arXiv:quant-ph/0112114v1 20 Dec 2001 yWee n igl[,p 4]adas yBl 7.W ontcnitwit conflict not do We [7]. Bell by also p. and [6, 148] variables p. hidden [5, Siegel on and theorem Wiener non eve famous by be any Neumann’s must Von In operators with Hermitian mechanics. conflict rand and seem quantum variables provides theory stochastic ransom between that in [5] rule in paths Siegel sample space and of probability stochastic space the Wiener of but space of whethe operators, probability known theory Hermitian the not space is ob enlarging differential mechanical It without The quantum out momenta. general carried and a coordinates be of both that of as function same the is distribution orsodn unu ehncloeaos o h ruds example. ground an the as For calculated explicitly a are operators. of mechanical case quantum st the corresponding been to in work has Shucker’s coordinates problem of desir this as generalization particle, is a well free It here the as present of We momenta space. case with the Hilbert For con a associated mechanics. sharp on be in operators can is ar by variables momenta position represented and of momentum being coordinates role the where both elevated than mechanics quantum role This of fundamental formalism theory. more a this describ plays in being position observable motion The its time-depen space, 2]. a in [1, is path continuous position a particle’s traverses a which mechanics quantum stochastic In Introduction 1 mi:[email protected],Wb www.spectelresearch.co Web: [email protected], Email: ∗ † twudb neetn ofida loih o osrcigarando a constructing for algorithm an find to interesting be would It admvralsaedfie eewoedsrbtosaetesame the are distributions whose here defined are variables Random etr nMteaia Physics Mathematical in Letters urn drs:SetlRsac oprto,87RreWa Rorke 807 Corporation, Research Spectel Address: Current eateto hsc,SnJs tt nvriy a Jose, San University, State Jose San Physics, of Department rirr oeta nryfntosaecniee.Teo The considered. are example. functions energy potential Arbitrary oetmi nlzda admvral nsohsi quan stochastic in variable random a as analyzed is Momentum oetmi tcatcqatmmechanics quantum stochastic in Momentum 5 18)523–529. (1981) akDavidson Mark Abstract 1 Copyright †
m c ,Pl lo A94303 CA Alto, Palo y, 1981 clao speetda an as presented is scillator .Rie ulsigCompany. Publishing Reidel D. aeo h siltrthey oscillator the of tate btaypotential. rbitrary ovdb hce 3 4]. [3, Shucker by solved A912 U.S.A. 95192, CA db akvprocess Markov a by ed lna nodrt avoid to order in -linear ob agrta the than larger be to s t h correspondence the nt, ntesm footing, same the on e bet e frandom if see to able uhapormcan program a such r evbe ngnrla general in servable, etrno variable random dent unu mechanics. quantum u mechanics. tum mvralsfrall for variables om 0] on made point a 209], rs oteusual the to trast catcquantum ochastic rta n other any than or o Neumann’s Von h aibewhose variable m stoeo the of those as ∗ theorem here since we do not assert that the sum of two non-commuting operators should be represented by the corresponding sum of their random variables. The random variable for momentum which is presented here is quite complicated except for the case of a free particle. We find that in general it cannot be determined analytically and that it depends upon the state of the quantum mechanical system. Nevertheless, its existence is guaranteed in a wide range of cases, and it may yield some insight into the stochastic interpretation of quantum mechanics. The results presented are derived for the generalized F´enyes–Nelson model [8]–[10] where the diffusion parameter is arbitrary. Various other applications of stochastic quantum me- chanics are discussed in [11]–[17].
