Dissipation in
Quantum Mechanics
Robert Dawes[1]
and Ajay Patrikar[2]
Abstract - In this paper we propose a possible
solution of the geometric quantization problem for dissipative systems and
present the results of numerical simulations for the damped harmonic oscillator. These simulations yield
close agreement with the classical system, but only with an unexplained factor of 1/p that begs for a better derivation from the first principles. Our solution relies on the
introduction of a nonlinear term into the potential energy field in a manner
reminiscent of the way in which soliton solutions are obtained from the
Schrödinger equation.
I. INTRODUCTION
The geometric quantization problem is the problem of
obtaining quantum mechanical equivalents of given classical systems [1]. The quantization of systems subject to an
external force has been a topic of investigation in physics literature
[2]-[3]. Recently there has been considerable
interest in quantization of control systems [4]-[5]. A standard approach to quantization is to construct a Lagrangian
of the system from its dynamical equations, obtain the corresponding
Hamiltonian, and use it in the Schrödinger equation [3, 5]. That approach, however, does not apply well
to dissipative systems. Ray [3] has
shown that the quantum representation obtained using that method does not
accurately describe the original dissipative system. For example, when the procedure is applied to the damped harmonic
oscillator, Ray shows that the resulting Schrödinger equation describes instead
a non-dissipative system with time-varying mass m(t) = m0eat rather than a system whose
amplitude damping results from the dissipation of its energy. Apparently, the
proper treatment of dissipation in quantum mechanics remains an open question
[2].
II. NONLINEAR SOLUTION
We have obtained a quantized representation of the
damped harmonic oscillator not by starting with the Lagrangian for that system,
but by incorporating nonlinear viscous damping into the potential energy field.
We begin with a nonlinear Schrödinger equation
(1)
in which the potential energy U(x,Y,t)
is the sum of two parts,
(2)
If l = 0,
and V(x,t) is given by
(3)
then the controlled harmonic oscillator, whose
classical dynamical equation is given by
(4)
results [5].
Let Q(Y) be an arbitrary quadratic form on Y. The
properties of a nonlinear Schrödinger equation in which Q(Y) is given by Q(Y)=G(|Y|2), for some
function G, are described by
Bialynicki-Birula and Mycielski [6]. We
consider an equation in which Q(Y) is a flux potential for Y.
That is, for any given Y, if
(5)
is the probability current density associated with Y, suppose that Q(Y) satisfies
(6)
A potential term in this form induces a component of
the Ehrenfest force that opposes the flow of the wave function in proportion to
its current density. The coefficient l in (2) then determines the
damping rate of the system. The properties of a nonlinear equation obtained
with this choice of Q(Y) are studied through simulation in the
following section.
III. SIMULATION RESULTS
We now compare the response of the classical damped
harmonic oscillator with the response of the nonlinear Schrödinger equation
obtained through numerical integration.
The equation of the classical system is given by
(7)
where x is the damping constant, w is the natural angular frequency and f(t)
is the external input. The response of
this system for x(0)=0.5, x = 0.1, w = 100 and f(t)
= 0 was obtained analytically and is plotted in Fig. 1.
A numerical method proposed by Goldberg, Schey and
Schwartz [7] was used to integrate the nonlinear Schrödinger equation. This
method has been proven to be stable, unitary and second-order accurate in space
and time. The equation was integrated
over a lattice of 1000 points in a spatial domain spanning the interval [-2, 2]
leading to a spatial quantization step e of 0.004.
The spatial boundary conditions were specified to be Y(‑2,t)
= Y(2,t)
=0 for all t. The integration step size, d, in the time domain was set to 0.00005. The
initial state of the particle was represented using a stationary Gaussian wave packet
centered at x0 given by
(8)
where the initial velocity k0 was chosen
to be 0 and x0 was chosen to be 0.5. This form of wave packet
ensures that r(x,t) = |Y(x,t)|2 has unit integral. The
variance of r(x,t), s02, was chosen to be
0.05. This choice of s02 leads to a narrow wave
packet and minimizes the effect the spatial boundary conditions may have on the
integration.
The potential energy U(x,Y,t) was calculated as below:
(9)
where jY(x,t) is given by (5). The
values of w and f(t) are the same as for
the classical system. In (5) the value
of ÑY(x,t) was approximated by the central difference,
(10)
The expectation of
x was calculated as
(11)
Fig. 1 shows the expectation of x plotted against wt
for l=2xw/p. The scaling factor of 1/p was empirically discovered in order to match
the response of the quantum system with that of the classical system and
clearly begs for a better derivation from the first principles. Further simulations revealed that as the
value of s0 is changed, the value of l also needs to be adjusted
slightly to make the damping rates match. This indicates that the scaling
factor is not equal to p for all values of s0 but remains within a vicinity of p.
Figure 1. Responses of the
classical (outer plot) and quantized systems for a damped harmonic oscillator.
III. DISCUSSION
These simulation results indicate that it may be
possible to model dissipative quantum systems with the help of a nonlinear
Schrödinger equation. The proposed nonlinear solution is based on
considerations similar to those described by Newell [8] for the use of the
cubic nonlinearity in the Schrödinger equation in the form, Q(Y) = |Y|2. In that form, with l < 0 in equation (2), the nonlinear term
supplements the linear potential field with a potential well that shadows any
wave packet and follows it wherever it goes.
The Ehrenfest force field of such a well counters the dispersive
tendency of the linear time dependent Schrödinger equation. In one spatial dimension, this cubic
nonlinearity exactly offsets the dispersive force, resulting in soliton
solutions (cf., Newell [8], pp. 24,
37, 46).
Similarly, the nonlinear term defined in (4) creates
an Ehrenfest force field that is proportional to the probability current
density and opposite to its direction.
This dissipates the energy of the wave function. Note that this treatment of dissipation applies
not only to the damping of the harmonic oscillator but to the general case as
well. It does not matter what form the
linear part, V(x,t) of the scalar potential may be.
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for under and critically damped oscillators, American Journal of Physics,
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[3] J. R. Ray, Lagrangians and systems they describe
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Physics, 47(7), July 1979.
[4] E. B. Lin, Quantum mechanical control systems,
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Computer-generated motion pictures of one-dimensional quantum-mechanical
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