Revista Mexicana de Física 31 No. 4 (1985) 551-574 SSl

THE STOCHASTIC ROAD TO QUANTUMMECHANICS: AN EXPERIENCE t

L. de la Peña and A.M. CettoInstituto de Fisica, UNAM

Apartado Postal 20-364, México 20, D.F., C.P. 1000 México

ABSTRACf

The work perforrned by our group in the last 16 years in connec-tion with the íoundations oí quantum mechanics can be roughly divided iníour stages: The preparatory or introduetory stage, the period oí stoc-hastie quantum meehanies, the period oí stoehastie electrodynamics andfina11y, the present stage. In this paper we present a surnmary oí thedevelopment and main results of the various stagesi however, in view oíthe minar significance of the first one, and of the excellent recen sum-mary by Brody(1) oí the second and third, we shall pay special attentionto our current work, which consists basieally in the construction of anew version of stochastic electrodynamics.

RESUt-lEN

El trabajo realizado por nuestro grupo en los últimos 16 añosrelativo a los fundamentos de la mecánica cuántica, puede dividirse agrandes rasgos en cuatro etapas: la preparatoria ° introductoria, la dedi

t Presentado en la asamblea general ordinaria de la SMF el 21 de junio de19B4.

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cada a la I~cánica cuántica estocástica, la dedicada a la electrodinámi-ca estocástica y, finalmente. la etapa actual. En este trabajo se pre-senta un resume~ breve del desarrollo y los resultados principales de c~da una de ellas; sin embargo. dados el significado menor de la primeraetapa y la publicación reciente de un excelente trabajo de Brody(l) enque se revisan la segunda y tercera etapas, aquí se presta atención esp~cial al trabajo actual del grupo, el quP consiste en 10 fundamental en1m primer intento de construcción de una nueva versión de la electrodin~mica estocástica.

1. l'RELI~Il:-'[ARYSTAGE: BROI\~lA~ A'IIAl0GY.

Our \\'01).; 00 the fOlmdations of qu;mtum mcchanics ini tiatcd inthe late 60' s, was J'OOtiv3tedby a general statc of intel1ectual dissatisfaetian with the orthoc~x views on quantum mcchanics. By that time ex-tensive Jiseussions and nl~eTOUS controversies on the intcrprctations ofquantum mech~ics h.1darisen(2); but therc existed fcw, scarcely k-nownand unaccomplisheJ efforts to construct a fundamental thcory that wouldexplain the qu:mtum fonnalism. On analysing the various facts of quan-turo mechanics it gradualIy became clcar that stochasticity should playan important role in any causal ano objective dcscription of the quantumphenomenon. This feeling '..•.as reinforccd by our rather accidental ac-quaintancc Id th Fénycs1 work(:;}, ,,-hich represents a serious effort to

underst.::mdqu;mtum~ChMics in teI1lL'iof a stachastic J-tarkovian behav-iour oí the electron.

This line of research was stimulatcd, but at the same tin~ limited, by the analo~~ with Brownian moticn, by far the best-known stochas-tic proccss at the tin'('. Indced, it leJ to the identification af COfTlnxm

fcatures of quantum mechanies and Markavproeesscs, and to the use of amcthodalogy that had been esscntially foreignt() qU<lIltWTl thcory(4). Ilowcver, the close analo&')'oet\\'ccn the sto<.:hastic clcetron nml the Browninnparticle raiscd several questions whieh in the framcwark of this primi-tivC' treatmcnt cOlJld not he solvcu satisfactorily, n~ly: Which is thecssential differenee bctwcen a classical anu a quantum Brownian JlX)tion?In the frietionless quantum rrotion,ho1" is it thlta stationary state canbe reached?

It soon bccame clear that so~ ftmdamcntal points wcrc wrong al"

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at least obscure, since any theory aimed at a bettcr understanding ofquantum mechanics should pp.rmit to distinguish clearly a quantum systemfrom a classical ene.

Unfortunately the confusio~ still prevails -it is present evenin Jammerls excellent book-(2) to the extent that special terminologyhas been suggested to describe the "Brownian IOOtion" oE the electron;this widespread confusion adds its share to the aprioristic attitude ofm1.I1y nonspecialist against the possibility of a causal, stochastic expl~nation of quantum mechanics.

