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Brownian motion of a quantum particle
__ __ l!!i!!J
&
EUEVIER
30 June 1997 PHYSICS
LETTERS
A
Physics Letters A 23 ! ( 1997) 52-60
Brownian motion of a quantum particle B.B. Kadomtsev, M.B. Kadomtsev Kurchatov Institute, Moscow 123182, Russian Federation
Received 26 February 1997; revised manuscript received 16 March 1997;acceptedfor publication 18 March 1997 Communicated by V.M. Agranovich
Abstract Brownian motion of a heavy particle in a light-atom gas is examined in the Schrodinger representation. We start from the presumption that the binary collisions are capable to make decoherent the wave functions of light atoms. A collapse of the wave function of a heavy particle is described. We derive an equation for a heavy-particle random wave function, resembling the Langevin’s equation and having the structure of a Schrbdinger’s equation supplemented with an additional operator. The latter is responsible for stochastic impacts exerted by the gas atoms and molecules. 0 1997 Published by Elsevier Science B.V.
1. Introduction
Whatever a quantum theory manual is, it surely formulates first of all the principle of wave-corpuscular dualism. It is well known that the Schriidinger equation describes the wavelike properties of particles, while their corpuscular nature is revealed in measurements. A characteristic example is the recording of an electron-wave interference pattern by means of a photographic plate. The emergence of a perfect image depends on the accumulation of many visible spots in the photographic emulsion. In general, any confirmation of the fundamental relation p= 1$12, which links the probability p with a squared modulus of a wave function I,!I,involves a sufficiently large number of elementary acts. Fig. 3 from a review by Namiki and Pascazio provides a lucid illustration of how this happens in practice [ll. The occurrence of each visible spot within the photographic emulsion is due to a collupse of the wave
function of an individual recorded electron, At this moment the electron wave function suddenly disappears everywhere outside the “spot” volume. Moreover, the actual mechanism of the detection (measurement) is of nearly no importance: the same collapses would occur if, instead of a photographic plate, a cloud chamber was chosen as an electronsensitive detector. In principle, it even makes no difference whether the particle is “measured” deliberately or spontaneously (e.g., due to atomic collisions by themselves). By definition a measurement is accomplished, if the considered particle had left behind some “memory” in the macroscopic medium of the detector it visited. What happens when a test particle penetrates into a macroscopic body (e.g., a gas bulk at room temperature)? The very first act of particle-gas interaction will immediately cause a collapse of the particle wave function. Somewhat later the test particle will experience the gas Brownian motion. Afterward it
03759601/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SO375-9601(97>00281-8
B.B. Kadomtsev. M.B. Kadomtsev/Physics
will achieve a thermal equilibrium with the gas and start to diffuse in space. The diffusion is described by a master equation for the density matrix. Quantum-particle Brownian motion has been treated by many authors (see, e.g., Refs. [2-51, and references therein). The relevant mathematical apparatus they used is rather sophisticated. It tends to the operator calculus, in other words, to the Heisenberg representation. However, there exist experimental situations where the initial state of the quantum Brownian particle is represented with a spatially continuous wave function. Evidently, the experimental data of this kind would hardly be accessible to theoretical interpretation in the framework of the conventional approach. Against this background, the invention of a simple and intuitively clear alternative description of the quantum-particie Brownian motion seems to be of paramount interest. In what follows we develop such a description in the Schrijdinger representation for the particular case of Brownian particle behavior. 2. Theory It was Kleinert and Shabarov who explored the possibility of describing the Brownian motion of a quantum particle by means of a Langevin’s equation for the coordinate operator [5]. Nevertheless, a wave description seems to be more elegant and intuitively clear. If so, there arises the question whether is it possible to compose a Langevin-type equation directly for a heavy-particle wave function Ilr(R, r> in R = {X, Y, Z] coordinates. In other words the problem can be formulated as follows. Is there a random wave function for which one can set a phenomenological equation, similar to a Langevin’s equation for a random velocity of a classical particle? Indeed, the new equation should be no other than a generalization of the Schrodinger equation. Hence, the wave function of a heavy-particle q should satisfy the equation a!P
fi2
53
Letters A 231 (1997) 52-60
experiences only one collision and, consequently, receives a momentum hq. The result of such an impact can be taken into account by multiplying the wave function p by the factor exp(iq - R). Evidently, the spatial region involved in a single-molecule impact cannot be large, because the motion of gas molecules is decoherent (i.e., highly chaotic). Since the momentum is transferred to the heavy particle as a whole, the wave function of the latter should be immediately nullified everywhere except in the spatial region of the momentum transfer. In a multiple-impact case the operator can be constructed as ifixs(r-
rN)@,,, exp(iq,
- R).
(2)
N
Here, t, and tiq,., represent the time of the Nth impact and the momentum transfer in the Nth impact, respectively. This equation includes a form factor rDN describing the collapse of the wave function that occurred in the vicinity of a random point R,. Of course, all the N point positions R, must be intercorrelated. The temporal distribution of the impacts is presumably Poissonian. Let r be the average time between successive impacts. The probability p,,, of the absence of any impact within the t - t, time span (starting at the moment of the Nth impact) is equal to P,,, = exp[ - (t - rN)/r]. Consider now the one after the Nth. The probability of the occurrence of the next impact within the r-r + At time interval is equal to PN
At/r-
From the gas molecule viewpoint, the appearance of the factor pN can be plausibly attributed to the attenuation of the amplitude of the heavy-particle wave function produced by the light atom scatterings. As time goes, this initial amplitude is being attenuated in accordance with the exp[ -(t - r, )/25-l law. Therefore, we express the operator K in the form of K=
7%
-iz+ihC6(r-f,)GN
exp(iqN’R).
N
VR2?P+K!P.
(1)
(3)
Here K represents a stochastic operator that describes the random kicks of gas atoms. Let us try to explain what operator K is, starting on a qualitative level. Thus, consider a particle that
So we have come back to (1) which is similar to the Langevin equation. Operator (3) is analogous to the random force F. The first term in the right-hand side of (3) represents the monotonic decay of the wave
itix
= - 2~
54
B.B. Kadomtsev.
