LETT]~I~E A]L NUOVO CII~I]~NTO VOL. 36, ~. 14 2 Apr i le 1983

Momentum from Sample Paths in Stochastic Mechanics (*).

D. DE FAT,CO, S. D~ 1V[ARTI~O and S. DE SIENA

Ist i tuto di Eisica della FaeoIt~ di Scienze dell' Universit~ - 84100 Salerno, I tal ia

@icevuto i l 13 Dicembre 1982)

PACS. 03.65. - Quan tum theo ry ; q u a n t u m mechanics.

Summary . - Accord ing to Shucker ' s analysis , quan tum-mechan i ca l m o m e n t u m can be read f rom the a sympto t i c behav iour of Nelson ' s sample paths. Here we use t he po s i t i on -momen tum unce r t a in ty p roduc t as a measure of the ra te of approach to the large t i m e asymptot ics .

S tochas t ic mechanics , in i ts or ig inal fo rmula t ion (~), heav i ly relies on the indi- v idua t ion of conf igurat ion observables. I n th is f r amework quan t i za t ion consists in fact in suitably r e in te rp re t ing the classical conf igurat ion observables as r a n d o m var iables whose t i m e evo lu t ion i s to be read f rom the Lag range equa t ions of mot ion rewr i t t en as dynamica l condi t ions on the corresponding s tochast ic process. F o r a t r anspa ren t analysis of th is approach we refer to ref. (2), on which we also re ly for t he no ta t ions no t expl ic i t ly defined in w h a t follows.

W e res t r ic t our a t t en t ion to the s imple case of a po in t par t ic le of mass m, m o v i n g on the real l ine in a po t en t i a l V.

To each solut ion F(x, t ) of the SchrSdinger equa t ion for such a sys tem, Nelson ' s procedure associates a diffusion process q(t) whose expec ta t ion , for eve ry con]igurational obse rvab le ](x) satisfies

(1) E( / (q( t ) ) ) = <v,(x, t ) , / (x) ~(~, t)>.

Namely , one can answer any ques t ion abou~ con/igurational observables at a g iven t i m e t by s imply pe r fo rming s ta t i s t i ca l e labora t ions (for shor t : (( measuremen t s >)) on the sample pa ths of t he process q at t i m e t (such as, for ins tance , genera ted by numer ica l s imulat ions) . Here we discuss t he fo l lowing p rob lem: how can one read in fo rma t ion about momentum observables f rom (< measurements) ) (in the above- res t r i c ted sense) on the sample pa ths of t he process q(t)?

(*) Research supported in part by Ministero della Pubblica Istruzione. (1) E. NELSON: Connection belween Brownian motion and quantum mechanics, in Einstein Symposium, edited by It. NELKOWSKI (Berlin, 1979). (~) F. GUERRs Phys. Rep., 3, 263 (1981).

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~5~ D. I)E FALCO, S. I)E 1VIARTINO and s. DE SIENA_

In ref. (a,4) D. SHVCK]~X~ presents an analysis of how information about momentum is encoded in the stochastic processes q(t) and simultaneously exhibits the intrinsic probabilistie significance of the quantum-mechanical procedure of reading the mo- mentum distribution from the Fourier transform of the wave function. Here we briefly review his approach.

i) If t is the t ime at which the momentum distribution is requested, construct the h'ee process q~~ <~ tangent ~> to the process q(t) at t ime t, as the process which evolves in the absence o] potential starting from init ial data at t ime t determined by the probabil i ty density and the mean forward velocity field associated to the process q(t) at t ime t;

ii) By Nelson's correspondence the process qfr~r can be studied by means of the associated solution ~fr,~ of the ]ree SehrSdinger equation which starts from the ini t ia l data ~f~,~(x, t) = ~(x, t).

iii) Under mild technical assumptions, for almost every sample path ~o(t) of the process qf~(t) the following l imit exists:

(2) zt(t, r = l i ra m ~.---~+,:o T

e(~ + T) -- e(~)

The key to the proof of the existence of this l imit is the observation that the meaa forward velocity field v~e'(x,t) for qf~r behaves for large t as

(3) ~7O~ t) = ~/t + o(t=~lr t)l=~).

iv) Granted the existence of the l imit 2), the fact that

(4) p2 T . T ) = t)

(where ~(p, t) is the Fourier transform of ~(x, t)) shows that the probabil i ty density of the random variable

(5) ~(~) -= ~(~, ~) is t~(p, bl ~-

The random variable =(t) is, therefore, to be identified with the image of quantum- mechanical momentum in Nelson's correspondence.

