PHYSICAL REVIEW VOL U MEl 5 0, N UM B E R 4 28 OCTOBER 1966

Derivation of the Schrodinger Equation from Newtonian Mechanics*

EDWARD NELSON

Department of Mathematics, Princeton University, Princeton, New Jersey(Received 21 April 1966; revised manuscript received 21 June 1966)

We examine the hypothesis that every particle of mass m is subje.ct to a Brownian motion with di~usion

coefficient ft/2m and no friction. The influence of an exte:nal field.IS expr~ssed .br~e~?s o~~e~tont~Ia~F=ma, as in the Ornstein-Uhlenbeck theory of macroscopic Brownl~n m?tIon WIt .nc .Ion.. e ypo .esisleads in a natural way to the Schrodinger equation, but t?e I?hysical interpretation 1.S e?tIrely classical.Particles have continuous trajectories and the wave function IS not a complete description of the state.Despite this opposition to quantum mech~nics, an ex:amination of the measurement process suggests that,within a limited framework, the two theones are equivalent.

I. INTRODUCTION

W E shall attempt to show in this paper that theradical departure from classical physics produced

by the introduction of quantum mechanics forty yearsago was unnecessary. An entirely classical .deriv~tion

and interpretation of the Schrodinger equation will begiven, following a line of thought w~ic? is a natu:aldevelopment of reasoning used in statistical mechanicsand in the theory of Brownian motion.

Consider an electron in an external field. The electronis regarded as a point particle of .mass m in. th~ senseof Newtonian mechanics. Our basic assumption IS thatany particle of mass m constantly undergoes a Brow.nianmotion with diffusion coefficient inversely proportionalto m. We write the diffusion coefficient as h/2m andlater identify h with Planck's constant divided by 21r.As in the theory of macroscopic Brownian motion, theinfluence of the external force is expressed by means ofNewton's law F=ma, where a is the mean accelerationof the particle. The chief differe.nce ~s that it; the ~t~dy

of macroscopic Brownian motion In a fluid, frictionplays an important role. For the electron we must as­sume that there is no friction in order to preserveGalilean covariance. The kinematical description ofBrownian motion with zero friction is the same as thedescription used in the Einstein-Smoluch~wskitheo;yl(the approximate theory of macroscopic Brownianmotion in the limiting case of infinite friction).

The picture which emerges is the following. If wehave, for example, a hydrogen atom in the ground state,the electron is in dynamical equilibrium between therandom force causing the Brownian motion and theattractive Coulomb force of the nucleus. Its trajectoryis very irregular. Most of the time the electron is ne~r

the nucleus, sometimes it goes farther away, but Italways shows a general tendency to ~ove ~owa:d thenucleus, and this is true no matter which direction wetake for time. This behavior is quite analogous to that

*This work was supported in part by the National ScienceFoundation. .

1 A. Einstein, Investigations on the Theory of the BroumianMovement translated by A.D. Cowper (Methuen and Company,Ltd., Lo~don, 1926); M. v. Smoluchowski, A~handlungen ube«die Broumsche Bewegung una uenuandte Erscheznungen (Akade­mische Verlagsgesellschaft, Leipzig, 1923).

150

of a particle in a colloidal suspension, in dynamicalequilibrium between osmotic forces and gravity. How­ever, the electron in the hydrogen atom has other statesof dynamical equilibrium, at the usual discrete energylevels of the atom.

The equations of motion which we derive are non­linear, but if the wave function 1/; is introduced, in away simply related to the kinematical description of themotion, we find that 1/; satisfies the Schrodinger equa­tion. Every solution of the Schrodinger equation arisesin this way.

Our theory is by no means a causal theory, but proba­bilistic concepts enter in a classical way. The descrip­tion of atomic processes is by means of classical ideasof motion in space-time, and so is contrary to quantummechanics. However, we show that for observationswhich may be reduced to position measurements, thetwo theories give the same predictions. This, and a dis­cussion of von Neumann's theorem" on the impossibilityof hidden variables, is contained in Sec. IV. The sameargument shows that some features of our descriptionare incapable of observation. It is well known" thatmacroscopic Brownian motion imposes limits on theprecision of measurements if the measuring instrumentsare subject to it. If, as we are assuming, every systemis subject to a Brownian motion, this implies an absolutelimit restricting some measurements.

