Download A Diffusion Model for the Schrodinger Equation*l

1889
Progress of Theoretical Physics, Vol. 49, No. 6, June 1973
A Diffusion Model for the Schrodinger Equation*l
L. BEsst>
Rutgers University, New Brunswick, New Jersey, 08903
(~eceived
October 30, 1972)
A diffusion process has been formulated to have a mass conservation equation and a
momentum conservation equation that together are equivalent to the Schrodinger equation of
quantum mechanics. It is suggested that this diffusion process be used as a physical model
to help describe the microphysical processes that underlie the Schrodinger equation.
§ I. Introduction
At least among certain physicists, a desired goal has always been to· find a
physical model**> that can provide a description of the microphysical processes
associated with quantum mechanics. An early example of this type of effort is
the work of Madelung. 1> A more recent example is provided by Bohm. 2> However, to date, no model has been devised that can accurately be described by the
Schri:idinger equation and not be quite artificial physically. In the work to be
reported here such a model may perhaps have been realized. The characteristic
feature of this model is a special type of diffusion process. It is to be hoped
that, at the very least this diffusion model will prove to be of value as a conceptual aid when dealing with quantum mechanical phenomena.
Actually, the association of diffusion processes with the Schri:idinger equation
is nothing new. A number of investigators 3>-B> have already noted a formal mathematical resemblance existing between the equations governing diffusion processes
and the Schri:idinger equation for a single particle in a force field. However, the
diffusion processes treated by most of these investigators 3>-s> cannot be relevant
here. This is because these diffusion processes involve an imaginary diffusion
constant, D, and therefore are not physically realizable (at least in the realm of
classical physics). Hence these diffusion equations can serve as mathematical
models only.
*l Paper of the Journal S=ries, New Jersey Agricultural Experiment Station, Rutgers, the State
University of New Jersey, Department of Environmental Sciences, New Brunswick, New Jersey.
t> Department of Environmental Sciences, Rutgers, The State University, New Brunswick, New
Jersey.
**> The term physical model is used here to mean those models employing the paraphernalia
of classical physics that can directly represent the physical processes they describe even though they
may be simplifications. An example is the use of a group of randomly moving elastic spheres in
the kinetic theory of gases .to represent the essential. processes there. Other types of models using
an indirect symbolic type of representation are to be called. mathematical models.
1890
L. Bess
The other investig ators 7h 8> treated diffusio n process es with real diffusio
n constants but present difficult ies in connect ion with their physica l viability
.*> The
work of one of these, Kersha w, 8> is of particul ar interest here since
it is to serve
as the starting point of the present treatme nt. In the followin g sections
, Kersha w's
treatme nt is to be analyze d in detail and will be shown to contain
a serious flaw,
even though his results were obtaine d in a very elegant manner ·
and are valid
from a strictly mathem atical point of view.
§ 2. Basic derivat ions
In this section the general governi ng equatio n set for a diffusion process
is
to be derived followin g the method of Kersha w (which in turn is
based on the
work of Chandr asekhar9>). These equatio ns are then compar ed with
the real equation set associat ed with Schrodi nger's equatio n so as to determi ne
the conditio ns
for the two equatio n sets to be equivale nt.
In the derivati on that follows , w (x.t, V.t, t) is the density in six dimensi
onal
phase space** ' of the ensemb le probabi lity fluid. p (x.t, t) is the density
in real
space, and V"' (xk, t) is the "drift" velocity . The followin g set of relation
s define
p and V"':
(1)
(2)
(3)
The propaga tion kernel E between the event points (x"'- ~k• t) and (xk,
t + Jt)
that is to be used here is the same as the one employ ed by Kersha
w and has
the form
E(xk,
t+Jtlx.~:-e:.t,
t)
= [ 4nDJt]- 812exp { -
:Ec ~lJ~t,Jq},
(4)
*l Regardin g the work of Weizel, the author feels that, although this work
is a worth while
attempt, the treatmen t is unsatisfa ctory for several reasons. One of these
is that he seems to have
confused the roles of the "drift" velocity and the "transpo rt" velocity
(the terms are to be defined
below). Another reason will be pointed out in § 5.
**l To ensure that this point is not missed, it is now stated that the treatmen
t here was devised
to complete ly fit into the methodol ogy develope d by. Gibbs and· Boltzman
n for statistical mechanic s.
Specifically, use is made of a point moving in a six dimensio nal phase
space to represent the change
of state of a particle with time. Moreove r, the concept of the ensemble
of particles is used in the
manner employed in statistical mechanic s (for example, see R. B. Lindsay,
Introduct ion to Physical
Statistics (]. Wiley and Sons, New York, 1941), pp. 108""-'119).
A Diffusion Model for the Schrodinger Equation
1891
where
Moreover, quasi-Huygens propagation relations for the mass density and momentum density are formulated just as was done by Kershaw. These take the
form
p(x~c, t+Lit) =
Sd 8~p(x~c-~k• t)E(x~c, t+Lit[x~c- ~1c, t),
(5)
p (xk, t +Lit) vk (xk, t +Lit) .
=
Sd 8~P (x~c- ~k• t)
X [V~c(x~c-~~c,
t)
+LIV~c(x~c, t+Lit[x~c-~~c,
t)]
xE(x~c, t+Lit[x~c-~~c, t).
(6)
Once the difference in notation has been accounted for, it can be seen that
Eq. (5) is identical with Kershaw's Eq. (6). Assuming Lit to be small, Kershaw
then converts his Eq. (6) to his Eq. (11). This result is valid here. Thus the
integral relation of Eq. (5) can be converted into the following differential equation. *l· **l
(7)
where Ci= Vi- D (fJip/ p) and is called the "transport" velocity s111ce pCi is the
probability current.
The quantity LIV~c(x~c, t+Lit[x~c-~~c, t) 111 Eq. (6) represents the increase 111
average velocity that the probability fluid undergoes between the event points
(x~c- ~~c, t) and (x~c, t +Lit), when acted on by various types of forces.
