bc scatter cd lnelas'tlcRlly leRvlng thc atom withenergy E and wave function, P (x).

We begin by writing down the initial wave function,4', (x, y, t). The incident particle, whose coordinates arerepresented by y must be associated with a wavepacket, which can be written as

fp(y, t) = t e'" f(k—ke)exp( —ihk't/2m)dk. (21)

product

@;=Ps(x)exp( iEet—/h) fe(y, t) (.22)

Let us now see how this wave function is to be under-stood in our interpretation of the theory. As pointed outin Sec. 6, the wave function is to be regarded as a mathe-matical representation of a six-dimensional but ob-jectively real held, capable of producing forces that acton the particles. We also assume R six-dimensionalrepresentative point, described by the coordinates ofthe two particles, x and y. We shall now see that whenthe combined wave function takes the form (22) in-volving a product of a function of x and a function of y,the six-dimensional system can correctly be regardedas being made up of two independent three-dimentionalsubsystems. To prove this, we write

Pe(x) =Re(x)expLiSo(x)/h] and

fe{y, t) =Me(y, t)expt i.V,(y, t)/h jWe then obtain for the particle velocities

dx/dt= (1/m) V'Sp(x); dy/dt = (1/m) V'Ee(y, t), (23)

and for the "quantum-mechanical" potential

O'I (7'.'+V'o')R(x, y) }

2m'(x, y)

—h' PRo(x) ~7sMo(y, t)t+ . (24)

2m Re(x) Me(y, t)

Thus, the particle velocities are independent and the"quantum-mechanical" potential reduces to a sum ofterms, one involving only x and the other involvingonly y. This means that the particles move independ-

The center of this packet occurs where the phase hasan extremum as a function of k, or where y= hkot/m

Now, as in the usual interpretation, me begin bywriting the incident wave function for the combinedsystem as a

ently. Moreover, the probability density, P=Ees(x)XMos{y, t), is a Product of a function of x and a func-tion of y, indicating that the distribution in x is sta-tistically independent of that in y. We conclude, then,that whenever the wave function ean be expressed as aproduct of two factors, each involving only the coordi-nates of a single system, then the two systems are com-pletely independent of each other.

As soon as the wave packet in y space reaches theneighborhood of the atom, the two systems begin tointeract. If we solve Schroedinger's equation for thecombined system, we obtain a.wave function that eanbe expressed in terms of the following series:

%'=+,+P„P„(x)exp(—iE„t/h) f„(y, t), (25)

where the f„(y, t) are the expansion coefficients of thecomplete set of functions, P„(x). The asymptotic formof the wave function is'4

iE„tq4=4;(x, y)+P„P„(x)exp~ —

}'

f(k—k,)h )

exp Lik r—(hk„s/2rs) tJX g„(8, &f, k)dk, (26)

h'k, ,s/2m = {hske'/2m)+ E(&—E„(conservation of energy). (2/)

The additional terms in the above equation representoutgoing wave packets, in mhich the particle speed,hk„/m, is correlated with the wave function, it'r (x),representing the state in which the hydrogen atom isleft. Thc center of the nth packet occurs at

r„=(hk„/m)t.

It is clear that because the speed depends on the hy-drogen atom quantum number, n, every one of thesepackets will eventually be separated by distanceswhich are so large that this separation is classicallydescribable.

When the wave function takes the form (25), thetwo particles system must be described as a single six-dimensional system and not as a sum of two independentthree-dimensional subsystems, for at this time, if wetry to express the wave function as 'f(x, y)=R(x, y)Xexpl iS(x, y)/h], we find. that the resulting expres-sions for R and 5 depend on x and y in a very compli-cated way. The particle momenta, pi ——VsS(x, y) andps= V'„S(x, y), therefore become inextricably interde-pendent. The "quantum-mechanical" potential,

U= — (7,'E+ t7„'R)2mB(x, y)

ceases to be expressible as the sum of a term involvingx and a term involving y. The probability density,

"N. F. Mott and H. S. W. Massey, The Theory of Atomic Cotltstows (Clarendon Press, Oxford, 1933).

QUANTUM THEORY IN TERMS OF ''H I D DEN'' VARIABLES. I

R'(x, y) can no longer be written as a product of afunction of x and a function of y, from which we con-clude that the probability distributions of the two par-ticles are no longer statistically independent. Moreover,the motion of the particle is exceedingly complicated,because the expressions for R and S are somewhatanalogous to those obtained in the simpler problem ofa nonstationary state of a single particle { see Eqs.(19) and (20)]. In the region where the scattered wavesP„(x)f„(y, t) have an amplitude comparable with thatof the incident wave, $0(x)fo(y, t), the functions R andS, and therefore the "quantum-mechanical" potentialand the particle momenta, undergo rapid and violentfluctuations, both as functions of position and of time.Because the quantum-mechanical potential has R(x, y, t)in the denominator, these fluctuations may becomevery large in this region where R is small. If the particleshappen to enter such a region, they may exchange verylarge quantities of energy and momentum in a veryshort time, even if the cls,ssical potential, V(x, y) isvery small. A small value of V(x, y) implies, however,a correspondingly small value of the scattered waveamplitudes, f„(y, t). Since the fluctuations become largeonly in the region where the scattered wave amplitudeis comparable with the incident wave amplitude andsince the probability that the particles shall enter agiven region of x, y space is proportional to R'(x, y),it is clear that a large transfer of energy is improbable(although still always possible) when V(x, y) is small.