2. STOOV\STIC QUANTUl-1~IECHANICS

As a rcsult of this work it seemed necessary to construct amore elaborate formalism that couId serve to describe a more generalstochastic process (in the ~mrkovian approximation), and to distinguishbetv.¡eenBrO\~ian ITOtion and quantum mechanics as two different physicalsituations. The intention was to show that quantum mechanics can indcedbe understood as the result oí a specific stochastic process added uponotherwise cIassical laws of mation, without yet inquiring into the ori-gin of stochasticity; stochastic quantum mechanics has therefore an cx-plicitly declared phcnomenological character.

The closest antecedent of our work in this direction is Nclson'swell-knawn Markovian theory of quantum mechanics(5,6), which had clear-Iy established the possibility oí a phenomenological treatment in coor-dinate spacc similar to the Einstcin-Smluchcwski treatment of Brov..nianITOtion. Based on the introduction of two different derivatives of alocal function with reference to the initial and to the final paint, wedevelopcd a kinematics that would serve to describe a general ~nrkovianprocess in coordinate space(7). In this description, in addition to thedrift or systematic time derivative:

a/at + V'Vthere appears the osmotic OY stochastic time derivative:

(la)

(lb)

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wherc

~=lW£np (2)

is the stochastic velocity,D is the diffusion coefficient and p is thedcnsity oí particles. Four different accelerations may therefore bedefined, in terms oí which ene can write the stochastic generalizatíanoí Newton's second law:

(3)

As a result oí imposing three physical conclítions, namely: The time-reversa! invariance of Eq.(3) when F(t)=F(-t), the continuity equationfor p, and the recovery oí the classical description in the Newtonianlimito Eq.(3) tranforms into

wi th A n2 sti!! undefined.

(4a)

(4b)

The system of Eqs.(4) admits a first integration and !inearization in terms oí the new variables

~,R, S/IA= 1/2 ,S/IA(5)e p e

where

~ ~ 2DVS. (6)u = 2DVR. V =

As a result oí this integratían Dne obtains the linear, uncou-

555

plcd equations

a~+.2mD T-'i.--at (7)

Since the free paramcter A appears only in the combinatíanD T-'i.one may take ,2 = l by an adequate seleetion of the phenomeno-logieal pararneter Do With, = -1 &jo (7) is parabolie and the ampli-tudes ~ are real; with , = +1 it is hyperbolie and the amplitudes ~ arein general complexo To stress the physical differcnces we speak in thefirst case oí an Einstein process (a classical, Brownian~type process

in the limit of negligible frietion) and in the seeond ease of a deBroglie (non-classical, \.¡ave-! ikc) process. Thc Schrodinger equation

is obtained froo &jo (7) for, = 1, with a suitable seleetion of the di.!:£Usían coefficient:

D = hl2m (S)

The aboye Tesults show that qUaJltlln ITIC'chanics can indecd be intcrpreted 35 a ~brkov proccss, but irreducible to Br~TIian-typestoch-astíe motian. This fo~,lism has allowed a varicty oí extensions andgeneralizations, 5uch as the introduction oí spin(8), relativistictrcatmcnts(9), the extcnsion to ~xed states(IO.ll) anu variatíanalformulations(6,12-14); the eorresponding path-integral formulation hasbeen developed(14.IS) and applied to the problem of barrier penetra-tion(IS) o ~bre general schcmcs have been developcd, either as a thco-ry of the eleetron(l6) or even as a field theory (17) o An impcrtant offspring oí stochastic quantummcchanics in the so~callcd stochasticquantization, a stochastic treatment cf ficId theory dcvcloped in re-eent years(17,IS) o

lhe theory has a150 rcccived much criticismo An analysis oíthe objections raised rcvcals that they are IOOst frcqucntly the resultoC a traditional, classical approach to stochastic proccsses and theconscqucnt lack of distinction between Einstein and de BrogUe proc-esscs. though in sorne cases they are relatcd to a spccific technical

556

detai 1, such as considcring thc back.\'Jardderi vat ive ~ - 05 as the timeinversion of Vc + Os' The maio shortcoming of stochastic quantum n~-

ehmlic5 is its phcnomelogical char3cter: Being a fotm.1.1 thcory, it isunable to elucidate thc physical mechanism responsiblc fay stochasti-cit)' and, in thc last instanee, far quantization. Iiowcver, it has themerit of indicating the way along which ene should look far a deeperamI JOOre flUldarncntal thcory; this is the subject oí the nex! section.

