Orthogonal jumps of wavefunction in white-noise potentials

0.06 MB·English

by  Lajos Diósi

KFKI-1985-28
HU ISSN 0368 5330
ISBN 963 372 361 2
March 1985
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ORTHOGONAL JUMPS OF WAVEFUNCTION
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IN WHITE-NOISE POTENTIALS
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L. Dio´si
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Central Research Institute for Physics

1 H-1525 Budapest 114, P.O.B. 49, Hungary
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0 ABSTRACT
1. Weinvestigatetheevolutionofthequantumstateforafreeparticleplaced
0 into a random external potential of white-noise type. The master equation
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for the density matrix is derived by means of path integral method. We
1
: propose an equivalent stochastic process where the wavefunction satisfies a
v
i nonlinearSchro¨dingerequationexcept forrandommomentsatwhichitshows
X
orthogonal jumps. The relation of our work to the usual theory of quantum
r
a noise and damping is briefly discussed.
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Since theearlyworks [1,2,3]onthetheoryofquantumnoise anddamping,
a great interest has been shown in quantummechanical systems affected by
random forces and also, new aspects have appeared [4,5,10]. Here we shall
investigate the effect of Gaussian white-noise potentials.
For brevity we restrict ourselves to the case of a single point-line particle
moving in one dimension.
LetusassumethatthepotentialV(x,t)actingontheparticleisastochas-
ticvariable(x,tstandforthecoordinateandtime). Wedefinetheprobability
distribution of V by the following generator functional G[h]:
G[h] ≡ hexp(i V(r,t)h(r,t)drdt)i =
Z (1)
1
′ ′ ′
= exp(− h(r,t)h(r ,t)f(r−r )drdr )
2 Z
where h is an arbitrary function. The symbol hi stands for expectation
values evaluated by means of the probability distribution of V. Functional
differentiation of G[h] gives rise to the two tipical relations of moments of
white-noise type:
hV(r,t)i = 0, (2)
′ ′ ′ ′
hV(r,t)V(r ,t)i = δ(t−t)f(r−r ). (3)
For further use, we introduce tho following notation:
g(r) = f(0)−f(r). (4)
Now we investigate the effect of the white-noise potential (1) onthequan-
tummechanical motion of a given point-like particle of mass m.
If we single out a given potential V then the wavefunction ψ (x) of the
t
particle will satisfy the Schro¨dinger equation of motion:
∂ i~ ∂2 i
ψ (x) = ψ (x)− V(x,t)ψ (x) (5)
∂t t 2m∂x2 t ~ t
where x is the spatial coordinate and t refers to the actual value of the time.
Taking the initial wavefunction ψ (x) at t = 0, one can express the solution
0
for ψ (x) by means of the Feynman’s path integral formula [7]:
t
t
i m
ψ (x) = exp{ [ x˙2 −V(x ,τ)]dτ}ψ (x )Dx . (6)
t Z ~ Z 2 τ τ 0 0 τ
t>τ≥0
0
2
In our case, V(x,t) is a stochastic variable, thus ψ (x) will evolve in time
t
according to a give stochastic process. We shall not derive the rules of this
stochastic process. Instead, we shall construct another stochastic process for
ψ (x) which leads to the correct statistical predictions and which has genuine
t
features from the viewpoint of measurement theory.
In the generic case, the quantum state of the particle is uniquely charac-
terized by the density matrix [6]
∗
ρ (x,y) ≡ hψ (x)ψ (y)i. (7)
t t t
Indeed, it can be shown [8] that ρ yields all the usual statistical predic-
t
tions of the quantummechanics. Namely,
O = O(y,x)ρ (x,y)dxdy (8)
t t
Z
b
whereOisthehermititianoperatorofanarbitrarilygivendynamicalquantity
and O stands for its predicted value at time t.