M.B. Kadomtsev / Physics Letters A 231 (1997) 52-60
function: this is an analog of the friction force appearing in the classical Langevin equation. The second term in (3) describes the stochastic impacts caused by gas atoms and molecules. These impacts are capable of making the considered particle (along with gas molecules) pass over from one to other Hilbert space. In this respect our approach is similar to the Machida-Namiki theory [6] involving multiHilbert spaces. Otherwise, it differs from this theory, because we assume the occurrence of collapses in individual events. The wave-function attenuation, described by the first term in (3), implicitly takes into account the disappearance of the coherence. This attenuation has an analog in theoretical nuclear physics. The optical model of neutron-nucleus collisions comprises a term responsible for the absorption of the neutron wave function. This absorption was introduced for purely phenomenological reasons. In order to obtain the form factor Qs, appearing in (31, it was necessary to.know what the form is of the wave functions of gas atoms (or molecules). Recently we showed that the wave function of any atom (or molecule) of a gas can be described as a wave packet, moving along straight segments of the atom (or molecule) trajectory [7,8]: the straight segment length depends on successive random collisions with other atoms. Each act of scattering (an intrinsic measurement) results into an additional shrinkage (contraction) of the wave packet. This piece of knowledge enabled us to find explicitly the form factor Qp, (see the second term in (7)). A corresponding derivation is briefly stated below. Consider now a heavy particle with mass M surrounded with light particles (whose mass is m> in a gas where m GZM. Presumably, the heavy particle size is much less than the light-particle free path. Let us denote the heavy particle coordinates as X, Y, 2 and assume that the heavy particle is in the vicinity of the coordinate origin. In the zero approximation (with respect to a small parameter m/M), the heavy particle remains at rest for some time. At this point, follow one of the light particles. Owing to its interaction with the heavy particle, the light particle undergoes a scattering. Let the wave function of the incident light particle be expressed in terms of a wave packet, rather similar to a plane wave I++= exp( - i o t + i ki - r), where k, represents the initial wavenumber vector (so far, we disregarded
the wave function normalization). Assuming the scattering to be spherically symmetrical, we can express the scattered S-wave in the form of
I,$= Aexp[-iot+ik,*(r-R)
+iqo].
(4 Here a and cp represent the scattered wave amplitude and phase, respectively. The wavenumber vector of the elastically scattered wave is assumed to be k,=k(r-R)Ir-RI-‘, where k= Ik,l. At the scattering point r = R the phase of the scattered wave must coincide with that of the incident wave, i.e., cp= ki R. Now we are going to take into account all the gas particles and define the atomic-molecular chaos hypothesis. -First, similar to the classical case, we presume that before the scattering UC? the light and heavy particles are not at all intercorrelated. -Second, after the light-heavy particle interaction the wave function of the scattered light particle undergoes a decoherence due to subsequent scatterings on other light particles, i.e. on the gas atoms and molecules. (The gas temperature is assumed to be sufficiently high for this effect.) Indeed, owing to the scattering on other light particles - atoms and molecules - the structure of the wave function of a given light particle becomes more and more complicated. -Finally, the given light-particle wave function splits into decoherent wave packets. Presumably, this particle certainly gets only into one of them. This is what is called collapse of a waue fincfion. Hereafter, any irreversible destruction of the coherence of a wave function will be attributed to a large number of random collapses of this wave function and respectively described. At this point it is appropriate to find out what happens to the wave function of the heavy particle. The collapse of the wave function of a light scattered particle is assumed to occur at a distance L from the coordinate origin. The heavy particle is in the vicinity of the coordinate origin. The X-axis is oriented along the direction of propagation of the scattered light-particle wave packet. At large distances L z+ R, l
B.B. Kadomtseu.
M.B. Kadomtsev/
the wave function of the scattered light particle can be approximately expressed as -R
-iwt+i(k,-k,) ikx
ik (y-Y)‘+(z-Z)* L
+z+z
(5)
I
Here we have taken into account that kx = k, R. Let a rise of decoherence make the wave function (5) be reduced (collapsed) over x, y. As a result, there appears a wave packet localized over y and z, namely
Here b represents the packet localization width. Let us “project” the scattered wave onto the collapsed state of i,!~r.This amounts to multiplying (5) by I+!+* and averaging the result over y, z and gives
k2b2(Y2
+Z’)
2( L2 + k2b4)
iL
x
( ii l-3
(6)
Presently, it should be taken into consideration that the joint wave function of both the heavy and light particles has a form of WR, r). Hence, a collapse over y. z automatically leads to the reduction of the wave function of the heavy and light particles, according to form factor (6). If so, we seemingly have a pair of correlated particles involved in the “thought experiment” of Einstein, Podolsky, and Rosen. One can state that - in this particular subensemble - a collapse of the scattered particle makes the heavyparticle wave function undergo a contraction over both the Z and Y directions, in accordance with form factor (6). It is sensible to assume that the length L, involved in the collapse of the light particle, characterizes the light-particle free path A in a gas consisting of light atoms. As to the localization width b, it turns out to be of the order of \Ih, [7,8]. Here A, represents the average de Broglie wavelength of gas atoms. Rigorous reasoning validates the relations b2k - h - L. However, it is more practical to assume
55
instead that L + 0. This would simplify the following formulas at the price of tolerating slightly wider “inciting” collapses. It turns out that there is an opportunity to additionally simplify the form factor (6). Indeed, our previous choice of the X-axis was arbitrary. In the general case, the localization of the light-particle wave function is permissible over any direction without restriction. Hence on the average, one can replace quantity Y * + Z* with fR* for any collapse. Taking advantage of this circumstance, we get approximately 2
@=exp
$=exp[-iwr+ikw-(y2+x2)/2b2].
-
Physics Letters A 231 (1997) 52-M)
-i(ki-kf)
*R-
$
Here R = (X, Y, Z) represents the radius vector of a heavy particle within a reference frame where the collapse occurs near R = 0. The first term in the exponent of (7) merely represents the momentum interchange. The light particle aquires a momentum hq = h(k, - ki). Meanwhile, the heavy-particle experiences a recoil equal to -hq. The second term in the same exponent corresponds to the form factor GN appearing in relation (3). This term asserts that a collapse of a light-particle wave function is accompanied with a similar collapse of the heavy-particle wave function. Thus, the role of the atomic-molecular chaos in the quantum realm is not restricted solely to cause random impacts exerted by colliding particles. It also ensures successive collapses of the heavy-particle wave function, making this particle behave in some respects like particles in classical physics. The very first collapse of the wave function of a scattered light particle causes a localization of the heavy-particle wave function in accordance with form factor (7): the localization scale is of the order of b. The further collapses of scattered light particles sharpen more and more the localization of the wave function of a heavy particle. The mechanism of the collapse-produced localization of wave functions will be clarified a little later. Now one might ask how to describe the motion of a quantum particle, if the observations are inaccurate and the resolution power of the instruments is insufficient for revealing the “intrinsic structure” of wave packets. The answer is trivial: there exists a sole description, and it is close to the classical one.
56
B.B. Kadomtsev, M.B. Kadomtsev/Physics
Normally, the scale of the path traversed by a particle in the course of Brownian motion substantially exceeds the wave-packet localization size. If so, the motion of a packet is describable in terms of packet-“center mass” coordinates R, = (X,, Y,, Z,) and of the average velocity V,. These quantities are defined as follows, R, = 19
* (R).RP(R)
d3R,
(8) d3R,
V,,= -i$lt*(R)RO,Y(R)
(9)
where T(R) is a wave function of the heavy particle. Due to random impacts exerted by light particles, R, and V, become random time-dependent functions. These functions need to be statistically described. In the framework of an approximation, close to a classical description, the effects of the impacts can be expressed with two equations =F,
M-z
dR0
dt
V
The substitution of the expression for ?P into (12) gives
G +V.
where F is a random force. Thus, we come back to the classical Langevin equation. The quantum properties of the considered particle are described with the nondiagonal elements of the density matrix
V,,A=
R,(R,
R’, V, I”, t)
=(A~R,
v, t)A(R’,V’,
R’, t) = (‘P*(R,
t)P(R’,
t)).
t)).