While interaction between the two particles takesplace then, their orbits are subject to wild fluctuations.Eventually, however, the behavior of the system quietsdown and becomes simple again. For after the wavefunction takes its asymptotic form (26), and the packetscorresponding to different values of m have obtainedclassically describable separations, we can deduce thatbecause the probability density is ~P~', the outgoingparticle must enter one of these packets and stay withthat packet thereafter (since it does not enter the spacebetween packets in which the probability density isnegligibly different from zero). In the calculation of theparticle velocities, Vi ——V,S/rN, V2 ——'7„5/m, and of thequantum-mechanical potential, U= (—k'/2mR) (V','R+V'„'R), we can therefore ignore all parts of the wave .

function other than the one actually containing theoutgoing particle. It follows that the system acts as ifit had the wave function

($E„f)e„=y„(x)exp{

~"f(k-k.)(a)~

exp{fLk„r—(Sk.9/2') t]}X g.(8, y, k)dk(2, 9)

where e denotes the packet actually containing theoutgoing particle. This means that for all practicalpurposes the complete wave function (26) of the sys-tem may be replaced by Eq. (29), which corresponds to

an atomic electron in its eth quantum state, and to anoutgoing particle with a correlated energy, E„'=k'k„'/2'. Because the wave function is a product of a func-tion of x and a function of y, each system once againacts independently of the other. The wave function cannow be renormalized because the multiplication of 4„by a constant changes no physically significant quan-tity, such as the particle velocity or the "quantum-mechanical" potential. As shown in Sec. 5, when theelectronic wave function is f„(x)exp( —iE„t/k), itsenergy must be E„.Thus, we have obtained a descrip-tion of how it comes about that the energy is alwaystransferred in quanta of size E —Ep.

It should be noted that while the wave packets arestill separating, the electron energy is not quantized,but has a continuous range of values, which fluctuaterapidly. It is only the final value of the energy, appear-ing after the interaction is over that must be quantized.A similar result is obtained in the usual interpretationif one notes that because of the uncertainty principle,the energy of either system can become definite onlyafter enough time has elapsed to complete the scatter-ing process. '5

In principle, the actual packet entered by the out-going particle could be predicted if we knew the initialposition of both particles and, of course, the initial formof the wave function of the combined system. "In prac-tice, however, the particle orbits are very complicatedand very sensitively dependent on the precise values ofthese initial positions. Since we do not at present knowhow to measure these initial positions precisely, wecannot actually predict the outcome of such an inter-action process. The best that we can do is to predictthe probability that an outgoing particle enters the mth

packet within a given range of solid angle, dD, leavingthe hydrogen atom in its eth quantum state. In doingthis, we use the fact that the probability density in

x, y space is ~P(x, y) ~' and that as long as we are re-stricted to the mth packet, we can replace the completewave function (26) by the wave function (29), corre-sponding to the packet that actually contains the par-ticle. Now, by definition, we have J'~P„(x)~'dx=1.The remaining integration of

exp {iLk„r—(hk„'/2m) t$}~~ f(k—1,) g.(e, y, k)dk

r

over the region of space corresponding to the eth out-going packet leads, however, to precisely the sameprobability of scattering as would have been obtainedby applying the usual interpretation. We conclude,then, that if P satisfies Schroedinger's equation, thatif v= VS/m, and that if the probability density of par-ticles is P(x, y) =R'(x, y), we obtain in every respect"See reference 2, Chapter 18, Sec. 19."Note that in the usual interpretation one assumes that

gothing determines the precise outcome of an individual scatteringprocess. Instead, one assumes that all descriptions are inherentlyand unavoidably statistical (see Sec. 2).

DA VI D BOB M

exactly the same physical predictions for this problemas are obtained when we use the usual interpretation.