t
b
First, we shall prove that the density matrix (7) satisfies a parabolic
differential equation of motion. Using eq. (6) along with eq. (7), one gets:
ρ (x,y) =
t
t
i m
= h exp{ [ (x˙2 −y˙2)−(V(x ,τ)−V(y ,τ))]dτ}ρ (x ,y )Dx Dy i
Z ~ Z 2 τ τ τ τ 0 0 0 τ τ
0 t>τ≥0
(9)
∗
where ρ is the initially given density matrix: ρ (x,y) = ψ (x)ψ (y). On the
0 0 0 0
rhs one can substitute
t t
i 1
hexp{− [V(x ,τ)−V(y ,τ)]dτ}i = exp{− g(x −y )dτ}, (10)
~ Z τ τ ~2 Z τ τ
0 0
which obviously follows from eq. (1) if we insert there h(r,t) = −(1/~)(δ(r−
x )−δ(r−y )) and use eq. (4). Thus we have path integral representation
τ τ
for the density matrix at arbitrary time t:
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ρ (x,y) =
t
t
m 1
= exp{ [i (x˙2 −y˙2)− g(x −y )]dτ}ρ (x ,y )Dx Dy .
Z Z 2~ τ τ ~2 τ τ 0 0 0 τ τ
0 t>τ≥0
(11)
Differentiating bothsides of eq. (11)by t, one getsthe equation of motion
for the density matrix in Gaussian white-noise potential (1):
∂ i~ ∂2 ∂2 1
ρ (x,y) = ( − )ρ (x,y)− g(x−y)ρ (x,y). (12)
∂t t 2m ∂x2 ∂y2 t ~2 t
This equation is sometimes called the master equation of the quantum
noisetheory. Ourpathintegralmethodseems tobeveryeffective forderiving
the master equation even in more general external noises.
The second term on the rhs of eq. (12) is a tipical damping term known
from the quantum theory of reservoir effects (c.f. coarse grained approxi-
mation in ref. 3). This term cannot be reproduced by any given stochastic
hamiltonian.
Beside damping, a very peculiar feature of eq. (12) is that it produces
mixed quantum state from a pure one in a continuous manner. Exploiting
the nature of this permanent quantum state mixing we shall construct the
stochastic process for the evolution of the wavefunction itself.
In order to make the calculation as simple as possible we suppose that
1
g(r) = A2r2 +higher order terms in r; A = const. (13)
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and we shall neglect the “higher terms” by assuming that the width of the
wavefunction will always be small enough.
Thus, eq. (12) takes the form
∂ i~ ∂2 ∂2 A2
ρ (x,y) = ( − )ρ (x,y)− (x−y)2 ρ (x,y). (14)
∂t t (cid:20)2m ∂x2 ∂y2 t 2~2 (cid:21) t
Ifattimettheparticleisinapurequantumstatewiththegivenwavefunction
ψ then [6]
t
∗
ρ (x,y) = ψ (x)ψ (y) (15)
t t t
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and eq. (14) yields
ρ (x,y) =
t+ǫ
ǫA2 ~ǫ ∂2 ~ǫ ∂2
= 1− (x−y)2 1+i ψ (x) 1−i ψ∗ (y) (16)
(cid:20) 2~2 (cid:21)(cid:20) 2m∂x2(cid:21) t (cid:20) 2m∂y2(cid:21) t
for infinitesimal ǫ > 0. Now, the rhs is not a single product like it was in eq.
(15). Nevertheless, it can be decomposed into the sum of two such diadic
terms:
∗
∗ ˜
ρ (x,y) = (1−ǫw)ψ (x)ψ (y)+ǫwψ (x)ψ (y) (17)
t+ǫ t+ǫ t+ǫ t+ǫ t+ǫ
where
A2
w = σ2 (18)
~2 ψ
is the mixing rate, the dominant wavefunction ψ is
t+ǫ
ǫA2 ǫ~ ∂2
ψ (x) = {1− (x−x )2 −σ2 +i }ψ (x), (19)
t+ǫ 2~2 ψ ψ 2m∂x2 t
(cid:2) (cid:3)
˜
and the contaminating wavefunction ψ is
t+ǫ
x−x ǫA2 a3
˜ ψ ψ
ψ (x) = ( + )ψ (x). (20)
t+ǫ σ 2~2 σ t
ψ ψ
We introduced the following notations:
x = x|ψ (x)|2dx,
ψ t
Z
σ2 = (x−x )2|ψ (x)|2dx, (21)
ψ ψ t
Z
a3 = (x−x )3|ψ (x)|2dx.
ψ ψ t
Z
˜
It is easy to verify that ψ , ψ (x) are normalized to the unity and or-
t+ǫ t+ǫ
thogonal to each other (in the lowest order of ǫ, of course).