(14)
With the aid of (13) we obtain z
+(v.
VRR+V. VRR’)PO
= -&(V;-
v;>p,.
(15)
It is suitable to introduce new variables v+v’ R,=
+(R+R’),
V,=
Q,=+(R-R’), p(R,
(13)
;&‘A,
where V = hK/M. If fi.--+0, we obtain an equation for a free transfer of amplitude A with velocity V. If there is a superposition of wave packets, having different wavenumber vectors, each value of K corresponds to its “own” amplitude A. Therefore, the amplitude A can be regarded as an A(R, V, t) function. Eq. (13) is valid for any of these amplitudes. Let us compose now a P,(R, R’, V, V’, t) matrix
( 10)
”
Letters A 231 (1997) 52-60
2,
(16)
W,,= y.
(17)
(II)
Here ly is a wave function, and the brackets mean averaging over the statistical ensemble. Instead of using the ordinary density matrix, it turns out more appropriate to describe the statistical properties of a quasi-free quantum particle with a somewhat modified matrix representation. Consider now the Schriidinger equation for the wave function of a free heavy particle,
After the substitution of these variables, Eq. (15) becomes dP0 at
=
* v~RoPo+w*
+
vo
-
g_yRo
VePo
~(VQ~O).
(18)
Assume PO to be a sufficiently smooth function of Then it is possible to. set h -+ 0 and to neglect the right-hand side of Eq. (18). Let us assign
R,. a!P ih-=--
at
g
v;!P.
(12)
Let us assume the wave function ?P(R, t> to be very similar to a plane wave ?P=A(R, t) exp(-Ot + iK - R), where 0 = iiK 2/2 M is the frequency and K represents the wavenumber vector. Function A( R, t) is an envelope weakly dependent on both R and t.
R,=F(R,,
V,, t)U(Q.
W, t).
Then, according to (18), the distribution function F
satisfies the free-motion equation dF x + Vo - VRoF= 0.
(19)
B.B. Kadomtseu, M.B. Kadomtsev/Physics
Having separated the variables, we obtain an equation for the function II, g
+
g&K-K’) *v,u=o,
(20)
where V = iiK/M. Now it is clear that U(Q, W) is an analog of the
Wigner function, i.e., a density matrix in a mixed representation. Let us replace the wavenumber vectors K and K’ in (20) with the corresponding operators -ifiV,r and -ifiVR, and pass from the K, K’ (momentum) representation to the ordinary one. The result consists in the equation
au ihx
h2 = - 2~
(21)
QQ21J.
It is clear that in the absence of collapses, the function U satisfies the free-motion Schrodinger equation. Thus, we are able to express the distribution matrix in the form of P,,(R,
R’, V,,, f) =F(R,,
R - R’ V,,, t>U 2
(
57
Letters A 231 (1997) 52-60
Since for some time velocity V is equal to zero, let us examine, first of all, the coordinate-dependent part of the distribution matrix. Already the very first collapse of the scattered light particle, occurring in the vicinity of R = 0, leads to a localization of the heavy-particle wave function, making it shaped according to (7). Thus, after the first collapse within the considered subensemble with a collapsed wave function we have
(23) Here A is a normalization coefficient. With the aid of (23) it is easy to obtain the density matrix of the collapsed subensemble PJR,
R’) = p(R)‘J’(R’) =A2exp
-g-
R=
(R-R’)’ 6b2
i
(24)
1
. (22)
Here, the distribution function F(R,, V,, t> corresponds to the diagonal part of the distribution matrix. On the other hand, the factor U[(R - R’)/2], normalized as U(O) = 1, describes the nondiagonal terms of the distribution matrix (i.e., of the envelope density matrix). Let us inspect the problems of the description of nondiagonal elements of the distribution matrix. In the m +z M approximation, the first term in the exponent of form factor (7) can be temporarily neglected and the heavy particle in the initial state can be regarded as motionless. Let ?PO(R) represent the initial wave function. Then, the initial distribution matrix (when V = 0) coincides with the initial density matrix, and both of them are proportional to the product !P * (R)P& R’). Accordingly, the distribution function F&R, V>, pertaining to the initial state, is given by the relation
Here, we used the notation R, introduced in (16). Evidently, p,,(R, R’) splits into two multipliers: diagonal factor F, and nondiagonal factor U = exp[ - (R - R’j2/6b2 I. The presence of the diagonal factor F, is due merely to a casual choice: we have chosen a subensemble, corresponding to the occurrence of a collapse near R = 0. In principle, a localization of this kind may occur near any arbitrary point R = R,. In order to remain within the initial ensemble, it is necessary to average expression (24) with a weight factor over all the possible collapse coordinates (A should be proportional to 1W,,(R,) I 2>. Consequently, a few early collapses transform the density matrix into the expression U()R-R’l).
V) = l*o(R)2)l
S(V).
Meanwhile, the diagonal part of the density matrix is p(R,
R’) = 19, I*.
(25)
Here the nondiagonal part of the density matrix is described by the function ,
F,(R.
I.
(26)
where A = b. The further collapses of the light-particle wave functions lead to a monotonic decrease in A as time
58
B.B. Kadomtsev. M.B. Kudomtsev/Physics
goes on. In order to understand how this happens, let us trace the behavior of a subensemble whose initial wave function is like (23). The centers of the successive collapses might slightly shift off the coordinate origin. However, this effect can be ignored because we are mostly interested in the nondiagonal part of the density matrix. Indeed, the positions of the nextstep collapse centers are predetermined at preceding steps and should be distributed proportionally to I?P I*. The latter makes us believe that all the successive collapses would be concentrated around the origin of coordinates. In this approximation, every new collapse manifests itself in a multiplication of the previous wave-packet form factor by an additional multiplier having the form of (23). To simplify matters, assume the effective cross sections of the light-light and light-heavy particle scattering to be the same. If so, the average time between collapses is equal to +r= h/v,, where h = l/cm. In addition, (+, n, and vr represent the cross section of the just mentioned scattering, the numerical density of light particles, and their thermal velocity, respectively. Because each new collapse adds another multiplier (23) to the packet form factor, the wave packet becomes progressively thinner as time goes on. Its size tends to zero according to the following law, ?P(R,t)=A,exp
(27)
where A, is a normalizing coefficient, whose value depends on the subensemble choice. Elementary calculations transform (27) in a!P = at
Letters A 231 (1997) 52-60
elasticity, the last term being responsible for the conservation of the wave fanction norm. In the long run, the solution of Eq. (29) becomes stationary and can be sought in the form of ?P= exd( - i fl t - R*/2 A*>]. Inserting this expression into (291, we obtain A-“=(l-i)Ai, R=-
A,=b
3h 2M4
(30)
’
Here we have assigned r = 3fi2/2 MAi for ensuring the frequency 0 to be a real quantity. The knowledge of A enables us to evaluate the quantity AE = h*/MA*. It expresses an increment in the heavy-particle kinetic energy, which is due to the wave-function collapse. The energy increment AE may be substantially less than the average energy transfer Tm/M per act of light particle scatter-. ing on the heavy particle. It is noteworthy that when analyzing the collapse mechanism for A > b, it is superfluous to worry about the energy balance. Let us remember now the first term in the exponent of the exponential form factor (7). It is responsible for the transfer of momentum to the heavy particle from the scattered light particles. The transferred momentum sets the heavy-particle wave packet in motion. The wave function of a moving wave packet has the following form, (R-R,)* TP‘b”o
2A2
+i-
MV, n
(31)
-kR*$l+m.