3. STOCIIASTI e ELECTRODYNAr-1les

The search fay 3Il cxplanation of quanturn stochasticity in tennsof a physical cause reprcscnts obviously a departure fram the orthodoxvicws on quantum mechanics. Stochastic electrodynamics (Sr~)in such anattcmpt: It5 purpose is to intcrpret the quantum phenomenonas origina-ting in the interaction with the allpervading zero-pointradiation fieldprodueeJ by the far matter in the lmiversc. Intuitively one can vicwthis field as the result of the superposition of the uncorrelated fieldscmitted be .111 aceelerated chargcs in the tmiversc. This assumption canin principIe help to salve <U1 oId dilcrrrna: Recall that at the beginningof the century, the planetary model of the atom was dismisscd because ofthc instability produccd by the radiation of the accelerated electron.SED attempts to revindicate this radiation as the ul timate cause of ato!!!ic stability and stochasticity at the same time, by proposing that astate of cquilibrit.un can he reached in which the energy absorbed by theclcctron from the stochastic field is compensatcd in the average by theradiatcd energy. In this picture, radiation is necessary to stabilizethe (stochastic) atomic orbits. As 0pposco to earlicr lTlodeIs, the atOln-le elcctron is now considcred an open system that continuously cxchangesencrgy with the random zera-point radiation [icld.

The idea of this ficId 15 certainly not ncw: It Has already cn-visaged by Planck(20) ano sorneof itsean smJlogical eonsequences werestudiedbyNernst(2l). In prcscnt-day physics it appears as the vaelIDlTlof ql.lantumelectrodynamics, usually-but not ah •.ays inpractiee(221. rcgars!.ed as a virtual field. though \oJith fluctuations giving rise to observa-ble effeets (see, c.g., the disclIssion in Ref. 23). The assLm1ptionof

557

rcality of thc vncl.UTlm fich.l is not free of difficultics. evcn ve!)'serious ones as the cosmological implications of its cnormous energydeosity; thi5 is an tmsolved problem, which strictly spcaking affectsquantum e lectrodymunics 3S we 11.

The idea oí a close cOfUlcction bctwecn qU3Iltum stochasticityand the zera-point radiation [¡cid is more thml thrce decades oId andhas beco rediscovered by several independent workers. It is mentionedexplicitly by Kalitsin(24) already in 1953. but the first attempts toelaborate a theoty based on it are due to Braffort and coworkers(25)and ~larshall (26). A short review o[ SED is presented in Re£. 27 and amore dctailed ene GUl be founJ in Ref. 28

The zera-poiot ficlJ is nonnally asslllncd to be a solution oí~L.1xwCll'5equations without sourccs, th3t can be exprcssed in tcnns ofFouricr components \vith stochastic amplitudes averaging to z.ero and haying independent Gaussian distributions. By thc rl~uirement oí Lorentzinvariance, its spectral dellsity must be of th€" fonu Aoo3. NithA = 4h/3nc

2, thc cnergy of th(' ficId loode w is -!-hü', as it should at zero

tcmpcraturc.

Refare entcring into thc dynamics, let us i Ilustrate the funcL'lmental role that the hypothcsis o[ the zero-point field may playas afOlmdational cornerstone of quantum theary.