In his
treatment Kershaw assumes that the net force on the probability fluid is just the
external force F~c(x~c, t), so that LIV~c= [Fdm]Lit. As will soon become apparent,
the main difference between Kershaw's treatment and the one being presented
here is the assumption about L1 V~c. Here, a more complex origin of L1 V~c is assumed. The exact nature of this L1 V~c origin is to be considered in § 3. In this
section, only the mathematical form of LIV~c is required to obtain a differential
equation from Eq. (6) that is to be the analog of Kershaw's Eq. (15). In addition to the obvious impulse from the external force F~c, it is assumed that there
*> The mass and the momentum conservation equations in Kershaw's treatment are his Eqs. (11)
and (15). As is seen, Kershaw's Eq. (11) is identical to Eq. (7) of the text. Kershaw's Eq. (15), the
momentum conservation equation, is (using the present notation):
8< V~c+ Vi (8i V~c) =[~ ]+D[ (1/ p) (8~i(PV~c))- (V~c/ p)
(8~io)].
..
**> Tensor notation is used here throughout, where the repetition of an index in a term means
summation on that index. Moreover,
8i=8/8xi;
8~1 =(8i8J);
and
8"=8/8t.
1892
L. Bess
are contribu tions from what might be called quantum mechani cal (or QM) "forces"
,
which are now to be represen ted by the most general mathema tical expressio
n
that can contribu te to the :first order in Jt. Thus, from the foregoin g considera
tions the explicit form for j vk is assumed to be
JVk(xk, t+ Jtjxk-g:k , t) = [Fk/m]J t+ akJt
+ Eikg:i + ri}kg:ig:,,
(8a)
(8b)
The quantitie s Fk, ak, 13tk and 'i'iik are all functions of xk- g:k and t while
the
quantitie s Fk, ak, {3;k and rt 1k are functions of xk and t. Moreove r, riik ;,_ r iik·
Upon performi ng the Taylors expansio n on the factors containin g Jt and g:,
in Eq. (6), then performi ng the indicated integrati ons, and ,:finally retaining
only
those terms that are of the :first order in Jt, it IS shown in Appendi x A that
the
followin g different ial equation results:
a, (p Vk) +a; [p V; V., -Do; (p Vk)]
=p[Fdm ] +pak+pV;{3;k
- 2D (a;p) ({3;k)
+ 2Dp [rm- a;{3;k].
(9)
Equation (9) is the momentu m conserva tion equation for the diffusion process
just
as Eq. (7) is its mass conserva tion equation . Togethe r, these two equation
s constitute the governin g equation set. It is of interest to note that if no QM "forces"
are assumed to exist (i.e., if ak = 0, f3tk = 0 and riik = 0), Eq. (9) can be obtained
from Kershaw 's results by combinin g his Eqs. (11) and (15).
A comparis on is now to be made between the governin g set, Eqs .. (7) and
(9), and the Schrodin ger equation . First, the Schrodin ger equation must be
converted from a single complex equation with a single variable ¢ into two
real
equation s having the variables p and Vk (or Ck). To do this, the usual relation
for this type of transform ation is used here also. This is
¢= VP exp(iS/h ),
(lOa)
and
(lOb)
111
The substitut ion of Eq. (lOa) is now made in Schrodin ger's equation , which
the present Jtotation is
(11)
where
A Diffusion Model for the Schrodinger Equation
1893
The real terms and the imaginary terms are collected together and each
group set equal to zero, thus making up two real equations. The equation resulting from the imaginary group can be shown to be identical with Eq. (7) above.
If the equation resulting from the real group is operated on by the space partial
derivative ak and the result is combined with Eq. (7), it can be shown7> that
the following relation is obtained:
a. (pCk)
+ a{pc ck- D
2
J P[ ~].
pai ( a~p) =
(12)
where it is assumed that D = h/2m.
For comparison purposes it is convenient to represent the momentum conservation equation in terms of the "drift" velocity, vk, rather than the "transport"
velocity, Ck. Hence by substituting, Ck=Vk-D(akp/p), Eq. (12) can be converted to the form
a. (p Vk)
+ ai [p vi vk- nai (p vk) J
=p[ ~] -2ai[Dp(aiCk)].
(13)
Equations (7) and (13) are the real equation set associated with the Schrodinger
equation.
In comparing the diffusion equation set with the Schrodinger set, differences
are observed only on the r.h.s. of Eqs. (9) and (13). The two equation sets
become identical if
pak + pVd3ik- 2D (aip) ({3ik)
= -2ai[Dp(aiCk)].
+ 2Dp Crm- ai{3ik)
(14)
With the derivation of Eq. (14), the mam objective of this section has been
attained.*> Before proceeding, it is of interest to clear up a point connected with
the statement that the diffusion equation set and the Schrodinger equation are
equivalent. As was seen, comparison was actually made with the equation set,
Eqs. (7) and (13), not the Schrodinger equation itself. Solving Eq. (7) and (13)
will provide the functions p and Vk (or Ck) not ¢, the actual solution of the
Schrodinger equation. To obtain ¢, Eqs. (10) must be invoked. As can be seen
from Eq; (lOb), the function S is not uniquely determined by Ck. Hence, strictly
speaking, the diffusion equations cannot be equivalent to the Schrodinger equation. They can only be consistent with it. However, what in effect amounts to
an equivalence can be made if the following relation**> is postulated as a sup*l It is interesting to note that Kershaw was not able to derive the more general time-dependent
Schrodinger equation but only the time-independent equation. The reason for this was his failure
to include the QM "forces", as is readily apparent from Eqs. (9) and (14).
**> Where
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L. Bess
plemen tary conditio n:
(15)
where E is the energy of the particle and P( = mv) is its moment
um. The validity of Eq. (15) is actually a very reasona ble postulat e since, in
the usual quantum mechan ical treatme nt, it is derived from the Schrodi nger equation
· to show
that there is agreem ent with classica l mechan ics and thus lend plausibi
lity to the
theory. With the assump tion of Eq. (15), it can be shown that the
S function
derived from Eq. (lOb) (which is S) is related to the S of Eq.
(lOa) by S=S
+ ft¢> 0, where ¢>0 is a constan t. Hence, the ¢ function which is derived from Eqs.
(7) and (13) (which is qi) is related to ¢ by qi = ¢etq,,. From a
physica l point
of view, ¢ and q; are equivai ent since all the physica l parame ters
of interest are
identica l when derived from the two ¢'s. In what follows the validity
of Eq.
(15) is assumed , so it is underst ood that there can be a real equival
ence between
the diffusio n equatio ns and the Schrodi nger equation .