The discussion in this paper is restricted to the non­relativistic mechanics of particles without spin, in thepresence of external fields.

Our work has close connections with some previouswork on classical interpretations of the Schrodingerequation. A comparison with other hidden-variabletheories is contained in Sec. V.

II. STOCHASTIC MECHANICS

Stochastic processes occur in a number of classicalphysical theories.' Statistical mechanics is based on a

2 J. von Neumann, Mathematical Fpundations. of .QuantumMechanics, translated by R. T. Beyer (Pnnceton University Press,Princeton, New Jersey, 1955).

3 R. B. Barnes and S. Silverman, Rev. Mod. Phys. 6,162 (1934).4 A more detailed account of this subject is contained in lecture

notes by the author, Dynamical Theories of Brownian Motion(Princeton University Press, Princeton, New Jersey, to bepublished).

1079

1080 EDWARD NELSON 150

If x(t) is differentiable, then of course Dx(t) = D*x(t)=dx/dt, but in general D*x(t) is not the same as Dx(t).

The Omstein-Uhlenbeck Theory

As an example, consider the Ornstein-Uhlenbecktheory of Brownian motion with friction in the presenceof a potential V. We denote by x(t) the position of theBrownian particle at time t, by vet) its velocity, by mits mass, and by

where E, denotes the conditional expectation (average)given the state of the system at time t. Thus Dx(t) isagain a stochastic process. The symbol 0+ means thatL\t tends to 0 through positive values. Similarity, wedefine the mean backward derivative D*x(t) by

x(t)-x(t-At)D*xV)= lim E, . (2)

At~O+ L\t

the acceleration of the particle produced by V. Weassume that the system is in equilibrium, having theMaxwell-Boltzmann distribution. We let m(j be thefriction coefficient. Then the Langevin equations are

dx(t) = v(t)dt , (4a)

dv(t) = -(jv(t)dt+K(x(t»dt+dB(t). (4b)

Here B is a Wiener process representing the residualrandom impacts. The dB(t) are Gaussian with mean 0,mutually independent, and

(7)

(8)

(9)

Dx(t) = D*x(t)= vet) ,

Dv(t) = -l3v(t)+K(x(t».

Similarly, we find from (6) that

D*v(t)= {3v(t)+K(x(t».

In the case of a free particle (K= 0), we see thatDv(t) = -D.v(t) = -(jv(t). Because of the dampingeffect of friction, the velocity has a general trend toward0, no matter which direction of time we take.

It follows from (8) and (9) that

!DD.x(t)+!D.Dx(t) = K(x(t». (10)

where E denotes expectation (average), k is Boltz­mann's constant, and T is the absolute temperature.The dB(t) are independent of all of the xes), yes) withs:$t (see Doob's discussion"), There is clearly an asym­metry in time here: we may also write the Langevinequations with (4b) replaced by

dv(t) = (jv(t)dt+ K(x(t»dt+dB*(t) , (6)

where the dB*(t) are independent of all of the xes), yes)with s,?:-t.

If we apply our definitions (1) and (2), we find

a(t) = !DD*x(t)+!D.Dx(t) . (11)

If x(t) is a position vector, we call Dx(t) the mean for­ward velocity, D*x(t) the mean backward velocity, andaCt) the mean acceleration.

Thus we see that in the Ornstein-Uhlenbeck theory,Newton's law F=ma holds if F is the external force anda is the mean acceleration.

We define the mean second derivative of a stochasticprocess to be

sincex(t) is differentiable with dx/dt= vet).By (5), dB(t) is of the order dt1/2, so that B(t), and

consequently vet), is not differentiable. However,DB(t) =0, sinceB(t+.1t)-B(t) for .1t>O, is independentof the pair x(t), vet) (the state of the system at time t),and has expectation O. Since D is a linear operation,(4b) implies that

Kinematics of Markoff Processes

For a time scale large compared to the relaxationtime trt, the macroscopic Brownian motion of a freeparticle in a fluid is adequately described by the Wienerprocess wet). The dw(t) are Gaussian with mean 0,mutually independent, and

(3)

(5)

K= - (11m) gradV ,

EdB(t)2= 6«(jkT/m)dt ,

stationary stochastic process, the measure-preservingflow in phase space. The Einstein...Smoluchowski'theory of Brownian motion involves a Markoff processin coordinate space, and the more refined Ornstein­Uhlenbeck" theory of Brownian motion is expressed interms of a Markoff process in phase space.

