haves according to the Schr¨dinger equation. Models based on deterministic

o

¬‚uid dynamics are known as quantum hydrodynamics or Madelung ¬‚uid. If

the ¬‚uid is represented by a statistical distribution of point particles, the

term Bohmian particle dynamics is often used. Particles constituting the

quantum ¬‚uid have also been called “beables,” in analogy and contrast to

the “observables” of traditional quantum mechanics (Bell 1976, 1982; Vink

1993). The quantum force is supposed to originate from a wave that accom-

panies the quantum particles.5

In addition to the interpretation in terms of a ¬‚uid or a distribution of

particles behaving according to causal relations, several attempts have been

made to eliminate the quantum force and ascribe its e¬ects to the di¬usional

behavior of particles that undergo stochastic forces due to some unknown

external agent. Such quantum stochastic dynamics methods (F´nyes 1952;

e

Weizel 1954; Kershaw 1964; Nelson 1966; Guerra 1981) will not be consid-

ered further in our context as they have not yet led to useful simulation

techniques.

It is possible to rewrite the time-dependent Schr¨dinger equation in a

o

di¬erent form, such that the square of the wave function can be interpreted

4 Exchange can be introduced into path integral methods, see Roy and Voth (1999), and should

never be applied to electrons in systems with more than one electron.

5 See for an extensive description of the particle interpretation, including a discussion of its

origins, the book of Holland (1993).

3.4 Quantum hydrodynamics 65

as the density of a frictionless classical ¬‚uid evolving under hydrodynamic

equations, with the ¬‚uid particles subjected to a quantum-modi¬ed force.

The force consists of two parts: the potential force, which is minus the

gradient of the potential V , and a quantum force, which is minus the gradient

of a quantum potential Q. The latter is related to the local curvature of the

density distribution. This mathematical equivalence can be employed to

generate algorithms for the simulation of the evolution of wave packets, but

it can also be used to evoke a new interpretation of quantum mechanics in

terms of hidden variables (positions and velocities of the ¬‚uid particles).

Unfortunately, the hidden-variable aspect has dominated the literature

since Bohm. Unfortunately, because any invocation of hidden variables in

quantum mechanics is in con¬‚ict with the (usual) Copenhagen interpreta-

tion of quantum mechanics, and rejected by the main stream physicists.

The Copenhagen interpretation6 considers the wave function of a system of

particles as no more than an expression from which the probability of the

outcome of a measurement of an observable can be derived; it attaches no

meaning to the wave function as an actual, physically real, attribute of the

system. The wave function expresses all there is to know about the system

from the point of view of an external observer. Any interpretation in terms

of more details or hidden variables does not add any knowledge that can be

subjected to experimental veri¬cation and is therefore considered by most

physicists as irrelevant.

We shall not enter the philosophical discussion on the interpretation of

quantum mechanics at all, as our purpose is to simulate quantum systems

including the evolution of wave functions. But this does not prevent us

from considering hypothetical systems of particles that evolve under speci-

¬ed equations of motion, when the wave function evolution can be derived

from the behavior of such systems by mathematical equivalence. A similar

equivalence is the path-integral Monte Carlo method to compute the evo-

lution of ensemble-averaged quantum behavior (see Section 3.3.9), where a

ring of particles interconnected by springs has a classical statistical behavior

that is mathematically equivalent to the ensemble-averaged wave function

evolution of a quantum particle. Of course, such equivalences are only use-

ful when they lead to simulation methods that are either simpler or more

e¬cient than the currently available ones. One of the reasons that interpre-

tations of the quantum behavior in terms of distributions of particles can be

quite useful in simulations is that such interpretations allow the construc-

tion of quantum trajectories which can be more naturally combined with

6 Two articles, by Heisenberg (1927) and Bohr (1928), have been reprinted, together with a

comment on the Copenhagen interpretation by A. Herrmann, in Heisenberg and Bohr (1963).

66 From quantum to classical mechanics: when and how

classical trajectories. They may o¬er solutions to the problem how to treat

the back reaction of the quantum subsystem to the classical degrees of free-

dom. The ontological question of existence of the particles is irrelevant in

our context and will not be considered.

3.4.1 The hydrodynamics approach

Before considering a quantum particle, we shall ¬rst summarize the equa-

tions that describe the time evolution of a ¬‚uid. This topic will be treated

in detail in Chapter 9, but we only need a bare minimum for our present

purpose.

Assume we have a ¬‚uid with mass density mρ(r, t) and ¬‚uid velocity

u(r, t). We shall consider a ¬‚uid con¬ned within a region of space such

that ρ dr = 1 at all times, so the total mass of the ¬‚uid is m. The ¬‚uid is

homogeneous and could consist of a large number N ’ ∞ of “particles” with

mass m/N and number density N ρ, but ρ could also be interpreted as the

probability density of a single particle with mass m. The velocity u(r, t) then

is the average velocity of the particle, averaged over the distribution ρ(r, t),

which in macroscopic ¬‚uids is often called the drift velocity of the particle. It

does not exclude that the particle actual velocity has an additional random

contribution. We de¬ne the ¬‚ux density J as

J = ρu. (3.82)

Now the fact that particles are not created or destroyed when they move,

or that total density is preserved, implies the continuity equation

‚ρ

+ ∇ · J = 0. (3.83)

‚t

J ·dS over the surface of a (small) volume

This says that the outward ¬‚ow

V , which equals the integral of the divergence ∇ · J of J over that volume,

goes at the expense of the integrated density present in that volume.

There is one additional equation, expressing the acceleration of the ¬‚uid

by forces acting on it. This is the equation of motion. The local acceleration,

measured in a coordinate system that moves with the ¬‚ow, and indicated by

the material or Lagrangian derivative D/Dt, is given by the force f acting

per unit volume, divided by the local mass per unit volume

Du f (r)

= . (3.84)

Dt mρ

3.4 Quantum hydrodynamics 67

The material derivative of any attribute A of the ¬‚uid is de¬ned by

DA def ‚A ‚A dx ‚A dy ‚A dz ‚A

+ u · ∇A

= + + + = (3.85)

Dt ‚t ‚x dt ‚y dt ‚z dt ‚t

and thus (3.84) can be written as the Lagrangian equation of motion

Du ‚u

+ (u · ∇)u

mρ = mρ = f (r). (3.86)

Dt ‚t

The force per unit volume consists of an external component

f ext = ’ρ∇V (r, t) (3.87)

due to an external potential V (r, t), and an internal component

f int = ∇ · σ, (3.88)

where σ is the local stress tensor, which for isotropic frictionless ¬‚uids is

diagonal and equal to minus the pressure (see Chapter 9 for details). This

is all we need for the present purpose.

Let us now return to a system of quantum particles.7 For simplicity we

consider a single particle with wave function Ψ(r, t) evolving under the time-

dependent Schr¨dinger equation (5.22). Generalization to the many-particle

o

case is a straightforward extension that we shall consider later. Write the

wave function in polar form:

Ψ(r, t) = R(r, t) exp[iS(r, t)/ ]. (3.89)

√

Here, R = Ψ— Ψ and S are real functions of space and time. Note that

R is non-negative and that S will be periodic with a period of 2π , but S

is unde¬ned when R = 0. In fact, S can be discontinuous in nodal planes

where R = 0; for example, for a real wave function S jumps discontinuously

from 0 to π at a nodal plane. The Schr¨dinger equation,

o

2

‚Ψ

=’ ∇2 Ψ + V (r, t)Ψ,