§ 3. The quantu m mechan ical "forces "--prel iminar y conside rations
Probabl y the most importa nt resul~ of the previou s section was to
show that
if there is to be an equival ence between the two real equatio n sets,
there must
be contribu tions to the velocity increme nt JVk, other than those arising
from the
externa l force Fk. This is to say, the so-calle d QM "forces " mu'st
exist. The
investig ation of the origin and the phenom enologi cal charact eristics
of these QM
"forces " is now to be underta ken and constitu tes the most importa
nt task of the
present work.
In the approac h adopted here, two postulat es about the QM "forces
" are to
be assumed . As will be seen, these postulat es are associat ed with
physica l properties that are so reasona ble as to almost be self-evid ent. The first
postulat e is
actually the equival ent of the relation
(16)
In the usual treatme nt of element ary quantum mechan ics, Eq. (16)
is derived from
the Schrodi nger equatio n again to show an agreem ent with classica
l mechan ics.
In the treatme nt adopted here, the mathem atical form of the first
postulat e (to
be called Postula te I hereafte r) is to be differen t from Eq. (16).
To present
this, it is necessa ry to define the quantiti es .d Vk and Qk. These are
JVk (xk + ~k• t + .dtjxk, t) =ak.dt+ ~d1tk+ ~t~JrHk
=.dVk- [Fk/m] .dt,
Qk(xk, t) = [m/ .dt]
(17a)
Sd8~.dVk (xk+ ~k• t + .dtjxk, t)
X
E(xk + ~_;, t+ .dtjxk, t).
(17b)
A Diffusion Model for the Schrodinger Equation
1895
The quantity Qk of Eq. (17b) represents the QM "force" acting on the pro bability fluid averaged at the event point (xk, t). Postulate I may now be ,written
entirely in terms of Qk as
(18)
Physically speaking, Eq. (18) states that the total ensemble average of the QM
"forces" (i.e., the total of these "forces" acting on the entire probability fluid)
must be zero. Hence only the remaining external forces can affect the total ensemble momentum (which is what Eq. (16) states).
The second postulate (to be called Postulate II hereafter) has to· do with
the velocity dispersion U 2(xk, t) which is defined as
U 2 (xk, t) =1/p
Moreover, let the quantity,
e (t),
s
(VtVt)Wd 8v-
vi vi.
(19a)
be defined as
(19b)
e
It is obvious from Eqs. (19) that (t) is the analog of the average gas temperature in the kinetic theory of gases. Postulate II states that there exists a value
@M SUCh that
@(t)<eM for all
t.
(20)
Equation (20) states, in essence, that the effective average temperature of the
diffusion process must always be finite. Postulate II is obviously necessary since
if it were violated the physical model of a given process would possess an infinite
kinetic energy and therefore could never be implemented.
It can readily, be shown that Kershaw's process satisfies Postulate I. It is
now to be shown, however, that it does not satisfy Postulate II. The reason for
performing this analysis is not only to provide a better understanding of Kershaw's
very interesting approach but also to have a working example from which the
requirements associated with Postulate II can be developed in a more explicit
mathematical form.
Kershaw does not actually give the detailed physical properties of the diffusion process on which his treatment is based. The only properties he mentions
are that the diffusion coefficient D is a constant and that .JVk = [Fk/ m] At. To
be able to calculate the velocity dispersion U 2 associated with Kershaw's treatment, it is necessary to investigate the details of the erratic diffusion motion of
a typical particle in the ensemble. In doing this, it is assumed, for simplicity,
that the typical particle is being acted on by random time sequence of delta function impulsive forces. This assumption has been used before and is actually not
very restrictive since by making the time interval r between successive impulses
1896
L. Bess
small enough, any arbitrarily shaped path in real space can be approximate d with
any desired accuracy. Stated more explicitly, the rapidly fluctuating force A" (t)
acting on the typical particle and causing it to undergo diffusion is assumed to
have a form given by
A" (t)
= I: (I").o [tv
(t).J,
(21)
o
where (t).+I>(t).; [t] is the Dirac delta function. The impulse (!")" at t = (t)v
is a random variable. The properties of the random impulse (I")v must now ·be
determined and to do this it is expedient to consider briefly the case of Brownian
motion. Here the form of (!")" is
(Ik)v = m (uk)v .
(22a)
In Eqs. (21) and (22a) both the time interval (r).= (t).+I- (t). and the velocity
(uK)v are random variables with probability distribution functions. For all values
of v and k, the velocities (u")" have the same distribution function fu (y). Similarly, for all values of v, the time intervals (r). have the same distribution function f,(y). The statistical properties of (uK)~ and (r)v are given by the following set of relations:
s
(22b)
S
(22c)
yf,.(y)dy= O,
Y2fu (y) dy = (u 2) , ,
J
yf,(y)dy= (r)1
•
(22d)
In Eqs. (22) both the velocity dispersion (u 2) 1 and the average time interval (r)
1
are functions of t and z"(t) (where z"(t) is the position of the particle at time
t).
Another set of statistical properties that must be stipulated is that the random
variable (u")" is completely uncorrelated with any other random variable (ut) 1,
if either V=/=1-! or i=/=k. Similarly, (Jr).= ('r)v- (r)1 is entirely uncorrelated with
(Jr)" if v=l= f.!.
Brownian motion is actually a special type of diffusion process that results
when the diffusing particle is acted on by the random sequence of impulses described by Eqs. (22) and by a viscosity force of the form - {3v". It is characterized by the property that the phase point describing the state of the particle
moves in a random :flight motion9l in three-dimen sional velocity space (which is
a sub-space of the six dimensional phase space). The existence of a viscosity
force is necessary for this diffusion process to be physically viable since it acts
as a kind of "force field" in velocity space restricting the position of the phase
point to the region of the origin with the result that (u 2) 1 always remains finite.
Without the viscosity force, the phase point would execute a free random flight
A Diffusion Model for the Schrodinger Equation
1897
motion in velocity space and it can be shown that for such a process, <u 2) 1 would
increase linearly with t and become infinite as t becomes infinite. 9l
In Kershaw's diffusion process there can be no viscosity force or the relation .JVk= [Fk/m].Jt is no longer valid. (It would be replaced by Langevins
equation.) If Eq. (22a) is used for (lk)., the velocity dispersion U 2 would have
the component <u 2)f> and this would now increase without limit as mentioned
above. Postulate II would hence be violated. There appears, however, a way
of avoiding this objection. This is to assume that (lk). has the form
(23)
Upon studying Eq. (23), it is seen that the diffusion process here is characterized
by having the phase point remain a distance, [ (ui). (u£).JI 2 from the origin in
three dimensional velocity space. Hence, for this process, <u 2) 1 remains finite.