Here k= 1/3b*T and I’=A,/A,. We assume r to be constant. When the localization of the wave function becomes sufficiently sharp, it is necessary to take into account the kinetic energy of the particle. For this purpose, the Schradinger equation can be generalized by the allowance of collapses, C?P h2 i?ix = - 2~ pkzP,- ihkR*?P+ ifiT?P.
(29)
This equation differs from (28) by the allowance of the kinetic energy operator. We see that (29) is the equation of a quantum oscillator with imaginary
Here K, represents the wavenumber vector; R, is the radius vector of the packet center mass, and V, = ?X,/M is its velocity. The wave function is seen to be labeled with indexes R, and V,. The exponential factor exp[ - (R - R,,>*/2 A*] can be regarded as the envelope of a plane tvave. Knowing the form of the wave function, we are able to include the distribution matrix into our analysis, P,(R,
R’, V,, t) =F(R,,
V,, t)U(R-R’). (32)
Here F(R,, V,, t) is a distribution function corresponding to the diagonal part of the distribution
B.B. Kadomtseo,
M.B. Kadomtsev /Physics
matrix. A nondiagonal factor U describes the wave packets. If the distribution function is a smooth one over distances comparable with the packet localization size, the kinetic equation is applicable to trace the time evolution of fuhction F. As to the nondiagonal factor, it is given by expression (26) where the localization width A is time-dependent. It follows from (27) that, as time goes, the packet shrinks in accordance with the A = 6(7/t)‘12 law until the packet size achieves its stationary value A,,. According to (30), the stationary packet size is A, = b(m/M)‘/4. Evidently, the increase in the heavy particle mass M makes the particle localization size tend to zero. In this way, a quantum wave bunch is transformed into a classical point corpuscle. In gases, the “intrinsic measurements” - by themselves - are capable to make the heavy particles approach the behavior inherent to classical physics in natural conditions. Above we derived relation (30) on the assumption that the cross section (+ of the atom-atom scattering is the same as the cross section o’B of the atomBrownian particle scattering. In reality, (TV> (T, and estimates (30) should be revised. It is necessary to increase the quantity k, appearing in Eq. (28) (~a/(+ times. On the other hand, the stationary localization size, mentioned in (30), should be diminished (a,/~)‘/~ times. It is noteworthy that the relations (30) are valid only for small-size particles, so that (+a < AZ.
3. Conclusion To summarize, we found that for an adequate description of the behavior of a heavy particle in a gas it is convenient to somewhat modify the density matrix approach. Namely, we introduced a density matrix for the wave packet envelopes (not to be mixed up with the ordinary density matrix). In order to avoid such a confusion, we named the former distribution matrix. If the distribution of the diagonal elements of the distribution matrix is rather smooth, Bq. (15) for this matrix spontaneously splits into two independent equations. The diagonal part of the matrix corresponds to a distribution function satisfying the kinetic equation. The nondiagonal part corresponds to the wave-packet form factor U(R - R’).
Letters A 231 (I 997) 52-60
59
The hypothesis of quantum atomic-molecular chaos brings us to Eq. (29), which describes the wave packet evolution. According to this equation, the successive collapses enhance the wave packet localization (progressively diminishing its size). It also follows from (29) that the wave packet has a Gaussian shape in real space. The obtained results reconstruct the scenario of the Brownian particle motion in gas as follows. Independently of its initial state (a coherent state inclusive), the particle evolves in accordance with the solution of a Schrodinger equation with allowance of absorption. The latter represents the disappearance of coherence in the initial state due to birth of the scattered waves. Against this background, there occur collapses of the wave functions in arbitrary individual statisticalensemble members. The very first collapse in each ensemble member destroys the initial wave function and gives birth to a wave packet with a b N \Ihh, size (here A and A, represent the free-path length of light particles and the average de Broglie wavelength of the same light particles, respectively). The collapses that occur afterward additionally reduce the nondiagonal matrix terms of the distribution matrix (in other words, they contract the wave function). The principle of multicollapse reduction of wave functions had been devised in Ref. [S] and was further developed here. As to the statistical behavior of a Brownian particle, it is governed by the classical kinetic equation for the distribution function (i.e., by the diagonal part of the distribution matrix). Our main goal here was to show that the gas medium can by itself impart the classical features to the quantum particle. The initial scatterings of light atoms lead to the wave function collapse. Later on the corresponding wave packet continues to shrink, reaching some constant shape at large time scale. The description of the Brownian motion itself has been done by many different ways [ l-6,9,1 01.
References [I] M. Namiki and W. Pascazio, Phys. Rep. 253 (1993) 301. (21 J. Schwinger, J. Math. Phys. 2 (1961) 407.
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B.B. Kadomtseu,
M.B. Kadomtsev / Physics Letters A 231 (1997) 52-60
[3] A.O. Caldeira and A.J. Leggett, Physica 121 A (1987) 587;
130 (1985) 374 (E). [4] A.J. Leggett, S. Chkravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg and W. Zwerger, Rev. Mod. Phys. 59 (1987) 1. [5] H. Kleinert and S.V. Shabarov, Phys. Lett. A 200 (1995) 224. [6] S. Machida, M. Namiki, Proc. Int. Symp. Found. quantum mechanics, Tokyo (1983) p. 127.
[7] B.B. Kadomtsev and M.B. Kadomtsev, JETP 81 (1995) 897. [8] B.B. Kadomtsev and M.B. Kadomtsev, Chaos 6 (1996) 399. [9] M.B. Mensky, Continious quantum measurements and path integrals (IOP Publishing, Bristol, 1993). [lo] B.L. Hu, J.P. Paz and Y. Zhang. Phys Rev. D 45 (1992) 2843.