Considcr a hlackhouy in equilibrilD1l at tempcrature 1. ItsIIDdcof frcquency w •••••i 11 have the average energy

mId assLU11ingthe ficId componcnts to be Gaussianly distributed with ze-ro mean, the variance of the ellergy is

Using thc h'ell-knOl\TI Einstein formula [or the fluctuations oftlle' encq.,'Y of a systcm in thcnn.'ll cquilibriwn \.,rith <l heat bath at t('m-

558

perature T:

2er B I/KT (9)

we get the differential equatían

with solution

o

where So is an integratían constant; since fer T ~ 00 (6 + O) <er>should become infinite, we mus! take BO = O and thus we get the classical fonnula

(ID)

which cerresponds te the Rayleigh-Jeans law. NO' let us add to theseelementary considerations a single new ingredient, namely, the zero-point ficId. For this we assUIDe the mas! natural hypotheses. namely:i) "Ihe meanenergy is a SlDTI oí the thennal and zero-poiot contributions:

ii) The zero~point ficId is Gaussian:

üi) The zero-point ficId and the thermal part are statistically inde-pendent, and thus the variance oí the energy is

e 2:lE <e> 2 = O'2 + 02e -r O

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whence

Inserting this result into Einstein's formula (9) we get thencw differential equation

(11)

The solution of Eq.(ll) th.t goes to (10) in the classical limit is

(12)

This is Planck's law for the blackbody for <cO> I O; it sufflces to take <co> = O as corresponds to the absence oí the zero.pointficId, to recovcr classical physics. This derivation, due to Boycr(29)and Theimer(30). allows a150 to lcarn a little more about <£0> fromfirst principIes. In faet, Wien's law fer the spectral density oíradiation in equilibrium with matter at temperatura T

3 (wp(w) = w f f)

gives when comp.red with Eq.(12), <cO>awand PO(w) aw3, in agreementwith the argL~ntsmcntioned aboye, In writing EO e ihw Planck's co£stant acquires • physical meaning: It measures the fluctuations of therandom field and hence .150 the fluctuations impressed by this fieldU90n the particle (as expresscd, c.g., in the Heisenberg inequalities).Other derivations of Planck' s law from SED are prcsend in Re£. 31-

To develop a theory oí thc moticn oí electrons in interactionwith the random ficId we can in principIe start for the Hamiltonian of

560

the entire (field plus partiele) s)'stcm:

1 ~ eA2 ~ lE 2H = 2m (p - e ) + V(x) + 2 nA (PnÁ

whith

(13)

~'£nÁ O. w = ell< l' + 1(PnÁ + iwnqnÁ)n n • anÁ nIlWn

2 2 2 = 1:. hw<p > w <qn),>nI. n 2 n

(14)

and derive the lIamiIton equations [ay both the particle and the fieIdvariables. An approxirnate procedure uscd to eliminate the lattcr leadsto a stochastic cxtension of the AbrahamLorcntz equation far the paT-

tiele:

mX = F(x) x + e(~ v+ -e x B) (15)

This highlynontrivial stochastic diffcrential equation with c£louTcd ncisc is somewhat simplified by neglecting the magnetic forceand taking the long-wavolength limit. in whieh E = E(t); Eq. (15) reducesthen to

(16 )

with T = Zc2/3mc3. This is the Braffort-f-tushall equation. usually

takcn as the starting poiot oí SED. It has beco applied to simple lin-ear problems, such as the free particlc(32), the paTtiele subject to aeonstant foreo(32) and the harmonie osci11ator(25,26.33.34) (see Ref.28

far a more complete list of rcfcrcnccs). Let us briefly mention sorneof the rcsul ts fer the hannonic osci llator: Owing te the Gaussian dis-tribution of the randomficld amplitudes, thc particle variables x anJ

561

pare Gaussian as wcll and the stationary phase-spacc distribution is

P(x,p) = n1h exp [- ';w 2 1 22-(E... + - J1LJ X )J2m 2

At temperatures 1>0 the average energy oí the ficId modes ist hw(I+6)/(1-6) with 6 = exp(-hw/kT) and the phase-space distributiontakes the fonn

P(x,p) 1ni, 1-6 r- 2 l + 1 2 2 l-eJ1+0 exp ~ hw (2m I J1LJ x ) 1+0 (17)

which is just the Wigncr distribution £01' a mixture in equilibriumwi th a hcat bath at tcn'q)craturc T. as dcscribcd by quantlDll statisticalmechanics (see, e.g., Ref. 35). The Lamb shift and the radiativedceay rate are obtained whcn the radiatían reaction i5 introduced as aperturbat ion.