There remains only one more problem; namely, toshow that if the outgoing packets are subsequentlybrought together by some arrangement of matter thatdoes not act on the atomic electron, the atomic electronand the scattered particle will continue to act inde-pendently. '" To show that these two particles will con-tinue to act independently, we note that in all practicalapplications, the outgoing particle soon interacts withsome classically describable system. Such a systemmight consist, for example, of the host of atoms of thegas with which it collides or of the walls of a container.In any case, if the scattering process is ever to be ob-served, the outgoing particle must interact with aclassically describable measuring apparatus. Now allclassically describable systems have the property thatthey contain an enormous number of internal "thermo-dynamic" degrees of freedom that are inevitably excitedwhen the outgoing particle interacts with the system.The wave function of the outgoing particle is thencoupled to that of these internal thermodynamic de-grees of freedom, which we represent as yi, y2, ~ ~ y, .To denote this coupling, we write the wave function forthe entire system as

+=K.-4-(x)exp( —~&-~/&)f-(y y4 y2 "3 ) (30)

Now, when the wave function takes this form, theoverlapping of difFerent packets in y space is not enoughto produce interference between the different f„(x).To obtain such interference, it is necessary that thepackets f„(y, yi, y2, y,) overlap in every one of theS+3 dimensions, y, yi, y2 y.. The reader will readilyconvince himself, by considering a typical case such asa collision of the outgoing particle with a metal mall,that it is overwhelmingly improbable that two of thepackets f (yi, yi, y2 y, ) will overlap with regard toevery one of the internal thermodynamic coordinates,yi, y2, ~ y„even if they are successfully made to.overlap in y space. This is because each packet corre-sponds to a difFerent particle velocity and to a difFerenttime of collision with the metal wall. Because themyriads of internal thermodynamic degrees of freedomare so chaotically complicated, it is very likely that aseach of the e packets interacts with them, it will en-counter difFerent conditions, which will make the com-bined wave packet f„(y, yi, .y,) enter very differentregions of y~, yg. ~ .y, space. Thus, for all practicalpurposes, we can ignore the possibility that if two of thepackets are made to cross in y space, the motion eitherof the atomic electron or of the outgoing particle willbe afFected. "

'7 See reference 2, Chapter 22, Sec. 11, for a treatment of asimilar problem.

'8 It should be noted that exactly the same problem arises inthe usual interpretation of the quantum theory for (reference 16),for whenever two packets overlap, then even in the usual inter-pretation, the system must be regarded as, in some sense, coveringthe states corresponding to both packets simultaneously. Seereference 2, Chapter 6 and Chapter 16, Sec. 25. Once two packets

8. PENETRATION OF A BARRIER

According to classical physics, a particle can neverpenetrate a potential barrier having a height greaterthan the particle kinetic energy. In the usual interpreta-tion of the quantum theory, it is said to be able, witha small probability, to "leak" through the barrier. Inour interpretation of the quantum theory, however, thepotential provided by the Schroedinger P-field enablesit to "ride" over the barrier, but only a few particles arelikely to have trajectories that carry them all the wayacross without being turned around.

We shall merely sketch in general terms how theabove results can be obtained. Since the motion of theparticle is strongly a&ected by its f-field, we must firstsolve for this fjeld with the aid of "Schroedinger'sequation. " Initially, we have a wave packet incidenton the potential barrier; and because the probabilitydensity is equal to ~f(x) ~', the particle is certain to besomewhere within this wave packet. When the wavepacket strikes the repulsive barrier, the P-field under-goes rapid changes which can be calculated" if desired, ,but whose precise form does not interest us here. At thistime, the "quantum-mechanical" potential, U= (—h'/2m)V'Z/Z, undergoes rapid and violent fluctuations,analogous to those described in Sec. 7 in connectionwith Eqs. (19), (20), and (25). The particle orbit thenbecomes very complicated and, because the potentialis time dependent, very sensitive to the precise initialrelationship between the particle position and the centerof the wave packet. Ultimately, however, the incidentwave packet disappears and is replaced by two packets,one of them a refiected packet and the other a trans-mitted packet having a much smaller intensity. Becausethe probability density is ~f~', the particle must endup in one of these packets. The other packet can, asshown in Sec. 1', subsequently be ignored. Since thereflected packet is usually so much stronger than thetransmitted packet, we conclud. e that during the timewhen the packet is inside the barrier, most of theparticle orbits must be turned around, as a result of theviolent Quctuations in the "quantum-mechanical"potential.

9. POSSIBLE MODIFICATIONS IN MATHEMATICALFORMULATION LEADING TO EXPERIMENTAL

PROOF THAT NEW INTERPRETATIONIS NEEDED

We have already seen in a number of cases and inPaper II we shall prove in general, that as long as we

have obtained classically describable separations, then, both inthe usual interpretation and in our interpretation the probabilitythat there will be significant interference between them is so over-whelmingly small that it may be compared to the probabilitythat a tea kettle placed on a 6re will happen to freeze instead ofboil. Thus, we may for all practical purposes neglect the possi-bility of interference between packets corresponding to the dif-ferent possible energy states in which the hydrogen atom may beleft.

'9 See, for example, reference 2, Chapter 11,Sec. 17, and Chapter12, Sec. 18.