Moreover, it is apparent that the phase point moves in a free random walk in
real space with ,the mean free path, *l O.k)u = ( vk). (r) •.
Even though the component <u 2) 1 remains finite when Kershaw's diffusion
process is associated with the relation (23)' the velocity dispersion U 2 still will
become infinite as t approaches infinity. This can be seen from the following
propagation relation which is derived in Appendix B:
p (xk, t + .Jt) U 2 (xk, t + .Jt)
=
sd3~p(xk-~k,
t)E(xk, t+ .Jtlxk-~k' t)
X [U 2 (xk-~k>
t) +~U~(xk, t+.Jtlxk-~k' t)]
s
+ d 3~p(xk-~k' t)E(xk, t+ .Jtlxk-~k' t).
X
L,:;
[Vi(xk-~k' t)- Vi(xk, t+.Jt) +.:1Vi(xk, t+.Jtlxk-~k, t)] 2 •
i'
(24)
In Eq. (24) ~U 2 (xk, t + .Jtlxk- ~k' t) is the average increment to the intrinsic
dispersion generated by the _fluctuating force Ax(t) acting on an ensemble particle
as it goes from the event point (xk-~k,t) to the point (xk,t+.Jt). For a diffusion process associated with Eq. (23), ~U 2 =3{[<u 2 ) 1]x,t+Jt- [<u 2 ) 1]x-~,,} (where
[ <u 2) 1 ]x,t is <u 2) 1 evaluated at the space point xk and time t).
As can be seen from Eq. (19b), to obtain an expression for (H) (t), both sides
*> To avoid possible confusion, it should be stated that neither Kershaw's diffusion process nor
the diffusion process of § 4 is Brownian motion. This is obvious since neither process includes a
viscosity force and both diffusion processes are characterized by a random flight motion in real space
rather than in velocity space. (Both real and velocity spaces are three dimensional sub-spaces of the
six dimensional phase space.) It is also to be noted that Kershaw's propagation kernel, B, (given
in Eq. (14)), is rigorously accurate only for diffusion processes where the phase point performs a
random flight in real space.
1898
L. Bess
of Eq. (24) must first be integrated over x space. The resulting x and ~ integrals can be evaluated to the first order in Lit by using various strategies. In
some cases, the x and ~ .integrals can be separated if the variable, xk, is
changed to 3\ = xk- ~k· Vi is expanded in a Taylors series, and Eq. (8) is used
to evaluate JV£. All of the ~ integrals and all but one of the x integrals can
readily be performed. The resulting equation is theri:
[8(t+Llt) -l9 0 (t+Llt)]
= [!9(t) -!9 0 (t)] +2(DL1t)
J fj
p
[ai,V1 -/1iiJd8x
(25)
where:
e
From Eq. (25) it is seen that, in _general, (t) becomes infinite as t becomes
infinite (since @0 (t) is always finite and.[!9(t)- !9 0 (t)] tends toward infinity).
This behavior arises from the variation of Vk with xk, which causes the integral
on the r.h.s. of Eq. (25) to always have a positive value (if {1£ 1 = 0 as in Kershaw's
case). Hence, even with the finite (u 2) 1 of Eq. (23), Kershaw's diffusion process
violates Postulate II. It can therefore
said that while Kershaw's derivl:!-tion
is perfectly correct in a m~thematical sense, the diffusion process he depicts cannot be used as a physical model. Finally, it is apparent from Eq. (25) that a
diffusion process with no viscosity forces can satisfy Postulate II only if it has
Q~ "forces" with a coefficient, {1ik• whose value is (ai Vk).
be
§ 4.
The quantum mechanical "forces"--p roposed formulatio n
The obj•ective of this section is to derive the specific form of the QM "forces"
which can make the diffusion equations equivalent to the Schrodinger equation.
In pragmatic terms, this means deriving specific forms of the coefficients, ak, {1ik
and f£Jk in Eq. (8) that can satisfy Eq. (14). As a first step in the derivation,
some consideratio n, however brief, must be given the problem of the physical
origin of the so-called QM "forces". Essentially the same problem has already
been considered by other investigators 10 )~ 12 ) who provide rather similar answers.
It ~s proposed by them that the QM "forces" a'rise· by the action of the "vacuum"
on any given ensemble particle. This type of solution is to be adopted here also
except that it is further postulated that th.e same forces (i.e., those of Eq. (21))
that give rise to the diffusion action also give rise to the QM "forces".
In order to be able to implement the forgoing ideas, it is found necessary
to use a new fluctuating force Ak (t) which is somewhat different from the fluctuating force Ak (t) of Eq. (21). Ak (t) must now be determined from the following relation (instead of Eq, (21)):
Ak (t)
+ Fk(zk(t), t) = L;
•
(lk).O' [t- (t).],
(26)
A Diffusion Model for the Schrodinger Equation
1899
where zk (t) is the position of the typical particle at time t.
It is also necessary to assume that the impulse (lk). associated with Eq. (26)
has the following form:
(27a)
The vector quantity Yk (z.) in Eq. (27a) denotes the vector field Yk (xk, t) when
xk=zk((t).) and t= (t).. Noting that (vk).(r).= (Ak). (the mean free path), (lk).
of Eq. (27a) can be rewritten to have a more suggestive form such as
(lk). ~m [ (uk). - (uk).--1 + (JYk_)
'tJt
(r).-1
v-1
J,
(27b)
where (iJYk/ot)._1= (a~Yk) + (aiYk) (vi) •. r, (aiYk) and (iJ~Yk) are evaluated at
(t).- 1 and xk.= zk ( (t)._ 1) . Moreover, it is again assumed that the diffusion
constant D(=iU12<r)1 ) does not vary with xk and t and that <r),~Jt.
In comparing (lk). of Eq. (23) with (lk). of Eq. (27a), it is seen that they
are essentially of the same form except that the impulse does not have a zero
ensemble average at a given point but instead has the value m [Yk (z.) - Yk (z.- 1) ] .