Let x(t) be a stochastic process (for example, thex coordinate of a particle at time t). It is well knownthat for many important processes x(t) is not dif­ferentiable. This is the case for the Wiener process,6

which is the process occurring in Einstein's theory ofBrownian motion. The same difficulty occurs with thevelocity in the Ornstein-Uhlenbeck theory. To discussthe kinematics of stochastic processes, therefore, weneed a substitute for the derivative.

We define the mean forward derivative Dx(t) by

x(t+L\t)-xtt)Dx(t) = lim E, (1)

At~O+ At

6 See S. Chandrasekhar, G. E. Uhlenbeck, and L. S. Ornstein,Ming Chen Wang and G. E. Uhlenbeck, and J. L. Doob, inSelected Papers on Noise and Stochastic Processes, edited by N.Wax (Dover Publications, Inc., New York, 1954).

6 The Wiener process is discussed under the name "Brownianmotion" in J.L. Doob, Stochastic Processes (John Wiley & Sons,Inc., New York, 1953).

(12)

where v is the diffusion coefficient kT/m(j. (We write vinstead of D to avoid confusionjwith mean forwardderivatives.) In general, if there are external forces orcurrents in the containing fluid, the position xC!) of the

150 DERIVAT'ION OF SCHRODINGER EQUAT'ION 1081

(15) where we define

This description is asymmetrical in time. We may alsowrite

(27)

(25)

(26)u= v(gradpjp) ,

u=!(b-b*).

The distribution pd3xdt is invariant on space-time. Forthis reason," (a/at+b· V+ vl1) and (-a/at- b.· V+vil)are adjoints of each other with respect to pd8x dt. Thatis,

or

where the superscript + denotes the Lagrange adjoint(with respect to d3x dt). If we compute the left-handside of (24) and use the forward Fokker-Planck equa­tion (17), we find

b*= b-2v(gradpjp)

According to Einstein's theory! of Brownian motion,(26) is the velocity acquired by a Brownian particle,in equilibrium with respect to an external force, tobalance the osmotic force. For this reason, we call uthe osmotic' velocity, Notice that by subtracting (17)from (18) we obtain

O=div(up)-vdp=div[up-v gradp], (28)

which also follows from (26).By (26),

(16)

(14)Dx(t) = b(x(t),t).

where w* has the same properties as w except that thedw*(t) are independent of the xes) with s?.t. Thus

Brownian particle satisfies

dX(I) = b(x(t),t)dt+dw(t) , (13)

where w is as before and b is a vector-valued functionon space-time. This is the description used in the ap­proximate theory of Brownian motion due to Einsteinand Smoluchowski, and is the limiting case of the Orn­stein-Uhlenbeck theory for large (j. The dw(t) areindependent of the xes) with s~t, so by (1) b is the meanforward velocity:

is the mean backward velocity.We wish to study this type of process in some detail,

as we shall use this kinematical description for the mo­tion of an electron.

Let p(x,t) be the probability density of x(t). Then psatisfies the forward Fokker-Planck equation

ap/iJt= -div(bp)+vl1p, (17)

where 11= V2, and the backward Fokker-Planckequation

iJpjiJt= -div(b*p)- vl1p. (18)

The average of (17) and (18) yields the equation ofcontinuity

plus terms of higher order. By (13), we may replacethedxi(t) by dWi(t) in the last term. When we take theaverage [given' x(t)], we may replace dx(t)· Vf(x(t),t)by b(x(t),t)· Vf(x(t),t), since dw(t) 'is independent ofx(t) and has mean 0.' Using (12), we obtain

We call v the current velocity.Let f be a function of x and t. To compute D!(x(t),t),

expand f in a Taylor series up to terms of order two indx(t) :

ajdj(xtt),t)=-(xV),t)dt+dxV)·V j(x(t),t)

atiJ2f

+! L:dXiV)dxi(t)--(x(t),t) , (21)i.1 aXiaXj

III. THE HYPOTHESIS OF UNIVERSALBROWNIAN MOTION

(33)v=h/2m.