2 Momentum as a random variable
Consider Schr¨odinger’s equation 1 ∂ − ∆+ V (x) ψ = i ψ (1) 2 ∂t where ∆ is the N-dimensional Laplacian, x is an N-dimensional coordinate, and V a potential 1 function. Units have been chosen to make the coefficient of ∆ equal to − 2 . Suppose that at time t0 the wave function is given by
ψ(x, t0)= ψ0(x). (2)
Then Schr¨odinger’s equation may be integrated, provided ψ0 is sufficiently regular. A stochastic model of Schr¨odinger’s equation is constructed by defining R and S by ψ = eR+iS (3) and identifying x with the stochastic process defined by the following stochastic differential equation dx = b(x, t)dt + dW (4) where b =2ν∇R + ∇S (5) and where the diffusion parameter ν is defined by
E (dWidWj)=2νδij dt. (6) In the most general version of stochastic quantum mechanics [8]–[10], this diffusion parameter 1 is arbitrary. For Nelson’s original theory [1, 2] ν in (6) must be 2 . Now, consider a free particle equation with the same starting conditions (2): 1 ∂ − ∆ψ = i ψ . (7) 2 F ∂t F Then this free-particle equation also admits a stochastic model defined by the equation
dxF = bF dt + dWF (8)
2 where RF +iSF ψF = e and bF =2ν∇RF + ∇SF . (9) Shucker [3] has proved the following result for the free-particle Schr¨odinger equation: x (t + T ) lim F 0 = P T →∞ T exists and has a probability density given by ∗ ψ (P )ψ(P ) 3 −iP ·x ρ(P )= 3 , ψ(P )= d x e ψ(x, t0). (10) (2π) Z Shucker’s results apply equally well to the general version of stochastic quantum mechanics as was shown in [4]. Note that ρ(P ) is the same quantum mechanical momentum density as the original problem with a potential at time t0 since the free and interacting wave functions are both the same at this time. Now consider (4) and (8). By fiat, require that the Wiener processes in these two equa- tions be the same. That is, impose the condition
W = W F . (11)
Then xF and x are related by
dx − b(x, t)dt = dxF − bF (xF , t)dt. (12)
By writing (12), as two equations
dxF = (−b(y, t)+ bF (xF , t))dt + dx (13)
dy = dx, y(t0)= x(t0) (14) then we obtain the standard form for a multidimensional stochastic differential equation, and since the sample paths of x are the same as those of a simple Wiener process, we may solve for xF in terms of x by iteration. This application of the Picard method was pointed out for a similar problem by Klauder [18] who also presented a perturbation method for finding a solution. Klauder’s interaction picture method may be useful for finding approximations to the momentum in cases of weak potential. The solution to (13) and (14) is obtained by iterating the following equation
t
xF (t)= x(t)+ [bF (xF , t) − b(x, t)] dt. (15) tZ0 Convergence of this iteration is ensured by the contraction mapping theorem if the following global Lipschitz condition is satisfied
|(bF (y1, t) − b(x1, t)) − (bF (y2, t) − b(x2, t))|
3 for all x1, x2,y1, and y2; and all t, and where k is a constant. See for example [2, p. 43]. Note that we have by fiat imposed the condition xF (t0)= x(t0) (17) which is consistent because both processes have the same probability density at t = t0. If we let (Ω, Σ,P ) denote the probability space for the process x(t), and if ω ∈ Ω, then x(ω, t) is a sample trajectory. The above analysis provides an iterative algorithm for calculating xF (ω, t). Thus the free particle’s sample trajectory is expressed as a random variable on the probability space of the interacting particle. Now we need only apply Shucker’s theorem. Let xF be calculated by (15) and consider the limit xF (ω, t0 + T ) P (t0) = lim . (18) T →∞ T Then, by Shucker’s theorem [4], P exists and has a probability density given by (10). Thus we have constructed a random variable for momentum whose distribution is the same as the quantum mechanical one for a system with an arbitrary potential. We next present a simple example to illustrate this technique. 