&
EUEVIER
30 June 1997 PHYSICS
LETTERS
A
Physics Letters A 23 ! ( 1997) 52-60
Brownian motion of a quantum particle B.B. Kadomtsev, M.B. Kadomtsev Kurchatov Institute, Moscow 123182, Russian Federation
Received 26 February 1997; revised manuscript received 16 March 1997;acceptedfor publication 18 March 1997 Communicated by V.M. Agranovich
Abstract Brownian motion of a heavy particle in a light-atom gas is examined in the Schrodinger representation. We start from the presumption that the binary collisions are capable to make decoherent the wave functions of light atoms. A collapse of the wave function of a heavy particle is described. We derive an equation for a heavy-particle random wave function, resembling the Langevin’s equation and having the structure of a Schrbdinger’s equation supplemented with an additional operator. The latter is responsible for stochastic impacts exerted by the gas atoms and molecules. 0 1997 Published by Elsevier Science B.V.
1. Introduction
Whatever a quantum theory manual is, it surely formulates first of all the principle of wave-corpuscular dualism. It is well known that the Schriidinger equation describes the wavelike properties of particles, while their corpuscular nature is revealed in measurements. A characteristic example is the recording of an electron-wave interference pattern by means of a photographic plate. The emergence of a perfect image depends on the accumulation of many visible spots in the photographic emulsion. In general, any confirmation of the fundamental relation p= 1$12, which links the probability p with a squared modulus of a wave function I,!I,involves a sufficiently large number of elementary acts. Fig. 3 from a review by Namiki and Pascazio provides a lucid illustration of how this happens in practice [ll. The occurrence of each visible spot within the photographic emulsion is due to a collupse of the wave
function of an individual recorded electron, At this moment the electron wave function suddenly disappears everywhere outside the “spot” volume. Moreover, the actual mechanism of the detection (measurement) is of nearly no importance: the same collapses would occur if, instead of a photographic plate, a cloud chamber was chosen as an electronsensitive detector. In principle, it even makes no difference whether the particle is “measured” deliberately or spontaneously (e.g., due to atomic collisions by themselves). By definition a measurement is accomplished, if the considered particle had left behind some “memory” in the macroscopic medium of the detector it visited. What happens when a test particle penetrates into a macroscopic body (e.g., a gas bulk at room temperature)? The very first act of particle-gas interaction will immediately cause a collapse of the particle wave function. Somewhat later the test particle will experience the gas Brownian motion. Afterward it
03759601/97/$17.00 0 1997 Published by Elsevier Science B.V. All rights reserved. PII SO375-9601(97>00281-8
B.B. Kadomtsev. M.B. Kadomtsev/Physics
will achieve a thermal equilibrium with the gas and start to diffuse in space. The diffusion is described by a master equation for the density matrix. Quantum-particle Brownian motion has been treated by many authors (see, e.g., Refs. [2-51, and references therein). The relevant mathematical apparatus they used is rather sophisticated. It tends to the operator calculus, in other words, to the Heisenberg representation. However, there exist experimental situations where the initial state of the quantum Brownian particle is represented with a spatially continuous wave function. Evidently, the experimental data of this kind would hardly be accessible to theoretical interpretation in the framework of the conventional approach. Against this background, the invention of a simple and intuitively clear alternative description of the quantum-particie Brownian motion seems to be of paramount interest. In what follows we develop such a description in the Schrijdinger representation for the particular case of Brownian particle behavior. 2. Theory It was Kleinert and Shabarov who explored the possibility of describing the Brownian motion of a quantum particle by means of a Langevin’s equation for the coordinate operator [5]. Nevertheless, a wave description seems to be more elegant and intuitively clear. If so, there arises the question whether is it possible to compose a Langevin-type equation directly for a heavy-particle wave function Ilr(R, r> in R = {X, Y, Z] coordinates. In other words the problem can be formulated as follows. Is there a random wave function for which one can set a phenomenological equation, similar to a Langevin’s equation for a random velocity of a classical particle? Indeed, the new equation should be no other than a generalization of the Schrodinger equation. Hence, the wave function of a heavy-particle q should satisfy the equation a!P
fi2
53
Letters A 231 (1997) 52-60
experiences only one collision and, consequently, receives a momentum hq. The result of such an impact can be taken into account by multiplying the wave function p by the factor exp(iq - R). Evidently, the spatial region involved in a single-molecule impact cannot be large, because the motion of gas molecules is decoherent (i.e., highly chaotic). Since the momentum is transferred to the heavy particle as a whole, the wave function of the latter should be immediately nullified everywhere except in the spatial region of the momentum transfer. In a multiple-impact case the operator can be constructed as ifixs(r-
rN)@,,, exp(iq,
- R).
(2)
N
Here, t, and tiq,., represent the time of the Nth impact and the momentum transfer in the Nth impact, respectively. This equation includes a form factor rDN describing the collapse of the wave function that occurred in the vicinity of a random point R,. Of course, all the N point positions R, must be intercorrelated. The temporal distribution of the impacts is presumably Poissonian. Let r be the average time between successive impacts. The probability p,,, of the absence of any impact within the t - t, time span (starting at the moment of the Nth impact) is equal to P,,, = exp[ - (t - rN)/r]. Consider now the one after the Nth. The probability of the occurrence of the next impact within the r-r + At time interval is equal to PN
At/r-
From the gas molecule viewpoint, the appearance of the factor pN can be plausibly attributed to the attenuation of the amplitude of the heavy-particle wave function produced by the light atom scatterings. As time goes, this initial amplitude is being attenuated in accordance with the exp[ -(t - r, )/25-l law. Therefore, we express the operator K in the form of K=
7%
-iz+ihC6(r-f,)GN
exp(iqN’R).
N
VR2?P+K!P.
(1)
(3)
Here K represents a stochastic operator that describes the random kicks of gas atoms. Let us try to explain what operator K is, starting on a qualitative level. Thus, consider a particle that
So we have come back to (1) which is similar to the Langevin equation. Operator (3) is analogous to the random force F. The first term in the right-hand side of (3) represents the monotonic decay of the wave
itix
= - 2~
54
B.B. Kadomtsev.