1 aa=- -,b+b*)+!(b·v)b.+!tb*·~)b

2 at-!'viltb- b*). -(31)

By (20) and (27),b= v+u and b*= v-u, so that (31)is equivalent to

,av/at=a- (v- V)v+(u· V)u+ vl1u. (32)

The constant h has the dimensions of action; we shallsee later that it can be identified with Planck's constantdivided by 2r.

7 E. Nelson, Duke Math. J. 25, 671 (1958).

We wish to examine the hypothesis that particles inempty space, or let us say the ether, are subject toBrownian motion. Macroscopic bodies do not appearto exhibit such behavior, so we shall assume that thediffusion coefficient v is inversely proportional to the­mass. We set

(19)

(20)

(23)

(22)

ap/at= -div(vp) ,

where we define vby

Df(x(t),t) = (a/at+b· V+ vil)f(x(t),t).

In the same way we obtain

D*f(x(t),t) =::(a/at+b*· V - vil)j(x(t),t).

1082 EDWARD NELSON 150

Then 1/; is real and p=1/;2. It is immediately seen that(37) is equivalent to the time-independent Schrodingerequation

By (26), the left-hand side is - f!mu2pd3x, so that

E=f ~U2Pd3X+f Vpd3x . (39)

for real 1/;. Thus for the hydrogen atom the hypothesisof Brownian motion leads to the correct energy levelsfor bound states of the atom, and interprets them asstates of dynamical equilibrium. This is discussedfurther in Sec. V.

where E is a censtant with the dimensions of energy. Ifwe multiply by mp and integrate, we obtain

f ~U2Pd3X-~f (u·gradp)d3x=f Vpd3x- E . (38)

(40)

(41)

(42)[ - (h2/2m)Li+ V ---E]1/;=O

R=! lnp

is the potential of mujh. Let

1/;=eR •

Thus E is the average value of !mu2+V, and so may beinterpreted as the mean energy of the particle. Noticethat in this case h= -b*=u, so we could replace u byb or b- in computing the mean kinetic energy. [If wecompute the mean of !m(dxjdt)2 there is the additionalinfinite term !m(dwj dt)2. Since this does not depend onthe potential V, differences of energy levels will not beaffected, and we can neglect it. In any case, the poten­tial V contains an arbitrary additive constant.]

The equation (37) is nonlinear, but it is equivalent toa linear equation by a change of dependent variable. By(29),

We cannot attribute any' friction to the ether, forthen we could distinguish absolute rest from uniformmotion. This means that the Brownian motion will notbe smooth, and velocities will not exist. Hence we cannotdescribe the state of a particle by a point in phase space.As in the Einstein-Smoluchowski theory, the motionwill be described by a Markoff process in coordinatespace, as in the previous section but with v=hj2m.

The mean acceleration a has no dynamical signifi­cance in the Einstein-Smoluchowski theory. That theoryapplies in the limit of large friction, so that an externalforce F does not accelerate a particle but merely impartsa velocity Fjm{3 to it. In our theory the dynamics willbe given by Newton's law F=ma. In other words, tostudy Brownian motion in a medium with zero frictionwe adopt the kinematics of the Einstein-Smoluchowskitheory but use Newtonian dynamics as in the Ornstein­Uhlenbeck theory.