Moreover, it is obvious from Eq. (27b) that the QM "forces" arise entirely from
the Yk field, since whatever momentum change the impulse m (uk). produces it is
cancelled out a short time, (r)., later. It is only the third impulse (oYk/ot).-1(r).~1
that produces a lasting change in momentum.
From Eq. (26) and from what has just been said about the Yk field, it would
follow that the total velocity increment JVk(xk+~\, t+Jtlxk> t) can be related
to Yk in the following way:
t=
"~ ( iJYk)
JVk=m I.;
- (r).,
•=.um
ot •
(28a)
where t< (t)" < (t)"+ 1 and IJ.M is the maximum fJ. while IJ.m is the minimum. Moreover,- it follows from the definition of JVk that the sum on the r.h.s. of Eq. (28a)
must be taken over the path of only those ensemble particles that have the same
initial event point (xk, t) and the same final event point (xk + ~k' t +.:It).
For the relationship of Eq. (28a) to be valid, it is necessary that the sum
on the r.h.s. be independent of the path of the particle and hence depend only
on the initial and final event points. That this is so follows from the consideration that JVk is obviously a quantity obtained by averaging over the entire ensemble while the sum in Eq. (28a) relates to a single ensemble particle. However, since only those particles need be considered that have the same initial and
final event points and since it is assumed . that the sum is independent of the
particle path, it is obvious that the sum will be the same for any particle chosen
and hence will be the same as the ensemble average. Thus, Eq. (28a} is then
a consistent relation in that both sides of Eq. (28a) are functions only of ~k and
Jt (and, of course, of xk and t).
Just as in the case of Eqs. (27a) and (27b), Eq. (28a) has an alternative
1900
form which
L. Bess
IS
(28b)
JJ=ftm
The sum m Eq. (28b) can now be evaluated and the result
4Vk=m[Y k(xk+ ~k, t+ Lit) -Yk(xk> t)].
IS
(29)
To the first order in Lit, LIVk can (with the aid of a Taylors expansion ) be
written as follows:
LIVk (xk + ~k' t + Lltlxk, t) = (aXk) Lit+~~ (atYk)
+ H~~~j) (a~jYk).
(30)
An important property of Eq. (30) to be noticed is that the r.h.s. depends
on ~k and Lit but not on the path shape of the particle. Thus, using the (iJYk/iJt)"
of Eq. (27b) has allowed the fulfillmen t of the basic restriction associated with
Eq. (28a).
Equation (30) for LIVk is now to be compared with Eq. (17a) which gives
the quantity LIVk. It is seen that the two expression s can become identical if
the two sets of coefficient s are related as follows:
ak = a"Yk- Fk/ m: f3tk = BtYk;
rijk = t (a~jYk).
(31)
From its definition given in Eqs. (17a) and (17b), it is now possible to calculate the averaged QM "force", Qk by using Eq. (30). The result is
Qk= [m(aXk) -Fk] +mV;(atY k)
(32)
At this point, a general mathemat ical propositio n which is necessary for the
present treatment is to be briefly stated. First, assume that Gk (xk> t) and Hk (xk, t)
are two vector fields and that
JHk+p+d 8x- JHkpd 8x=Llt Jckpd 8x,
(33a)
where p+ =p (xk> t +Lit); p=p (xk, t); Hk + =Hk + (xk, t -t- Lit) and Hk=Hk (xk, t). It is
shown in Appendix C that if the integral relation of Eq. (34a) is valid then the
following differentia l relation exists between G,k and Hk.
Gk = d* Hk -t- W; (a;Tk)
d*t
+ D (a~tTk),
(33b)
where
d*Hk=a" Hd- V;(atHk)
d*t
-t-D(a~tHk)
and W;=D (a 1 p/ p). Tk in Eq. (33) is an arbitrary vector indicating that there
is no unique relationsh ip between Gk and Hk. This, however, is to be expected
A Diffusion Model for the Schrodinger Equation
1901
m view of the fact that Eq. (33a) is an integral equation.
If it is assumed that rk = 0, the proposition of Eqs. (33) can be applied to
Eq. (32) so that this equation implies the validity of the following relation:
(34)
Postulate I is now to be invoked. However, the form to be used here is to
be different from that of Eq. (18). A modification of the original form (i.e.,
Eq. (16)) is now adopted (where it is to be noted that <Pk)~ = mf Vkpd 8x and
<Fk) ~ = JFkpd 3x). If Lit is very small, then it follows from Eq. (16) that
Jcvk+- ttWk+)p+d 3x-
J
(Vk- ttWk)pd 3x=
(~)
J
Fkpd 3x.
(35)
The constant tt in Eq. (35) is arbitrary and is to play a role analogous to
a Lagrange multiplier. It is possible to introduce ,ll because JWkpd 3x=O for
all t.
Upon subtracting Eq. (35) from Eq. (34) the following relation results:
J
(Yk+- Vk++ ttWk+)p+d 3x-
s
(Yk- Vk+ ttWk)pd 3x=
(~)
s
Qkpd 3x.
(36)
If the proposition of Eqs. (33) Is now applied to Eq. (36), a relation -for
Qx develops which has the form
d*
d*t
Qk=m-[Yk- Vk+ ttW.J
2
+ Wi(airk) +D(aiirk)•
(37)
The expression for Qk as given by Eq. (37) is to be compared with the one
given by Eq. (32). As a trial choice, rk, is assumed to have the value so that
(8£k = (aJ'k). It is seen that Eqs. (32) and (37) become identical if
(38)
So far, not only the arbitrary constant tt but also the function Yk Is completely undetermined. To find Yk it is first noted (from Eq. (3)) that Vk=<vk)E
(where the symbol
)E denotes the ensemble average at a given point). Moreover, from the properties of a diffusion process as given by Eqs. (26) and (27a),
it follows that here vk=uk+Yk. Since <uk)E=O, it is apparent that Yk= Vk-ak.
(The arbitrary constant ak is introduced here because Yk + ak could be substituted
for Yk in Eq. (27a) without affecting any of the essential results.) The relation
between yk and vk can also be stated as follows:
<
(39)
smce /3ik = aiYk (Eq. (31)), one important result immediately anses from this
1902
L. Bess
develop ment. This is that Postula te II is automa tically satisfied without
having
to' be invoked as an extra conditio n (siii'ce, as was shown, the·nec essary
criterio n
here is that f3u,
V~c).