M.B. Kadomtsev / Physics Letters A 231 (1997) 52-60
function: this is an analog of the friction force appearing in the classical Langevin equation. The second term in (3) describes the stochastic impacts caused by gas atoms and molecules. These impacts are capable of making the considered particle (along with gas molecules) pass over from one to other Hilbert space. In this respect our approach is similar to the Machida-Namiki theory [6] involving multiHilbert spaces. Otherwise, it differs from this theory, because we assume the occurrence of collapses in individual events. The wave-function attenuation, described by the first term in (3), implicitly takes into account the disappearance of the coherence. This attenuation has an analog in theoretical nuclear physics. The optical model of neutron-nucleus collisions comprises a term responsible for the absorption of the neutron wave function. This absorption was introduced for purely phenomenological reasons. In order to obtain the form factor Qs, appearing in (31, it was necessary to.know what the form is of the wave functions of gas atoms (or molecules). Recently we showed that the wave function of any atom (or molecule) of a gas can be described as a wave packet, moving along straight segments of the atom (or molecule) trajectory [7,8]: the straight segment length depends on successive random collisions with other atoms. Each act of scattering (an intrinsic measurement) results into an additional shrinkage (contraction) of the wave packet. This piece of knowledge enabled us to find explicitly the form factor Qp, (see the second term in (7)). A corresponding derivation is briefly stated below. Consider now a heavy particle with mass M surrounded with light particles (whose mass is m> in a gas where m GZM. Presumably, the heavy particle size is much less than the light-particle free path. Let us denote the heavy particle coordinates as X, Y, 2 and assume that the heavy particle is in the vicinity of the coordinate origin. In the zero approximation (with respect to a small parameter m/M), the heavy particle remains at rest for some time. At this point, follow one of the light particles. Owing to its interaction with the heavy particle, the light particle undergoes a scattering. Let the wave function of the incident light particle be expressed in terms of a wave packet, rather similar to a plane wave I++= exp( - i o t + i ki - r), where k, represents the initial wavenumber vector (so far, we disregarded
the wave function normalization). Assuming the scattering to be spherically symmetrical, we can express the scattered S-wave in the form of
I,$= Aexp[-iot+ik,*(r-R)
+iqo].
(4 Here a and cp represent the scattered wave amplitude and phase, respectively. The wavenumber vector of the elastically scattered wave is assumed to be k,=k(r-R)Ir-RI-‘, where k= Ik,l. At the scattering point r = R the phase of the scattered wave must coincide with that of the incident wave, i.e., cp= ki R. Now we are going to take into account all the gas particles and define the atomic-molecular chaos hypothesis. -First, similar to the classical case, we presume that before the scattering UC? the light and heavy particles are not at all intercorrelated. -Second, after the light-heavy particle interaction the wave function of the scattered light particle undergoes a decoherence due to subsequent scatterings on other light particles, i.e. on the gas atoms and molecules. (The gas temperature is assumed to be sufficiently high for this effect.) Indeed, owing to the scattering on other light particles - atoms and molecules - the structure of the wave function of a given light particle becomes more and more complicated. -Finally, the given light-particle wave function splits into decoherent wave packets. Presumably, this particle certainly gets only into one of them. This is what is called collapse of a waue fincfion. Hereafter, any irreversible destruction of the coherence of a wave function will be attributed to a large number of random collapses of this wave function and respectively described. At this point it is appropriate to find out what happens to the wave function of the heavy particle. The collapse of the wave function of a light scattered particle is assumed to occur at a distance L from the coordinate origin. The heavy particle is in the vicinity of the coordinate origin. The X-axis is oriented along the direction of propagation of the scattered light-particle wave packet. At large distances L z+ R, l
B.B. Kadomtseu.
M.B. Kadomtsev/
the wave function of the scattered light particle can be approximately expressed as -R
-iwt+i(k,-k,) ikx
ik (y-Y)‘+(z-Z)* L
+z+z
(5)
I
Here we have taken into account that kx = k, R. Let a rise of decoherence make the wave function (5) be reduced (collapsed) over x, y. As a result, there appears a wave packet localized over y and z, namely
Here b represents the packet localization width. Let us “project” the scattered wave onto the collapsed state of i,!~r.This amounts to multiplying (5) by I+!+* and averaging the result over y, z and gives
k2b2(Y2
+Z’)
2( L2 + k2b4)
iL
x
( ii l-3
(6)
Presently, it should be taken into consideration that the joint wave function of both the heavy and light particles has a form of WR, r). Hence, a collapse over y. z automatically leads to the reduction of the wave function of the heavy and light particles, according to form factor (6). If so, we seemingly have a pair of correlated particles involved in the “thought experiment” of Einstein, Podolsky, and Rosen. One can state that - in this particular subensemble - a collapse of the scattered particle makes the heavyparticle wave function undergo a contraction over both the Z and Y directions, in accordance with form factor (6). It is sensible to assume that the length L, involved in the collapse of the light particle, characterizes the light-particle free path A in a gas consisting of light atoms. As to the localization width b, it turns out to be of the order of \Ih, [7,8]. Here A, represents the average de Broglie wavelength of gas atoms. Rigorous reasoning validates the relations b2k - h - L. However, it is more practical to assume
55
instead that L + 0. This would simplify the following formulas at the price of tolerating slightly wider “inciting” collapses. It turns out that there is an opportunity to additionally simplify the form factor (6). Indeed, our previous choice of the X-axis was arbitrary. In the general case, the localization of the light-particle wave function is permissible over any direction without restriction. Hence on the average, one can replace quantity Y * + Z* with fR* for any collapse. Taking advantage of this circumstance, we get approximately 2
@=exp
$=exp[-iwr+ikw-(y2+x2)/2b2].
-
Physics Letters A 231 (1997) 52-M)
-i(ki-kf)
*R-
$
Here R = (X, Y, Z) represents the radius vector of a heavy particle within a reference frame where the collapse occurs near R = 0. The first term in the exponent of (7) merely represents the momentum interchange. The light particle aquires a momentum hq = h(k, - ki). Meanwhile, the heavy-particle experiences a recoil equal to -hq. The second term in the same exponent corresponds to the form factor GN appearing in relation (3). This term asserts that a collapse of a light-particle wave function is accompanied with a similar collapse of the heavy-particle wave function. Thus, the role of the atomic-molecular chaos in the quantum realm is not restricted solely to cause random impacts exerted by colliding particles. It also ensures successive collapses of the heavy-particle wave function, making this particle behave in some respects like particles in classical physics. The very first collapse of the wave function of a scattered light particle causes a localization of the heavy-particle wave function in accordance with form factor (7): the localization scale is of the order of b. The further collapses of scattered light particles sharpen more and more the localization of the wave function of a heavy particle. The mechanism of the collapse-produced localization of wave functions will be clarified a little later. Now one might ask how to describe the motion of a quantum particle, if the observations are inaccurate and the resolution power of the instruments is insufficient for revealing the “intrinsic structure” of wave packets. The answer is trivial: there exists a sole description, and it is close to the classical one.
56
B.B. Kadomtsev, M.B. Kadomtsev/Physics
Normally, the scale of the path traversed by a particle in the course of Brownian motion substantially exceeds the wave-packet localization size. If so, the motion of a packet is describable in terms of packet-“center mass” coordinates R, = (X,, Y,, Z,) and of the average velocity V,. These quantities are defined as follows, R, = 19
* (R).RP(R)
d3R,
(8) d3R,
V,,= -i$lt*(R)RO,Y(R)
(9)
where T(R) is a wave function of the heavy particle. Due to random impacts exerted by light particles, R, and V, become random time-dependent functions. These functions need to be statistically described. In the framework of an approximation, close to a classical description, the effects of the impacts can be expressed with two equations =F,
M-z
dR0
dt
V
The substitution of the expression for ?P into (12) gives
G +V.
where F is a random force. Thus, we come back to the classical Langevin equation. The quantum properties of the considered particle are described with the nondiagonal elements of the density matrix
V,,A=
R,(R,
R’, V, I”, t)
=(A~R,
v, t)A(R’,V’,
R’, t) = (‘P*(R,
t)P(R’,
t)).
t)).