Consider, then, a particle of mass m in an externalforce F. The particle performs a Markoff process, soits stage at time to is given by a point x(to) in coordinatespace. However, to know the particle's motion we alsoneed to know what the Markoff process is. That is, weneed to know h(x,t) and h*(x,t), or, equivalently,u(x,t) and v(x,t) for all t. But u and v satisfy (30) and(32) and the term a in (32) is known; it is Fjm. Thusu and v satisfy

oujot= - (hj2m) grad(divv)-grad(v·u), (34a)

ovjot= (ljm)F - (v- V)v+ (u- V)u+(hj2m)Liu. (34b)

Consequently, if u(x,to) and v(x,t o) are known. and wecan solve the Cauchy problem for the coupled nonlinearpartial differential equations (34), then the Markoffprocess will be completely known. Thus the state of aparticle at time to is described by its position x(to) attime to, the osmotic velocity u at time to, and the currentvelocity v at time to. Notice that u(x,to) and v(x,to) mustbe given for all values of x and not just for x(to).

which is the Euler equation of motion for a nonviscousincompressible fluid of unit density, if p is the pressure,Thus this situation appears to be related to diffusionprocesses in fluids with currents flowing in them. We donot want this, so we make the opposite assumption thatv is a gradient, and set .

The Time-Dependent Schrodinger Equation

There are in general other stationary (that is,aujat=o, ovjat=O) solutions of (34) for which v¢O(magnetic atoms). However, it is just as easy to discussthe general time-dependent case.

At this point we need a further kinematical assump­tion. Suppose we were to assume that divv= o. Since uis always a gradient, we could then write (34b) as

avjot=-(v·V)v-gradp, (43)

The Real Time-Independent Schrodinger Equation

Let us seek some special solutions of (34) in the casethat the force comes from a potential, F= -gradY.Suppose first that v= O. This implies, by (19) and (2.6),that p and u are independent of t and that the solutionis stationary. In this case (34a) says that aujat= 0 and(34b) becomes

u- Vu+ (hj2m)Liu= (ljm) grad V . (35)

By (29), .u is a gradient, so that (u- V)u=! grad u2 andLiu= grad(divu), Thus (35) becomes

grad(iu2+~ diVU)=~ gradV , (36)2m m

orIi 1 1

!u2+-divu=-V--E, (37)2m m m gradS= (mjh) v . (44)

150 DERIVATION OF SCHRGDINGER EQUATION 1083

We wish to show that 1/1 satisfies the Schrodingerequation

at/; It 1-=i-Lit/;-i-Vt/;+ia(t)l/J. (46)at 2m It

If we divide by t/;, take gradients, and separate real andimaginary parts: this is equivalent to the pair ofequations

[Since It/;12= p, if we multiply byit, integrate over space,and take real parts, we see that if (46) is satisfied thenaCt) is real. We can always choose the potential S sothat the phase factor a(t) is O.J That is, we wish to showthat

(58)e e (1 )-vX(curlv)=-vX(curlA)=- -vXH .me m c

where v is the current velocity. (The force should besymmetrical under t -1- -to Under time inversion,v~ -v and H~ -H, so that this is the case. Notethat u -1- u under time inversion.) To solve (34) withthis force, we proceed as before except that instead ofassuming the momentum mvto be a gradient, we assumethe generalized momentum mv+eAje to be a gradient.This assumption is gauge-independent. We set

gradS= (mjh) [v+ (ejme)A] , (52)

and define t/;=exp(R+iS) as before, where R is givenby (40).

We claim that l/I satisfies the Schrodinger equation

aif;= -~(-ihV _:A)2y;_ iecpif;+iaV)Y;. (53)at 2mli c It

[As before, aCt) must be real, and we can choose S sothat a(t)=O.] That is, we claim that

(aR as) Ii-+i- t/!=i-(LlR+iLlS+[grad(R+iS)]2)if;at at 2m

e 1 e+-[A· grad(R+ is)Jl/I+- - divAt/;

me 2 me

Now curI(v+eAjme)=O, so that

ie2 ie__-A2if;--cpy;+ia(t)l/I. (54)

2mlic2 Ii

Divide (54) by y; and take gradients. The real part thenyields (34a), as the two terms involving grad divAcancel and the two terms involving grad(A·u) cancel.If we use (50), the imaginary part yields, aftersimplification,

iJv e It-=-E+-Llu+! grad(u2)-! grad (v2) • (55)at m 2m

By the vector identity

! grad(v2)= vX (curl v)+ (v- V)v, (56)

and the fact that u is a gradient, this becomes

av e Ii-=-E-vX (curl v)+ (u'v)u- (v·V)v+--Liu. (57)at m 2m

(45)