Q~c is now defined as that value of Q~c corresp onding to
the Y~c which is
determi ned from Eq. (39). Hence, when the conditio ns of Eq. (39}
are applied
to Eq. (32), the result IS
=a,
-
d*V
d*t
Q~c=m--/c -F~c.
(40)
When Eq. (39) is applied to Eq. (37), it takes the followin g form:
-
Q~c
d*W
= 11m _
_1c + w, (a, V~c) + D (a~, V~c).
d*t
Moreov er, the explicit form of
d*W~/d*t
(41a)
is
(41b)
where
There now remains the task of determi ning the value of the arbitrar
y constant /1. in Eq. (41a). To do this, an extra restriCt ion on the form
of Q~c is
invoked which arises when Eq~ (34) is studied for a possible physica
l interpre tation. One alternat ive here is to interpre t the q1;1antity mf Y~cpd 8x
as a sort of
total potentia l moment um of the "vacuu m". A natural extensio n
of this is to
assume the existanc e of the function mA (x~c, t) which can be interpre
ted as a
potentia l energy associat ed with the "vacuum"~ mf Apd8x would thus
be the total
"vacuu m" potentia l energy at time t. In analogy to the moment um
relation of
Eq. (34), the energy relation is then
(42)
where
d*A
- - B.A+ V,(a,A)
d*t
+D(a~,A).-
Equatio n ( 42) results from an analysis very similar to one conduct ed
in Append ix
C in associat ion with Eqs. (33) . Since the pote~tial energy field
is closely connected with the moment um field mY,., it is related to the total "force"
R~c ( =F~c
+ Qk) acting on the particle . This relation can be obtaine d from Eq. ( 42) where
the quantity d* A/d*t is interpre ted as the intrinsi c rate of change
of the energy
field plus the rate of work done on the ensemb le by the "vacuu m"
at a given
A Diffusion Model for the Schrodinger Equation
1903
point. From its primary definition, *l the rate of work can be shown to be [ R; IT;
+ D (a;R;)]. Hence it follows from Eq. ( 42) that
(43)
Since it has already been assumed that F K = -a K(f), where (f)
tential function, Eq. ( 43) leads to the result that
IS
a scalar po(44)
Equation ( 44) thus sets a restriction on the form of Qk in that it must be the
gradient of a scalar. It is this condition that is to determine the value of the
arbitrary constant fl.. From Eqs. ( 41a) and ( 41b) it is readily seen that the condition of Eq-. ( 44) can only be satisfied if fl.= 1. For this value of 11, Qk = Qk,
and Eq. ( 40) takes the form
(45)
Upon adopting the values fl.= 1; a;Yk = 0; vk; and arYk =Or vk; it IS apparent
that (with the aid of Eqs. (41b) and (45)) Eq. (38) is satisfied. Hence, the
· initial trial assumption that a;Tk = aiYk was justified.
Since the values of fl. and Yk have been completely determined, it is seen
(Eq. (31)) that the values of the coefficients ak, (3;k and riJk are also determined.
These can now be substituted in Eq. (14) to see they are able to satisfy it. If
they do, then the diffusion process equations developed here can be considered
equivalent to the Schrodinger equation and the objective of the present work has
been attained. This is indeed found to be so and can readily be verified if the
following two alterations in expression form are noted:
a;[Dp(a;Ck)] =a;[Dp(a; Vk)]- (p/2m)Qk,
(46a)
ak=a" Vk-Fk/m= Qk/m- F;(a; Vk) -D(a~; Vk).
(46b)
Equation ( 46b) was obtained by substituting Eq. ( 45) in Eq. (31).
Finally, it is at least a matter of pedagogical interest to note that Eq. ( 45)
is the equivalent of the momentt~m conservation relation (13). [Equation ( 45)
results when Eqs. (7) and ( 46a) are used in Eq. (13) .] In attempting a physical
interpretation of Eq. (45), it must be realized that a special kind of time derivative of Vk (denoted by the symbol d*Vdd*t) is being taken at a given space
point. This kind of derivative takes into account the diffusion motion of the
particles constituting the probability fluid. Moreover, Qk is now the averaged
QM "force" and can be rewritten to have the form mDak [ (8~; ~p/Jp]. It is
very interesting to note that with D=h/2m, Qk can be recognized as having the
*l The result here follows from the evaluation of the primary expression for the mean work
done on the ensemble by the "vacuum" at a point in space. In the time interval At this is f[R;e-t]sdse-.
Incidently, the specific form of the intrinsic change in field energy is J CilrA) pd 3x.
1904
L. Bess
same form but twice the
point of interest here is
in the present treatment,
had to be assumed as an
value of the QM "force" proposed by Bohm. 2l Another
that the form of Qk (as a function of p) is derived
but in Bohm's treatment the form of the QM "force"
extra condition.
§ 5. Suppleme ntary comment s
It is recognized that the interpreta tion of the Schroding er equation as given
by the diffusion model of § 4 is still incomplet e in that the physical origin of the
impulses (Ik). is not specified but only some of their phenomen ological characteristics are described. A brief preliminar y atte.mpt is now to be made in this section to deal with this neglected feature using a treatment that ,is still only rough
and qualitative .
Actually, the origin of the impulses (lk). has already been considered to the
extent of making the assumptio n that they arose from the interactio n of the "vac"
uum" with a given particle. In bringing support to this view, the work of several investigat ors was cited. However, for the view of some of these investigators,11l'12> saying that the particle is acted on by the "vacuum" is an over-simpl ification. Their interactio n mechanism is much more specific and requires that the
"vacuum" be filled with a stochastic electroma gnetic field associated with the
zero point energy ihw of each frequency mode w that arises from quantum electrodynami cs. They assume that the random impulses on the particle arise when
it interacts with the stochastic field according to the laws of classical electrodynam1cs.
In the proposal being made here, the interactio n between the "vacuum" and
a given particle is assumed to arise from a stochastic zero point field just as in
the proposals of Surdin and Boyer. However, the particle is assumed to interact
with the field according to the laws of quantum electrodyn amics instead of classical electrodyn amics. The interactio n itself is to consist of a random time sequence of interactio n events. A typical interactio n ev·ent begins by the particle
emitting a photon and acquiring a recoil impulse of the amount m (uk).. At a
time (-r). later the particle absorbs the same type of photon f~om the stochastic
field and receives an equal and opposite impulse -m (uk). which brings it back
to its initial state of energy and momentum . It is also possible to have an 111teraction event where the photon is first absorb~d and the same type of photon
emitted a time (-r). later.