(14)
With the aid of (13) we obtain z
+(v.
VRR+V. VRR’)PO
= -&(V;-
v;>p,.
(15)
It is suitable to introduce new variables v+v’ R,=
+(R+R’),
V,=
Q,=+(R-R’), p(R,
(13)
;&‘A,
where V = hK/M. If fi.--+0, we obtain an equation for a free transfer of amplitude A with velocity V. If there is a superposition of wave packets, having different wavenumber vectors, each value of K corresponds to its “own” amplitude A. Therefore, the amplitude A can be regarded as an A(R, V, t) function. Eq. (13) is valid for any of these amplitudes. Let us compose now a P,(R, R’, V, V’, t) matrix
( 10)
”
Letters A 231 (1997) 52-60
2,
(16)
W,,= y.
(17)
(II)
Here ly is a wave function, and the brackets mean averaging over the statistical ensemble. Instead of using the ordinary density matrix, it turns out more appropriate to describe the statistical properties of a quasi-free quantum particle with a somewhat modified matrix representation. Consider now the Schriidinger equation for the wave function of a free heavy particle,
After the substitution of these variables, Eq. (15) becomes dP0 at
=
* v~RoPo+w*
+
vo
-
g_yRo
VePo
~(VQ~O).
(18)
Assume PO to be a sufficiently smooth function of Then it is possible to. set h -+ 0 and to neglect the right-hand side of Eq. (18). Let us assign
R,. a!P ih-=--
at
g
v;!P.
(12)
Let us assume the wave function ?P(R, t> to be very similar to a plane wave ?P=A(R, t) exp(-Ot + iK - R), where 0 = iiK 2/2 M is the frequency and K represents the wavenumber vector. Function A( R, t) is an envelope weakly dependent on both R and t.
R,=F(R,,
V,, t)U(Q.
W, t).
Then, according to (18), the distribution function F
satisfies the free-motion equation dF x + Vo - VRoF= 0.
(19)
B.B. Kadomtseu, M.B. Kadomtsev/Physics
Having separated the variables, we obtain an equation for the function II, g
+
g&K-K’) *v,u=o,
(20)
where V = iiK/M. Now it is clear that U(Q, W) is an analog of the
Wigner function, i.e., a density matrix in a mixed representation. Let us replace the wavenumber vectors K and K’ in (20) with the corresponding operators -ifiV,r and -ifiVR, and pass from the K, K’ (momentum) representation to the ordinary one. The result consists in the equation
au ihx
h2 = - 2~
(21)
QQ21J.
It is clear that in the absence of collapses, the function U satisfies the free-motion Schrodinger equation. Thus, we are able to express the distribution matrix in the form of P,,(R,
R’, V,,, f) =F(R,,
R - R’ V,,, t>U 2
(
57
Letters A 231 (1997) 52-60
Since for some time velocity V is equal to zero, let us examine, first of all, the coordinate-dependent part of the distribution matrix. Already the very first collapse of the scattered light particle, occurring in the vicinity of R = 0, leads to a localization of the heavy-particle wave function, making it shaped according to (7). Thus, after the first collapse within the considered subensemble with a collapsed wave function we have
(23) Here A is a normalization coefficient. With the aid of (23) it is easy to obtain the density matrix of the collapsed subensemble PJR,
R’) = p(R)‘J’(R’) =A2exp
-g-
R=
(R-R’)’ 6b2
i
(24)
1
. (22)
Here, the distribution function F(R,, V,, t> corresponds to the diagonal part of the distribution matrix. On the other hand, the factor U[(R - R’)/2], normalized as U(O) = 1, describes the nondiagonal terms of the distribution matrix (i.e., of the envelope density matrix). Let us inspect the problems of the description of nondiagonal elements of the distribution matrix. In the m +z M approximation, the first term in the exponent of form factor (7) can be temporarily neglected and the heavy particle in the initial state can be regarded as motionless. Let ?PO(R) represent the initial wave function. Then, the initial distribution matrix (when V = 0) coincides with the initial density matrix, and both of them are proportional to the product !P * (R)P& R’). Accordingly, the distribution function F&R, V>, pertaining to the initial state, is given by the relation
Here, we used the notation R, introduced in (16). Evidently, p,,(R, R’) splits into two multipliers: diagonal factor F, and nondiagonal factor U = exp[ - (R - R’j2/6b2 I. The presence of the diagonal factor F, is due merely to a casual choice: we have chosen a subensemble, corresponding to the occurrence of a collapse near R = 0. In principle, a localization of this kind may occur near any arbitrary point R = R,. In order to remain within the initial ensemble, it is necessary to average expression (24) with a weight factor over all the possible collapse coordinates (A should be proportional to 1W,,(R,) I 2>. Consequently, a few early collapses transform the density matrix into the expression U()R-R’l).
V) = l*o(R)2)l
S(V).
Meanwhile, the diagonal part of the density matrix is p(R,
R’) = 19, I*.
(25)
Here the nondiagonal part of the density matrix is described by the function ,
F,(R.
I.
(26)
where A = b. The further collapses of the light-particle wave functions lead to a monotonic decrease in A as time
58
B.B. Kadomtsev. M.B. Kudomtsev/Physics
goes on. In order to understand how this happens, let us trace the behavior of a subensemble whose initial wave function is like (23). The centers of the successive collapses might slightly shift off the coordinate origin. However, this effect can be ignored because we are mostly interested in the nondiagonal part of the density matrix. Indeed, the positions of the nextstep collapse centers are predetermined at preceding steps and should be distributed proportionally to I?P I*. The latter makes us believe that all the successive collapses would be concentrated around the origin of coordinates. In this approximation, every new collapse manifests itself in a multiplication of the previous wave-packet form factor by an additional multiplier having the form of (23). To simplify matters, assume the effective cross sections of the light-light and light-heavy particle scattering to be the same. If so, the average time between collapses is equal to +r= h/v,, where h = l/cm. In addition, (+, n, and vr represent the cross section of the just mentioned scattering, the numerical density of light particles, and their thermal velocity, respectively. Because each new collapse adds another multiplier (23) to the packet form factor, the wave packet becomes progressively thinner as time goes on. Its size tends to zero according to the following law, ?P(R,t)=A,exp
(27)
where A, is a normalizing coefficient, whose value depends on the subensemble choice. Elementary calculations transform (27) in a!P = at
Letters A 231 (1997) 52-60
elasticity, the last term being responsible for the conservation of the wave fanction norm. In the long run, the solution of Eq. (29) becomes stationary and can be sought in the form of ?P= exd( - i fl t - R*/2 A*>]. Inserting this expression into (291, we obtain A-“=(l-i)Ai, R=-
A,=b
3h 2M4
(30)
’
Here we have assigned r = 3fi2/2 MAi for ensuring the frequency 0 to be a real quantity. The knowledge of A enables us to evaluate the quantity AE = h*/MA*. It expresses an increment in the heavy-particle kinetic energy, which is due to the wave-function collapse. The energy increment AE may be substantially less than the average energy transfer Tm/M per act of light particle scatter-. ing on the heavy particle. It is noteworthy that when analyzing the collapse mechanism for A > b, it is superfluous to worry about the energy balance. Let us remember now the first term in the exponent of the exponential form factor (7). It is responsible for the transfer of momentum to the heavy particle from the scattered light particles. The transferred momentum sets the heavy-particle wave packet in motion. The wave function of a moving wave packet has the following form, (R-R,)* TP‘b”o
2A2
+i-
MV, n
(31)
-kR*$l+m.