Keeping R as before [Eq. (40)J, we let

1/I=eR+i S •

(aR as) Ii-+i- t/;=i-(LlR+iLlS+[grad(R+iS)]2)1fat at 2m

1-i-Vl/J+ia(t)t/;. (47)

It

au It-= --Llv-grad(v.u), (48a)at 2m

av Ii 1-=-Llu+! grad~u2)-!gradl.v2)-- grad]". (48b)at 2m m

Since u and v are gradients, (48) is equivalent to (34).Conversely, if we have any solution to the Schrodinger

equation (46), normalized so that f Iif; 12d3x= 1, we maywrite if;=exp(R+iS), u=hgradR/m, v=hgradS/m,b=v+u, b*=v-u, and p= 1if;12. The Markoff processwith diffusion coefficient h/2m, forward velocity b,and backward velocity b* has probability density p

and mean acceleration a= -gradV/m. That is, everysolution of the Schrodinger equation arises in this way.

The same considerations apply to systems of severalparticles with n-body potentials.

The Schrodinger Equation in an ExternalElectromagnetic Field

Finally, we consider the Brownian motion (withoutfriction) of a particle of charge e and mass m in thepresence of an external electromagnetic field. As usual,we denote by A the vector potential, by cp the scalarpotential, by E the electric field strength, and by H themagnetic field strength, so that

H=curl A,

1 aAE+--= - grade ,

c at

(49)

(SO)

Therefore (57) is (34b) with the force (51). As before,every solution of (53) arises in this way.

IV. COMPARISON WITH QUANTUMMECHANICS

where c is the speed of light. The force on the particle is

F= e[E+ (lje)vXH], (51)The theory we are proposing is so radically different

from quantum mechanics, and the latter is so well

1084 E I) WAR D N E L SO N 150

verified, that it should be a simple matter to show thatit is wrong. Consider, however, an experiment whichmight distinguish between the two theories. It canbe maintained. that all measurements are reducible toposition measurements (pointer readings). We consideran experiment which can be described within the frame­work of the nonrelativistic mechanics of systems offinitely many degrees of freedom.

At time to we have N particles with wave function tf;.We assume that they interact through given two-bodyand n-body potentials. We apply an external field andat a later time tl we measure the positions of the Nparticles (pointer readings, which may include a recordof readings made at previous times). We call U theunitary operator (depending on the external fieldchosen) which takes t/; into tJ;(tI), and if x is any coordi­na te of one of the particles we call the value of x attime ii, which we denote by ~, a primary observable. Ifit is true that all measurements can be reduced toposition measurements, then only primary observableshave an operational meaning as observables. In quantummechanics, ~ is represented by the self-adjoint operatorU-1xU. In stochastic mechanics it is represented by therandom variable X(tl). If ! is a bounded real function,we can make the measurement ~ and then compute !(~).

Quantum mechanics represents !(~) by the self-adjointoperator !(U-1xU) = U-lj(X) U, so that the expectedvalue of !(~) is [t/;,U-lj(X) U)] = [t/;(tl),!(X)tJ;(t l)], whichis equal to

(59)

where z represents the remaining coordinates. Stochasticmechanics represents !(~) by the random variable![X(tl)], so that the expected value of !(~) is again(59). So far, therefore, the statistical interpretations ofthe two theories agree. If one accepts the view that allmeasurements are reducible to position measurements,this means that the two theories give the samepredictions.

Suppose now that we had performed a differentexperiment, applying a different external field givingthe unitary operator V, and observing a possibly dif­ferent coordinate y at time is. Call this observation 'YJ.Quantum mechanics represents this observable by theself-adjoint operator V-lyV, stochastic mechanics by arandom variable y(t1), and again the statistical in­terpretations agree.