The necessity of including both emission and absorption of photons in an
interactio n event becomes apparent when it is noted that, when a particle emits
(or absorbs) a single photon so that momentum is conserved , energy cannot be
conserved . However, in quantum mechanics there can be a violation of the conservation of energy for short periods of time. More precisely, time varying purturbation theory of quantum mechanics 18> provides the following relationsh ip, JEJt
1905
A Diffusion Model for the Schrodinger Equation
= bh (JE is the energy discrepancy of the quantum mechanical system at a time,
Jt, after the initiation of the purturbation.
b is a numeric constant with a value
around unity.) In the interaction events described here it is seen that there is
a period (r), when there is an energy discrepancy. However, the second impulse
of the event restores energy balance.
For the system being considered here, the,energy discrepancy is JE=<CJE),)E
)E is the
(where (JE), is the energy discrepancy for a single particle and
ensemble average at the event point, (xk, t) .) If it is assumed that the interaction event beginning with a photon emission occurs with the same frequency as
events beginning with an absorption, it can be shown that JE= (3m/2) (ux), 2)E
= (3m/2) <u 2) 1 . Moreover, for the present system, the quantum mechanical energy
discrepancy relation would take the form
<
<
Noting that the diffusion constant D=i<u 2) 1 <r)1 , it is seen that the proposed
interaction mechanism leads to an important result which had to be assumed as
an extra condition in all other diffusion process treatments. This is that D= (b/3)
X (h/m), which states that D is a constant depending only on two physical
parameters (i.e., on h and m).
It is apparent that the interaction process just described accounts for the
components m (uk). and - m (uk).- 1 of the impulse (lk), in Eq. (27a) (since
corresponding to the component m (uk), in the impulse (lk), there is an equal
and opposite impulse component - m (uk), in (lk),+l). It is to be noted that no
mention was made of the components mYk (z,) and mYk (z,_ 1) . This is felt to
be justified in an initial rough analysis such as was conducted above. The main
reason for this is that the components m (uk). and m (uk),_ 1 are completely uncorrelated and hence it is reasonable to account for them by interactions with two
different photons. The components mYk (z,) and mYk (z,_ 1) , on the other hand,
are practically completely correlated and, since they occur at the same time,
nearly always cancel each other out. The residue m (iJYk/iJt),_ 1 (r),_ 1 is- much
smaller than mYk (z.) and mYk (z,- 1) and even smaller than m (uk), and m (uk\- 1•
Hence, to a first approximation, the effects of the term m (?JYkj(Jt). (r), can be
neglected in describing momentum changes associated with the calculation of the
diffusion constant D and there is no need to account for the effects of the Yk
field by any photon interaction model.
It is important to note, however, that the inclusion of the impulse component
m (iJYk/iJt), (r), in (lk), of Eq. (27b) is vitally necessary if a physically viable
diffusion process is to be devised that can be governed by the Schrodinger equation in a rigorously accurate manner. As is seen, (iJYk/iJt). must be dependent
on the instantaneous particle velocity as well as its position. Furthermore, one
of the most important results of the present work is to show that the desired
1906
L. Bess
type of diffusion process cannot be obtained by merely g1vmg the. value, h/2m
to the diffusion constant D and assuming that the net force at .any point is the
classical force Fk as has been attempted. 7 ),B) It is also necessary that the net
force includes the component Qk which is strictly of quantum mechanical origin
(and arises here from the impulse component m (fJYk/fJt).).
Some comments on a different topic, which could lead to. some new physical
insights, are now to be included in this section. This topic has to do with the
physical basis of Eqs. (26) and (28a). A study of these equations would indicate
that one possible interpretation here is that the random impulses (lk). acting on
a typical particle not only describes how the fluctuating force Ak (t) affects the
particle but also how the external force Fk (supposedly not quantum mechanical
in origin) affects it. More precisely, this means that, unlike classical physics
where Fk directly affects the motion of the particle (by determining its accelera"
tion), in the present case Fk only indirectly affects the particle motion. It does
this by influencing the generation of the impulses (lk).. It is only these impulses
that can directly affect the particle motion (i.e., only (lk). directly controls the
kinematic parameters). The force Fk itself can directly affect only the momentum
field Yk.
This interpretation allows a rather gratifying simplification in basic theoretical conception. It leads to the conclusion that both the "classical" force Fk
and the QM "force" Qk have the same physical origin (i.e., they both arise
as a consequence of the action of the impulses (lk). on the particle). Both Fk
and Qk can be considered to be just different components of the same total "force"
1?+
Actually, the idea that the force Fk arises from an interaction with the "vacuum" by a random time sequence of impulses is not really new. It is already
implied in the conventional treatment of quantum mechanics when charged particles interact with an electromagneti c field (i.e., in second quantization). A
charged particle interacting with a static coulomb field is explained by assuming
that the particle is constantly emitting and absorbing longitudinal photons (i.e.,
the virtual photons). This type of process can readily be interpreted as an interaction of the particle with the "vacuum" through a very rapidly fluctuating
interaction force and is thus seen to be, at least qualitatively, very similar to the
mechanism for the origin of the force Fk proposed here.
It is to be emphasized now that the physical interpretation of .the origi,n of
Fk described above is only one possibility and is included only as supplementary
material. It is not to be considered a part of the main treatment presented above.
It may be of interest to point out that the, interpretation of quantum mechanical phenomena as given by the physical model described here has the same ideological basis as the interpretation proposed by Lande. 14) Both interpretations are
"unitary" in that the particle aspects of the microphysical processes are considered to be the primary entity and the wave aspects arise (almost fortuitously)
A Diffusion Model for the Schrodinger Equation
1907
:as a consequence of the various laws governing the particle motion. The details
of the two treatments, however, are quite different.
In conclusion, it is suggested that the demonstration that a physical model
can exist for the Schrodinger equation could be a result of more than pedagogical
interest. The physical model could be of value in primary research just as it
has been in classical physics where it· worked hand in hand with mathematical
development as an analytical tool.
The author gratefully acknowledges the valuable discussions that were held
with Dr. Louis Chandler.