In spite oí the rcm1rkable coincidence with quantum mechanics.there are conceptural diffcrcnccs that descrve a close attention. Forinstanee, £01' SED the grOlUlu statc is the statc in which the rates of<1h501'pt10nand radiatían of cncrgy are eqml. The lIeiscnherg inequa1lty rcf1ects thc fluctuat ions impressed by the fieId lIpanx and p in thestationary statc. o..•.ing to the rclatively long ("or'":cl~~1"i.ontime of thesingle ficId components, thc particle fluct~~tions are highly correlat-ed, which cxplains the n3rraw cmission and absorption lines. Here wehave a theot). that can cxplain in principIe the successes of the fonnalstochastic theories oí quanttunmechanics, but wc also face sorneproblems.For instanee, in Eq.(17) thc excited states aprear simply 35 mathemati-cal components of the cquilibrium distribution, devaid oí a direct phy~icaI meaning. Also a ftUlJamcntal epistcJOOlogical qucstion, namely, whySchOdingcr's equation provides a correct description of the system,remoins unanswered. It is cleur that sornebasic elements are stilllacking in the theory.

In addition to those mentioned above, other problems havebecn successfully approachcd, such 3S the study of the free radiationficld(26,34,36) the van der W"lIs forccs(37) and diamagnetism(26,3R,39).SED has also served to suggcst an explanatíon of the origin and meaning

562

of the eleetron spin(39).H~cvcr, the few attempts to salve nonlinear problems have

becn unsuccessful. The usual proccdurc is to construct a Fokker-Planckequation for the (multiply periodie) elassieal system perturbed by therandom field and the radiation rcaetion(40.4l) (See also Refs. 28 and41 for noTe complete references). For the hydrogcn atom, for instanee,this proccdure leads to a ground state of zera energy, which Unplicsspontaneous ionization. [\breover, the system turns out to be non-ergodic and the Fokker-Planck equation admits several coexisting statiQnary solutions(39.4l.42) (See also Refs. 1 and 28).

For the quartic oscillator, the results are no! as aberrant:Thc ground-state energy is correctly predicted to first arder in theperturbation, but the second arder result is incorrect(41). Anothcrunsatisfactory result, cornmon to all nonlinear problems, is the lack ofdctailed energy balance at every single frcquency of radiation; itwould thus seem that the mechanical system pumps energy from sorne fieldmodes into others, contradicting Kirchhoff's law on the universalityof the equilibrium speetral density. Ihis problem was first diseussedby Boyer(43) and has been rediseussed afterwards(44).

4. lllE PRESE,'.¡r SfAGE: SED ANEW

A general feeling oí írustration has involved SED in recentycars, duc to its inability to produce new positive results. But acareful rcvision oí the situation tends to suggest that the difficul-ties encountcred are methodologieal rathcr than a matter oí principIe.Evcn though we cannot afirm at this stagc that SED is an essentially ,correet theory, there are two points that strengthen our confidence.First, the simplicity and physical elarity oí its postulates and second.the positive results it has furnished, which can hardly be considered amere coincidence with quantlUllmechanics. It seems therefore opportuncto explore anew the possibilities of SED and try to obtain a betterunderstanding of its implications. Ihis has been the purpose of ourmostreeent work(45).

563

Let liS start by examining a flUldarrentalpoint, namely, theorigín oí quantization. For this purpose, we first consider a systemsubject to an external binding force F(x,x,t):

mX = F (18)

such that it can reach a stationary periodic state oí motian (we shallconsidcr ene-dimensional motion, for simplicity). By writing

x = A sin e, x=nAcos e

with

A = A(t], e ¡¡t + Ht)

one obtains

«F + mlx)x>¡¡ o 2 • o (19)«F + nílx)x>¡¡ =

where < >n denotes averaging ayer a period T = 2n/Q, under the assump-tion that A and $ are essentially constant during that periodo Thisapproximate description in terms oí a harmonic oscillation is validonly in the stationary state oí motian. Wecan therefore speak oí alocal 1inearization , that changes from state to statc: For every valueof the amplitude (or the energy), there will be a different value of¡¡for Eqs. (19) to hold, which reflects the asynchrony of the system.