Now consider ~+1]. This has no clear operationalmeaning. If we perform an experiment to measure ~, itis not clear that the question of what would have hap­pened had we performed a different experiment tomeasure 1] has meaning. However, in quantum me­chanics it. is customary to consider the operatorU-1XU+ Y-lyY as corresponding to the "observable"~+1]. (Considerations of domains of operators areirrelevant here. We may replace x and y by bounded

functions of x and y.) The statistical interpretation isthe same as for primary observables: If ! is a boundedreal function, the expected value of f(~+'YJ) is[tJ;, !(U-1xU+ y-lyV)tJ;J.No operational meaning isassigned to this definition of ~+1]. It is the naturaldefinition in the framework of operators on Hilbertspace.

Stochastic mechanics also leads to a natural interpre­tation of ~+1]. The observable ~ is represented by therandom variable x(tt) and 'YJ is represented by the randomvariable y(tl), so we may represent ~+1] by the randomvariable X(tl)+y(tl). The random variables X(tl), y(tl)depend on the different external fields, but they aredefined on the same probability space, that of theWeiner processes for the N particles, so that it ismeaningful to add them. If f is a bounded real function,the expectation of ![X(tl)+y(tl)] is well-defined. Nooperational meaning is assigned to this definition of~+'YJ. It is the natural definition in the framework ofstochastic processes.

However, the statistical interpretations of the twodefinitions of ~+ 1] do not agree. In fact, we may easilyfind U and Y (unitary operators, arising from externalfields, which connect the wave functions at time toandtl) such that U-1xU and Y-lyV do not commute. Thismeans (and this is the content of von Neuman's proofof the impossibility of hidden variables) that for asuitable t/; there is no pair of random variables X(tl),y(tl) such that aU-1xU+fjY-lyY in the state withwave function t/; has the same probability distributionas ax(tl)+fjy(tl), for all real a and {3.

The situation is clarified by considering the motionof a free particle. Let ~(t) be the position of the particleat time t, In the quantum-mechanical description, letX(t) be the position operator at time t and in thestochastic description, let x(t) be the. random variablegiving the position at time t, The x(t) are the randomvariables of a Markoff process, the precise form of whichdepends on the wave function t/; at a given time.

Now consider two different times tl and 12- We know,by von Neumann's theorem," that aX(tI) + (1-a)X(t2)

cannot give the .same statistics as ax(tl) + (1-a)x(t2)

for all O!. On the other hand, since the particle is free,

where t3=atl+(1-a)t2, and we have seen that X(ta)gives the same statistics as xC/a). There is no contradic­tion because ax(tl)+ (1-a)x(t2) and X(t3) are differentrandom variables, although their expectations are thesame, for any t/;.

The quantum and stochastic descriptions give dif­ferent statistics for the triple~(tl), ~(t2), ~(ta), but thequestion as to which description is correct is moot unlessan operational meaning can be given to a~(tl)+ (1-a)X~(t2). Von Neumann, in his proof of the impossibilityof hidden variables, explicitly assumes that there is aone-to-one correspondence between observables and self-

150 DERIVATION OF SCHRODINGER EQUATION 1085

adjoint operators, but. this is not a necessary require­ment for a physical theory. It should be remarked,however, that the class of Hamiltonians that we havebeen able to treat by the stochastic method is quitelimited, all of them being of second order in themomentum.

One conclusion which may be drawn from this isthat the additional information which stochasticmechanics seems to provide, such as continuous tra­jectories, is useless, because it is not accessible to experi­mental varification.

Another conclusion which this analysis suggests isthat the practice of regarding the sum of two non­commuting operators corresponding to two observablesas being the operator corresponding to the sum of theobservables is a matter of convention which is not ac­cessible to experimental verification. If one wishes, forconvenience, to add observables, stochastic mechanicsprovides a simpler (although by now less familiar)framework than does Hilbert space, a framework inwhich observables are represented by random variableswhich may be freely added and multiplied together,and which all have joint probability distributions.