OF ''H I D DEN'' VARIABLES. I

assume that f satisfies Schroedinger's equation, thatv=VS(x)/m, and that we have a statistical ensemblewith a probability density equal to ~f(x) ~, our in-terpretation of the quantum theory leads to physicalresults that are identical with those obtained from theUsuRl lntclplctatlon. Evidence lndlcatlng thc nccd fox'

adopting our interpretation instead of the usual onecould therefore come only from experiments, such asthose involving phenomena associated with distances ofthe order of 10 "cm or less, which are Qot now ade-quately understood in terms of the existing theory. Inthis paper we shall not, however, actually suggest anyspeci6c experimental methods of distinguishing betweenour interpretation and the usual one, but shall conhneourselves to demonstxating that such experiments areconceivable.

Now'» thcI'c Rrc Rn IQ6nlte number of wRys of Inodlfy-

ing the mathematical form of thc theory that are con-sistent with our interpretation and not with the usualinterpretation. We shall conhne ourselves here, however,to suggesting two such modi6cations, which have al-ready been indicated in Sec. 4, namely, to give up theassumption that v is necessarily equal to VS(x)/m, andto give up the assumption that P must necessarilysatisfy a homogeneous linear equation of the generaltype suggested by Schroedinger. As we shall see, givingup either of those first two assumptions will in generalalso require us to give up the assumption of a statisticalensemble of particles, with a probability density equalto lk(x) I'.

Ke begin by noting that it is consistent with ourinterpretation to modify the equations of motion of aparticle (Sa) by adding any conceivable force term tothe right-hand side. Let us, for the sake of illustration,consider a force that tends to make the diGcrence,

p —V'5(x), decay rapidly with time, with a mean decaytime of the order of v=10 ia/c seconds, where c is thevelocity of light. To achieve this result, we write

d'x h2 V2E.= —V I (x)— +f{p—VS(x)), (31)

dP 2m E

where f(p —VS(x)) is assumed to be a function whichvanishes when p=V'5(x) and more generally takessich a form that it implies a force tending to make

p —VS(x) decrease rapidly with the passage of time.It is clear, moreover, that f can be so chosen that it islRI'ge only ln ploccsscs lnvolvlng vcI'y sholt dlstanccs(where V'S(x) should be large).

If the correct equations of motion resembled Eq. (31),then the usual interpretation wouM be applicable onlyover times much l.onger than ~, for only after such timeshave elapsed will the relation p= V'5(x) be a good ap-proximation. Moreover, it is clear that such modifica-

tions of the theory cannot even be described in theusual interpretation, because they involve the preciselydehnable particle variables which are not postulatedin the usual interpretation.

Let us now consider a modihcation that makes theequation governing P inhomogeneous. Such a modifica-tion lS

(32)

Here, H is the usual Hamiltonian operator, x;, representsthe actual location of the particle, and P is a functionthat vanishes when p=V'5(x;). Now, if the particleequations are chosen, as in Eq. (31),to make p —V'5(x,)decRy rapidly with tln1c» lt follows thRt ln Rtomlcprocesses, the inhomogeneous term in Eq. (32) willbecome negligibly small, so that Schroedinger's equationis a good approximation. Nevertheless, in processesinvolving very short distances and very short times, theinhomogeneities would be important, and the P-fieldwouM, Rs lQ thc CRsc of tlM clcctroIQRgnctlc flcM, de-pend to some extent on the actual location of theparticle.

It is clear that Eq. (32) is inconsistent with the usualinterpretation of the theory. Moreover, we can con-template further generalizations of Eq. (32), in thedixection of introducing nonlinear terms that are largeonly for processes involving small distances. Since theusual interpretation is based on the hypothesis oflinear superposition of "state vectors" in a Hilbertspace, it follows that the usual interpretation could notbe made consistent with such a nonlinear equation for aone-particle theory. In a many-particle theory, opera-tors can be introduced, satisfying a nonlinear generaliza-tion of Schroedinger's equation; but these must ulti-mately operate on wave functions that satisfy a linearhomogeneous Schroedinger equation. .

Finally, we repeat a point already made in Sec. 4,namely, that if the theory is generalized in any of theways indicated here, the probability density of particleswill cease to equal ~P(x) ~'. Thus, experiments wouldbecome conceivable that distinguish between ~P(x)~'Rnd this pxobablllty; RQd ln this wRy wc could obtRlQan experimental proof that the usual interpretation,which gives ~f(x) ~' owly a probability interpretation,must be inadequate. Moreover, we shall show inPRpel II thRt modi6catlons llkc those sUggcstcd hcx'c

would permit the particle position and momentum tobe measured simultaneously, so that the uncertaintyprinciple couM be violated.

ACKNOWLEDGMENT

The author wishes to thank Dr. Einstein for severalinteresting and stimulating discussions.