I t is interesting, from the point of view of analysing how realistically the previous approach could be implemented in a numerical simulation of the stochastic process q(t),. to estimate the rate of convergence in eq. (2). This problem we study first of all oa an explicit example.

Let the wave function at t ime t ~ 0 be

(6)

where

(7)

~(x, 0) = (2xa)-i exp [-- x~/4a],

= ( A x p .

(~) D. SHUOKEa: Stochastic mechanics, Princeton Thesis (1978). (4) D. SHUO]KER: J. Funct. Anal., 38, 146 (1980).

MOMENTUM FROM SAMPLE PATTIS IN STOCHASTIC MECHANICS 4 5 9

The mean forward ve loc i ty field for the process qfr~r evo lv ing f rom the in i t ia l condi- t ion corresponding to eq. (6) is

(s) v~'~(x, t) = t~/m [ R e { 1 L \~.oo(x,

) t) a x ~ ~ +

+ I m ~fr~(X, t) = x ~ ( x , t) ~x h2t 2 § 4 . 2 m 2 "

The process q~rr162 evolves therefore according to the s tochast ic d i f ferent ia l equa t ion

h 2 t - - 2mh~ (9) dq~r -- h~t 2 + 4~.2m 2 qfr~(t) dt + dw

(where dt > 0, and dw represents inc rements of Brownian mo t ion wi th diffusion coef- f icient h/2m) .

F r o m eq. (9) i t fol lows t h a t t he two-po in t func t ion of the process satisfies for t > 0 the different ial equa t ion

(lo) d h ~ t - - 2mh~ dt E(qfr~176 ) -- h 2 t 2 + 4~2m 2 E(q~o(O)q~,,oe(t) ) ,

whose expl ic i t solut ion, w i th the in i t ia l condi t ion E(qf~r 2) = c~ is

(1t) E(q~,~e(O)qfr~o(~) ) = ~(1 + t~2t2/4c~2m2)~ exp [ - - a r e t g (ht /2me)] .

Defining

(12)

f rom eq. (11) we ob ta in

(Aygt)2 ~ J~ ( (~b (q f r ee ( t ) - qfree(0))/t)2) ,

( )~ (13) AxAT~ = h/2 1 + 2 (T-12(1 - - ~/1 + (t/T) 2 exp [-- arc tg (t/T)]) , \ t]

where

(14) ~ = 2 m ( A x ) e / h .

The main features in eq. (13) are the following:

a) as t--+ + co, AxAJr t goes to t he q u a n t u m - m e c h a n i c a l u n c e r t a i n t y p roduc t A x A p (which in the pa r t i cu l a r case at hand is h/2);

b) as t - ~ 0% AxA~r t diverges as t - l ;

c) t he l im i t in a) is r eached ] tom below;

d) a f te r a t i m e of order T = 2m(Ax)2/h , AxA~ t is (wi thin a few percent) v e r y close to A x A p .

Going beyond the expl ic i t example considered up to now, we observe t h a t a, is a genera l consequence of Shueker ' s analysis. I t can also be der ived d i rec t ly f rom the

4 ~ 0 D. DE FALCO, S. DE MARTINO a n d s . DE SI:EN~K

explicit relation

(15) (Az~t)2 - - (Ap)~ : + o) - g oo(t)) t 2

where v~(x , t ) is the current velocity field associated to the process q~o(t). Equat ion (15) also shows the generality of observation b), which can be traced

back to the Brownian nature of the random noise present in Nelson's theory. As to the generali ty of the intr iguing point c), we observe that a necessary and

sufficient condition for Azt to reach its l imit Ap from below is that , for every t large enough

t

ee< froo, , , fre ) (16) E v~ - - , , ++ (+reo(0) 0 ) ) dr > C 2 , ~ .

o

Condition (16) lends itself to the suggestive interpretat ion that , at least in some cases, such as the explicit example discussed above, part of the uncertainty in momentum builds up as a cumulat ive effect of small fluctuations over large times.

As to observation d), we observe that , inserting AxAp = h/2 in eq. (14), it can be rephrased to say that , in order to read from the sample paths of the process q~,.ee(t) momentum and, therefore, energy with the accuracy permit ted by quantum mechanics, one must follow the sample paths of q~roe for a t ime of order ~ such that z A E ~ h . This is strongly reminiscent of the t ime-energy uncertainty relations, with the addi- t ional identification of the t ime At necessary to identify the quantum state of the system with the t ime T after which Brownian fluctuations become negligible enough that one can read momentum from the finite t ime incremental ratio m(co(t + z)--~o(t))/v. As such, the generali ty of d) presents itself as a st imulating conjecture.