Here k= 1/3b*T and I’=A,/A,. We assume r to be constant. When the localization of the wave function becomes sufficiently sharp, it is necessary to take into account the kinetic energy of the particle. For this purpose, the Schradinger equation can be generalized by the allowance of collapses, C?P h2 i?ix = - 2~ pkzP,- ihkR*?P+ ifiT?P.
(29)
This equation differs from (28) by the allowance of the kinetic energy operator. We see that (29) is the equation of a quantum oscillator with imaginary
Here K, represents the wavenumber vector; R, is the radius vector of the packet center mass, and V, = ?X,/M is its velocity. The wave function is seen to be labeled with indexes R, and V,. The exponential factor exp[ - (R - R,,>*/2 A*] can be regarded as the envelope of a plane tvave. Knowing the form of the wave function, we are able to include the distribution matrix into our analysis, P,(R,
R’, V,, t) =F(R,,
V,, t)U(R-R’). (32)
Here F(R,, V,, t) is a distribution function corresponding to the diagonal part of the distribution
B.B. Kadomtseo,
M.B. Kadomtsev /Physics
matrix. A nondiagonal factor U describes the wave packets. If the distribution function is a smooth one over distances comparable with the packet localization size, the kinetic equation is applicable to trace the time evolution of fuhction F. As to the nondiagonal factor, it is given by expression (26) where the localization width A is time-dependent. It follows from (27) that, as time goes, the packet shrinks in accordance with the A = 6(7/t)‘12 law until the packet size achieves its stationary value A,,. According to (30), the stationary packet size is A, = b(m/M)‘/4. Evidently, the increase in the heavy particle mass M makes the particle localization size tend to zero. In this way, a quantum wave bunch is transformed into a classical point corpuscle. In gases, the “intrinsic measurements” - by themselves - are capable to make the heavy particles approach the behavior inherent to classical physics in natural conditions. Above we derived relation (30) on the assumption that the cross section (+ of the atom-atom scattering is the same as the cross section o’B of the atomBrownian particle scattering. In reality, (TV> (T, and estimates (30) should be revised. It is necessary to increase the quantity k, appearing in Eq. (28) (~a/(+ times. On the other hand, the stationary localization size, mentioned in (30), should be diminished (a,/~)‘/~ times. It is noteworthy that the relations (30) are valid only for small-size particles, so that (+a < AZ.
3. Conclusion To summarize, we found that for an adequate description of the behavior of a heavy particle in a gas it is convenient to somewhat modify the density matrix approach. Namely, we introduced a density matrix for the wave packet envelopes (not to be mixed up with the ordinary density matrix). In order to avoid such a confusion, we named the former distribution matrix. If the distribution of the diagonal elements of the distribution matrix is rather smooth, Bq. (15) for this matrix spontaneously splits into two independent equations. The diagonal part of the matrix corresponds to a distribution function satisfying the kinetic equation. The nondiagonal part corresponds to the wave-packet form factor U(R - R’).
Letters A 231 (I 997) 52-60
59
The hypothesis of quantum atomic-molecular chaos brings us to Eq. (29), which describes the wave packet evolution. According to this equation, the successive collapses enhance the wave packet localization (progressively diminishing its size). It also follows from (29) that the wave packet has a Gaussian shape in real space. The obtained results reconstruct the scenario of the Brownian particle motion in gas as follows. Independently of its initial state (a coherent state inclusive), the particle evolves in accordance with the solution of a Schrodinger equation with allowance of absorption. The latter represents the disappearance of coherence in the initial state due to birth of the scattered waves. Against this background, there occur collapses of the wave functions in arbitrary individual statisticalensemble members. The very first collapse in each ensemble member destroys the initial wave function and gives birth to a wave packet with a b N \Ihh, size (here A and A, represent the free-path length of light particles and the average de Broglie wavelength of the same light particles, respectively). The collapses that occur afterward additionally reduce the nondiagonal matrix terms of the distribution matrix (in other words, they contract the wave function). The principle of multicollapse reduction of wave functions had been devised in Ref. [S] and was further developed here. As to the statistical behavior of a Brownian particle, it is governed by the classical kinetic equation for the distribution function (i.e., by the diagonal part of the distribution matrix). Our main goal here was to show that the gas medium can by itself impart the classical features to the quantum particle. The initial scatterings of light atoms lead to the wave function collapse. Later on the corresponding wave packet continues to shrink, reaching some constant shape at large time scale. The description of the Brownian motion itself has been done by many different ways [ l-6,9,1 01.
References [I] M. Namiki and W. Pascazio, Phys. Rep. 253 (1993) 301. (21 J. Schwinger, J. Math. Phys. 2 (1961) 407.
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M.B. Kadomtsev / Physics Letters A 231 (1997) 52-60
[3] A.O. Caldeira and A.J. Leggett, Physica 121 A (1987) 587;
130 (1985) 374 (E). [4] A.J. Leggett, S. Chkravarty, A.T. Dorsey, M.P.A. Fisher, A. Garg and W. Zwerger, Rev. Mod. Phys. 59 (1987) 1. [5] H. Kleinert and S.V. Shabarov, Phys. Lett. A 200 (1995) 224. [6] S. Machida, M. Namiki, Proc. Int. Symp. Found. quantum mechanics, Tokyo (1983) p. 127.
[7] B.B. Kadomtsev and M.B. Kadomtsev, JETP 81 (1995) 897. [8] B.B. Kadomtsev and M.B. Kadomtsev, Chaos 6 (1996) 399. [9] M.B. Mensky, Continious quantum measurements and path integrals (IOP Publishing, Bristol, 1993). [lo] B.L. Hu, J.P. Paz and Y. Zhang. Phys Rev. D 45 (1992) 2843.