Appendix A
The object of this appendix is to derive Eq. (9) in the text from Eqs. (6)
and (8). In order to evaluate the integral of Eq. (6), the various factors are expanded in a Taylors series in a manner shown by the following relations:
(Al)
and a similar relation for Vk (xk, t +Lit) obtained by substituting Vk for p in Eq.
(Al):
(A2)
and a similar relation for Vk (xk- ~\, t) obtained by substituting Vk for p in Eq.
(A2):
(Vi Vi) Lit+ (~i~j) (Vi Vj)
4D
8D2
J
'
where
N(~)
= [ 4nD]- 312 exp[-
(~i~J /
( 4DL1t)].
(A3)
In the above equations, the various factors were expanded to include only
those terms that could contribute to the integral of Eq. (6) to the first order in
jt, If Eqs. (Al), (A2), (A3) and (8) are substituted in Eq. (6), the integral
can readily be evaluated by straight forward algebraic manipulation and by making use of the following relations:
and
(A4)
where
(JiJ IS
the kronecker delta.
1908
L. Bess
When the operatio ns indicate d above have been carried out, it is noted
the
terms not having .dt as a factor cancel out. If, then, only those terms
with a
.dt of the first order are retained , the result readily become s convert
ed into Eq.
(9) of the text.
Appen dix B
The purpose of this appendi x is to derive Eq. (24) in the text. This
derivation is actually to be obt!lined by general izing a more simple case.
_It is assumed
that g (~, v) is a distribu tion function in a one-dim ensiona l velocity
v that is also
a function of ~. a one-dim ensiona l space variable . The charact eristics
of g(~, v)
are such that fg(~, v)dv= l. The quantiti es V(~) and V 2 (~) are
defined as
V(~) =fvg(~, v)dv and V 2 (~) =fv'g(~, v)dv.
Moreov er, let a(~) be a space
distribu tion function such that fa(~)d~=l. Now, let the "over-a
ll" average s,
(V) and {V 2) , be defined by
(V)==
(V')==
Ja(~) V(~)d~·,
Ja(~) V' (~) d~
(Bl)
.
It is obvious that <V> and <V') are not function s of~.
dispers ion ( (JV) 2)=<Y 2) -<V) 2 is then
By adding and subtrac ting
tion can be convert ed to
<CJV)2)=
fa(~) [V(~)]'d~
(B2)
The "over-a ll" velocity
to the r.h.s. of Eq. (B3), this rela-
Ja(~)JV'(~)d~+ Ja(~) V(~)
[
J
V(~)- a(~) V(~)d~Jd~
(B4)
=
where: .d P (~) Y 2 (~) - [ V (~) ] 2•
result for (.d V)') coverts into:
<
((JV)' )=
s
Using the definitio n of ( V) in Eq. (B4), the
J
a(~)JV 2 (~)d~+ a(~)[V(~) -<V)] 2d~
(B5)
(Note that fa(~)<V)[V(~) -(V)]d~=O). When Eq. (6) of the text
is compar ed
to Eq. (Bl), it is seen to be an obvious general ization of this relation
where the
variable , ~. is now in three dimensi ons and all the variable s are depende
nt on xk
in addition to their other depende nces. In fact, the followin g set of
correspond~
ences is seen to be in force here:
'
A Diffusion Model for the Schrodinger Equation
1909
and
(B6)
In order to be able to generalize Eq. (B5) in the same manner that was used
with Eq. (B1), it is necessary to find correspondences to the dispersion quantities,
AP(~) and <CAVY). Using V(~) and <V) as analogies, the choice obviously
IS:
and
(B7)
When the correspondences given by relation sets, (B6) and (B7), are used
m Eq. (B5), the result is Eq. (24) of the text.
Appendix C
In this appendix, Eq. (33b) of the text is to be derived from Eq. (33a).
When Eq. (5) is used in Eq. (33a) one of the terms on the l.h.s. can become:
JmHk+p+d x=m JJ[Hk+ (a"Hk)At]p(xk-~k' t)
3
(C1)
A change of variables is now to be introduced such that xk = xk- ~k Moreover, d 3x=d 3x, 'since it can be shown that the Jacobian J(x/x) = 1. Hence Eq.
(C1) can be rewritten (to the first order in At):
smHk+p+d x~m ss[Hk+~;(a;Hk) +!~;~1 (a~1Hk)
3
+ (a"Hk) At] p (:xk, t)
x8(xk+~k>
t+Atlxk,
t)d 8 ~d 3 x.
(C2)
It is to be noted in Eq. (C2) that the quantity Hk(xk+ ~k, t) has been expanded
in a Taylors series in ~k· The ~ integration in Eq. (C2) can readily be performed if it is noted that to the first order in At, 8 (xk + ~k' t + Atlxk, t) = 8 (xk, t
+Atlxk-~k' t).
Hence, the results of Appendix A can be used here. (Incidentally, note that f 8d 8 ~ = 1). When the ~ integrations in Eq. (C2) are performed,
and the result is substituted in Eq. (33a) the following relation arises:
[m
JHkpd x+mAt J(a<Hk)pd x+mAt JV;(a;Hk)pd x
3
8
+ mDAt
J
(a7;Hk) pd 8x
=At
SGkpd x
8
J- m
3
J
Hkpd 8x
(C3)
1910
L. Bess
In Eq. (C3), the "bar" has been dropped from
tegratio n. Eq. (C3) now simplifi es to the form
m
Xx
smce it
J[d;~k Jpd x= JGkpd x,
8
8
IS
a variable of in-
(C4)
where
It is obvious that no single relation betwe_en Gk and d*Hk/d *t satisfies
Eq.
(C4) but rather there is a family of relation s of the form
(C5)
-
where
One possible choice of Xk
IS
xk = w, ca,rk) + n ca:,rk),
· (C6)
where
is an arbitrar y vector. It has not been determi ned here whethe r
or
not the choice of Xk given in Eq. (C6) is unique. When Eq. (C6)
is used m
Eq. (C5), the result is Eq. (33b) of the text.
rk
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2)
3)
4)
5)
6)
7)
8)
9)
10)
11)
12)
13)
14)
E. Madelung , Z. Phys. 40 (1926), 332.
D. Bohm, Phys. Rev. 85 (19S2), 1966.
J. Meladier , Compt. rend. hebdom. B270 (1931), 193.
R. Furth, Z. Phys. 81 (1933), 143.
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