Let liSnow lcok at Eq. (16) from point of view. We can rewriteit by adding mn2x on both sides:

(20)

lf we select ¡¡so as to comply with Eqs.(19), then the effectoí the tcnms within square brackets is simply a periodic alteration

564

which averages to zera in the stationary state and Eq.(20) can be ap-proximated by the linear equation

•. 2 ...mx = - mn x + XTX + cE (21)

Here we are taking into account that the radiation reactionand the rmldom electric force are small compared with the externalforce and herree do no! alter the stationary periodic maticn signifi-cantly. Thesc terms play actually a stabilizing role: As secn fromEq.(20) the field mode of frequency ~ sustains the oscil1ation and theradiation reaction force prevents ~1 infinite resonan! response to thisrode.

The solution of the linearized Eq.(21) can be uscd to derivesorne interesting results, such as the l~i5enbergrelation

far the dispersions. oí x and p, and

<K>

(22)

(23)

far the average kinetic energy per dcgrce oí freedom. (The system is3ssumed to havc the right ergodic properties for the time averages tocoincide with the ensemble averages). We recall that these resultsho1d only in the stationary state. This mean S t~,t Eq.(23) must holdin addition to the relation between <K> and n obtained from the line-arization condition: The value of n (and that oí <K» is thusuniquely fixcd and the motion be comes quantized. Ihis is the propos-ed mechanism of quantization. In other words, of a11 classically al-lowed states, onIy one is stationary under thc influence of the randomeIectromagnetic ficId.

Consider, e.g., the circularzorbits oí thc hydrogen atoro.Eq.(20) goes into (21) for F(x) e3 x and Eqs.(19) give

a

n (e2/ma3) 1/2

565

The kinetic energy associated to this orbital motian is ther~fore, according to clussical mechanics,

<K> (24)

On the other hand, Eq.(23) gives

<K> 1=2M1 (25)

for two degrees of freedom. Combining Eqs.(24) and (25) we obtain

(:= - < K > (26)

which are the quantum mechanical results for the ground state.By virtue of the nonlinearity of the force, it is natural to

assume that the system cml respond preferentially a150 to ene of theharmonics, say nn, oí the ficId; in this case one has for two-dimen-sional motion:

<K> = i hlln (27)

instead of Eq.(25), and this leads to the correct values for the energylevels of the excited states:

(28)

These results can be generalized to elliptic ortibs and sim-ilar methods can be applied to other simple nonlinear prohlews, withsatisfactory results. They can a150 be rephrascd in a more famaliarform, by noticing that thc average kinetic energy is related to the

566

action variable:

<K>

whence from Eq.(27):

J nh (29)

This approximate calculatían suggests a physical explanatían oí thephenomenological Wilson-Sommerfeld rules.

Notice also that by assigning a wavelength to the orbital mo-tion oí velocity v sustained by the ficld hanoonic rú.l:

and using once more F4.(27):

<K>

one obtains

112= "2 hIúl ="2 mv

hA =-mv (30)

which indicates that de Broglie's wavelcngth can be intcrpreted as adynamical (average) property acqtlired by the particle through its inte~action with the V3CUum ficId.

Let us now develop a fonnalism bascd on the heuristic approachoutlined above. It is clear from Eq.(21) that the properties of theficId variables will reflect themselves in the stationary solution. Thestatistical properties are reflccted in the calculatían oí averages anddispersions, such as Eqs.(22) and (23). Similarly, the canonical pro-pertíes oí the ficId determine a symplectic structure for the particle

567

variables. Te see this, we introduce the Poisson bracket with rcspectto the random field amplitudes, which we shall call Poissonian:

< > " EmA [aa:, aaar -mA

and calculate the Poissonian oí x and p corresponding to the sationarysolution of Eq.(21); the result is

(31)

<x. ;x.> = <p. ;p.> o1 J 1 J

to lowest"order in T. Note that these results are independent oí n andhence they apply for any binding force and any (stationary) state of ffiQ

tion. Using Eqs.(31) one obtains for any phase-space variables f,g:

<f;g> =ihlf,g] xp (32)

where [ f ] xp is the usual Poisson bracket with respect to x and p.Two particular instances of Eqs.(32) are

ih afapi <p. ;f>

1_ ih af

aXi(33)