Now, let us read out the statistical meaning of eq. (17): in an infinitesi-
mally short time ǫ, the quantum state ψ of the particle either evolves contin-
t
uously into the neighbouring state ψ , or, with the infinitesimal probability
t+ǫ
˜
ǫw, jumps to the state ψ , which is orthogonal to ψ .
t+ǫ t+ǫ
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Thus, the stochastic process governing the evolution of the wavefunc-
tion is as follows. The wavefunction ψ (x) of the given particle satisfies the
t
following non-linear equation of motion [9]:
∂ ~ ∂2 A2
ψ (x) = i ψ (x)− (x−x )2 −σ2 ψ (x);
∂t t 2m∂x2 t 2~2 ψ ψ t
(cid:2) (cid:3)
x = x|ψ (x)|2dx, (22)
ψ t
Z
σ2 = (x−x )2|ψ (x)|2dx,
ψ ψ t
Z
apart from discrete orthogonal jumps
x−x
ψ
ψ = ψ (x) (23)
t+0 t
σ
ψ
which occur at random in time. A jump is performed in an infinitesimal
interval (t,t+ǫ) with probability
A2
wǫ = σ2ǫ. (24)
~2 ψ
By its construction, this stochastic process leads to the same physical
predictions in average as eq. (8) did. Namely, given the initial density
matrix ρ (x,y), we can decompose it as
0
(r) ∗(r)
ρ (x,y) = p ψ ψ (y) (25)
0 r 0 0
Xr
(r)
where ψ ’s form an orthonormal system. Let us regard equality (25) as
0
(r)
if the particle were in the pure state ψ with probability p . Starting the
0 r
stochastic process (22,23,24) from these initial wavefunctions, each of them
will give rise to the quantity
(r) ∗(r)
p ψ ψ (y). (26)
r t t
Xr
(r)
The stochastic average of this expression over the histories ψ is equal to
t
ρ (x,y).
t
We have to note that many other stochastic processes for ψ can be con-
t
structed with the same ρ (x,y). Nevertheless, we would like to underline that
t
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the orthogonality of the stochastic jumps (23) is very crucial from the view-
point of measurement theory: It can be shown that if we know the wavefunc-
tion ψ at t = 0 then, by means of von Neumann measurements [8], we can
t
registrateallstochasticjumps, without disturbing the measured particle. Thus,
in every moment, we are able to find out the wavefunction of the system, if
it satisfies indeed the eqs. (22,23,24). Of course, it is not obvious at all how
should we realize the proper measuring apparatuses in practice.
Finally, we note the in the Langevin approach of quantum damping [2]
also stochastic process is constructed but this process is related to the evo-
lution of the density matrix, not to the pure quantum state of the system.
I wish to thank to the authors of ref. 4 and also to Dr. P. Hrask´o for
illuminating discussions.
REFERENCES
[1] W.H. Louisell and L.R. Walker, Phys. Rev. 137, B204 (1965)
[2] M. Lax, Phys. Rev. 145, 110 (1966)
[3] W.H. Louisell, Quantum Statistical Properties of Radiation
(Wiley, New York, 1973)
[4] F. Ka´rolyh´azi, A. Frenkel and B. Luka´cs, in: Physics as
Natural Philosophy, eds. A. Shimony and H. Feshbach
(MIT Press, Cambridge, 1982)
[5] J.M. Ziman, Models of Disorder (Cambridge University Press,
Cambridge, 1977)
[6] L.D. Landau and E.M. Lifshits, Quantum Mechanics (Pergamon,
Oxford, 1977)
[7] R.P. Feynman and A.R. Hibbs, Quantum Mechanics and Path
Integrals (McGraw-Hill Book Company, New York, 1965)
[8] J. von Neumann, Mathematische Grundlagen der Quantenmechanik
(Springer Verlag, Berlin, 1932)
[9] The equation (22) in itself, possesses solitonlike solutions.
For similar mechanism see, e.g., L. Dio´si, Phys. Lett.
105A, 199 (1984);. I. Bialynicki-Birula and J. Mycielski:
Ann. Phys. 100, 62 (1976)
[10] N. Gisin, Phys. Rev. A28, 2891 (1983)
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