v. DISCUSSION

In this paper we have made no attempt at mathe­matical rigor. In deriving the Schrodinger equation weassumed u, v, and p to be smooth functions. This is thecase if and only if 1/1 is smooth and p= 11/1/ 2 does notvanish [see Eq. (40)]. For real solutions of the time­independent Schrodinger equation, other than theground-state solution, p has nodal surfaces on which ubecomes infinite. The definition (41) produces an equiva­lence between Eqs. (37) and (42) only within a regionbounded by nodal surfaces. However, it can be shownthat the associated Markoff process is well defined ineach such region, and that a particle performing theMarkoff process never reaches a nodal surface. Thequestion remains as to the proper definition of pthroughout space, which is essential for the identifica­tion of the constant E as the mean energy. Let us takethe smooth solution 1/1 of the Schrodinger equation (42),let Y;o be the ground-state solution, .and let

(61)

The solution of the time-dependent Schrodinger equa­tion with 1/IE as initial value corresponds to a Markoffprocess in which u becomes infinite on a surface onlyat isolated times, and there is no problem with unique­ness. The probability density is PE = I1/;EI2. As E tends to0, this Markoff process becomes approximately station­ary, and has as limit the Markoff process associated withif; and probability density P= /1/11 2• We could also think.of taking a different wave function Y;1 which in eachregion bounded by nodal surfaces is a constant multiple

of expR, but then if we approximate 1/;1 by a smooth Y;1ewith nonvanishing 1Y,IE I2, the corresponding Markoffprocess will no longer be approximately stationary. Forthis reason we take the usual probability densityp= 11/11 2

, and so have the usual energy levels.An interpretation of the Schrodinger equation in

terms of particle trajectories was first proposed by deBroglie" and later developed by Bohm," In this work theparticle velocity was identified with what we call thecurrent velocity v, and Bohm interpreted the deviationfrom the Newtonian equations of motion as being due toa quantum-mechanical potential associated with thewave function. Bohm and Vigier'? introduced 'the notionof random fluctuations arising from interaction with asubquantum medium. Since completing this work, theauthor became aware of the work of Fenyes," Weizel,"and Kershaw.PFenyes showed that instead of assuminga quantum-mechanical potential the motion could beunderstood in terms of a Markoff process. This workwas developed byWeizel, who also proposed a model forthe random aspects of the motion in terms of interactionwith hypothetical particles, which he calls zerons. Thecase v=O was also discussed by Kershaw. The theorywhich we have developed is just the Fenyes-Weizeltheory from a different point of view. Our aim has beento show how close it is.to classical theories of Brownianmotion and Newtonian mechanics, and how theSchrodinger equation might have been discovered fromthis point of view.

A formal analogy between Brownian motion and theSchrodinger equation was noticed by Furth'! and de­veloped recently by Comisar." In this work the diffusioncoefficient is imaginary. A parallelism between quantummechanics and stochastic processes has been noticed ina number of recent papers."

Only a very small part of the subject matter of quan­tum mechanics has been discussed here. Relativity,spin, statistics of identical particles, and systems ofinfinitely many degrees of freedom have all beenignored. Consequently, no firm conclusions can bedrawn. However, it appears that the phenomena whichfirst led to the abandonment of classical physics admita simple classical interpretation which is only in alimited sense equivalent to quantum mechanics.

8 L. de Broglie, Compt. Rend. 183,447 (1926); 184,273 (1927);185,380 (1927); Etude critique desbases de l'interpretation actuellede la mecanique ondulatoire (Gauthiers-Villars, Paris, 1963).

9 D. Bohm, Phys. Rev. 85, 166 (1952); 85, 180 (1952).10 D. Bohm and J. P. Vigier, Phys. Rev. 96, 208 (1954).11 1. Fenyes, Z. Physik 132,81 (1952).12 W. Weizel, Z. Physik 134, 264 (1953); 135, 270 (1953); 136,

582 (1954).13 D. Kershaw, Phys. Rev. 136,B1S50 (1964).14 R. FUrth, Z. Physik 81, 143 (1933).15 G. G. Comisar, Phys, Rev. 138,B1332 (1965).16 P. Braffort and C. Tzara, Compt. Rend. 239 157 (1954); P.

Braffort, M. Surdin, and. Adolfo Taroni, ibid. 261, 4339 (1965);T. Marshall, Proe. Cambridge Phil. Soc. 61, 537 (1965); R. C.Bourret, Can. J. Phys. 43, 619 (1965).