Let us now attempt to describe the dynamics. Since the effectsoí both the radiatían reaction and the random electric force are smallwhen the system is close to a stationary sta te, the dynamics will beessentially dctermined by the externa! force, which means that theequation oí evolution for anyftn:tion f( x,P. t) is basical1y the classi-cal equat ion

568

2with H .. ¥ro + V. Hov,ever, since x and pare random variables that mustsatisfy Eq. (31), the phase-spacc functions f and 11are subject to theconstriction expressed in (32):

ih [f,1I1 = <f;H>xp

By introducing this into the aboye equation we obtain

ihf = ih af + <f'H>at ' (34)

which we propase as the equation of evolution far f. It is intcrestingto observe that even though OUT dcscription has been strictly causal,

,.in \~iting F~(3~) any refercncc to the cause of stochasticity has becnlos!; this suggcsts explaining the apparently non-causal statisticalbchaviour of quantlUTI mechanics as an <Jl'tifact of the dcscription.

[4.(34) app1ied to the probability dcnsity p gives

ih £P- = <H;o>at (35)

since p is aevolution inwi th vectors

conserved quantity and therefore p .. o.tenms oí stationary states we introduce~n such that

To describe thea Hilbert space

p • (36)

It fo11ows from [cs. (35) and (36) that the ~n are the solutions of thceigenvalue equ3tion

(37)

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and the eigenvalues are given by

N* <H;t > d~n n(38)

¡"'hcrc "fJ is the accesible phasc spacc J asstDlling the tP to be properlynnormalized: f~*~ = l. Now, supposc we wan! the quantity

n n

flj;* <H;\jJ>dfj

to attaio an extremal value under small independent variations oíl/J(or ¡p), subject to the nonnalization condition

The corresponding variational problem reads (l\'hen ooly thc llJ*

are varied)

fóV«H;~> -A~)d~ = O

ffildits solutions are, according to Eqs. (37) and (38), ~ = $n' A = en'This means that thc energy parameter acqui res the extrema.l values £nwhen \jJ corresponds to the stationary solutions 4>0. This fonnalism hasallowed liS to transform the problem oí calculating the quantizeJ valuesoí n corrcsponding to the stationary sta tes oí matico, ioto aneigenvalue problem.

As a further step in this attcmpt to extricate the connectionbetwccn SED and quantum mechanics, we observe that the present descrip-tion lends itself to an operator fonnalism, in which the operators areassociated to the corresponding dynamical variables by means of therelation

<A;4J> (39)

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From thc propertics of the Poisson brackets one can derive use.fuI propcrties for the operators. For instanee, thc identity

<A;BC> ~ <A;B>C + B<A;C>

lcads to

or interch;m~, ::'.& A :md B:

The~c equations can be combined and rcwritten as

IA,B 1 = IA,B 1 = <A;B> (40)

Thc first (s~c"nd) exprcss ion is the conmutator of A and B in the A(B)representation; ro<¡. (40) tells us therefore that thc value of thc co","!!tatar is indcp.:-ndcntoí the representation. Thus for instanee, thccommutator of x and p is

[~,fl 1 = ih

in both the x and thc p rcprcsentation. From Eqs.(33) ,

-iha/ax.1

in the x reprcsentation and

in the p representacion.

571

On may further write the Jacobi identity:

<A;<B;41» - <Bj<A;4»> = «A;B>;lf¡>

In terms of operators:

whence

IA,E IA

<A;B> ( 41)

For instanee J farTelation gives

the components of the orbital angular momentum this

Finally, the eigenvalue equation (37) takes the form

and the general solution (36) satisfies the equation

_ 3' A

lh __7. = lti;atA2

with n = ¥ro + V(x) in the x representation. We thus recover the usualquantwn formulation in Hilbert spacc.

The results presented in this section suggcst that a properconsideration of thc specific dynamical effects of the ranrlom electro-n~gncticficId may salve the problems faced by SED in connection withenergy balance and the existence oí stationary states. OUT argumentshave beco mainly heuristic and it is obviously necessary to develop amore rigorous tratment in which every assumption is clearly justified.

572

If this can be done the new theory thu~developed should not only ex-plain the Kell-lmo\l,TI quantlIDI phenomenology, but a150 allo,," us to extcndour knowledge beyond prcsent-day quantum mechanics.

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