3 The Oscillator Consider the ground state of a one-dimensional harmonic oscillator which satisfies the equa- tion 1 ∂2 1 − + x2 ψ = Eψ. (19) " 2 ∂x2 2 # Up to a normalization constant, the solution to (19) is exp −x2/2 (20) h i so that b(x)= −2νx. (21) Now consider the free particle solution with initial conditions at t = t0 given by (20): 1 2 ψF (x, t) = exp − x /(1 + i(t − t0)) (22) 2 so that from (9) we have 1 R = − x2 /(1+(t − t )2) (23a) F 2 F 0 1 S = x2 (t − t )/(1+(t − t )2) (23b) F 2 F 0 0 2ν − (t − t0) bF = −xF . (24) " 1+(t − t0) # 4 Then applying (12) yields 2ν − (t − t0) dx +2νx dt = dxF + xF 2 dt (25) 1+(t − t0) which may be directly integrated to yield x(t) −γ(t) −γ(t) γ xF (t) = e x(t0)+ e e dx + x(Zt0) t ′ +2νe−γ(t) eγ(t )x(t′) dt′ (26) tZ0 where t ′ 2ν − (t − t0) ′ γ(t) = ′ 2 dt 1+(t − t0) tZ0 1 = 2νA tan(t − t ) − ln(1 + (t − t )2). (27) 0 2 0 The momentum may be calculated from (26). The result, after some manipulation, is ∞ −νπ γ(t) ∂γ P (t0)= e x(t)e 2ν − dt. (28) ∂t ! tZ0 One can check that this expression for P has the appropriate density by noting first that it has a Gaussian distribution and then by calculating its variance. Using (27) together with the fact that 1 E(x(t )x(t )) = exp [−2ν|t − t |] (29) 1 2 2 1 2 2 1 one finds that E(P )= 2 as expected. 4 Conclusion The technique presented here allows one to consider momentum as a random variable on an equal footing with position in the stochastic interpretation of quantum mechanics. It is easily generalized to allow for magnetic fields also. The momentum depends not only on the sample path in question, but also on the state of the system. Of course, this momentum variable satisfies the Heisenberg uncertainty principle with the coordinate. It is hoped that these results will facilitate the construction of random variables for still more general operators involving both coordinates and momenta. 5 References [1] Nelson, E., Phys. Rev. 150, 1079–1085 (1966). [2] Nelson, E., Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, 1967. [3] Shucker, D., J. Funct. Anal. 38, 146–155 (1980). [4] Shucker, D., Lett. Math. Phys. 4, 61–65 (1980). [5] Wiener, N., Siegel, A., Rankin, B., and Martin, W.T., Differential Space, Quantum Systems, and Prediction, MIT Press, Cambridge, 1966. [6] Von Neumann, J., Mathematical Foundations of Quantum Mechanics, Princeton Uni- versity Press, Princeton, 1955. [7] Bell, J.S., Rev. Mod. Phys. 38, 447–452 (1996). [8] Davidson, M., Physica 96A, 465–486 (1979). [9] Davidson, M., Lett. Math. Phys. 3, 271–277 (1979). [10] Davidson, M., Lett. Math. Phys. 4, 475–483 (1980). [11] Yasue, K., Prog. Theor. Phys. 57, 318 (1977). [12] Moore, S.M., J. Math. Phys. 21, 2102 (1980). [13] Caubet, J.P., Le Mouvement Brownien Relativiste, Lecture Notes in Mathematics Vol 559, Springer-Verlag, Berlin, 1976. [14] Guerra, F. and Ruggiero, P., Phys. Rev. Lett. 31, 1022 (1973). [15] Dohrn, D. and Guerra, F., Lett. Nuovo Cimento 22, 121 (1978). [16] De Angelis, G.F., De Falco, D., and Guerra, F., ‘Probabilistic Ideas in the Theory of Fermi Fields, I. Stochastic Quantization and the Fermi Oscillator’, Salerno-Princeton- Rome preprint, October, 1980. [17] Guerra, F. and Loffredo, M.I., ‘Thermal Mixtures in Stochastic Mechanics’, Rome preprint, November 1980. [18] Klauder, J., ‘Interaction Picture for Stochastic Differential Equations’, in L. Streit (ed.), Quantum Field-Algebras Processes, Springer-Verlag, Berlin, 1980, p. 53. 6