‚t 2m

can be split into an equation for the real and one for the imaginary part. We

then straightforwardly obtain two real equations for the time dependence of

R and S, both only valid when R = 0:

‚R 1 R2

= ’ ∇R · ∇S ’ ∇ S, (3.91)

‚t m 2m

‚S 1

=’ (∇S)2 ’ (V + Q), (3.92)

‚t 2m

7 We follow in essence the lucid treatment of Madelung (1926), with further interpretations by

Bohm (1952a, 1952b), and details by Holland (1993).

68 From quantum to classical mechanics: when and how

where Q is de¬ned as

∇2 R 2

def

Q(r) = ’ . (3.93)

2m R

The crucial step now is to identify ∇S/m as the local ¬‚uid velocity u:

∇S

def

u(r, t) = . (3.94)

m

This is entirely reasonable, since the expectation value of the velocity equals

the average of ∇S/m over the distribution ρ:

∇S

2 2

—

Ψ ∇Ψ dr = R2

v= k= R(∇R) dr + dr.

m im im m

The ¬rst term is zero because R vanishes over the integration boundaries,

so that

∇S

v = R2 dr. (3.95)

m

Applying this identi¬cation to (3.91), and writing

ρ(r, t) = R2 , (3.96)

we ¬nd that

‚ρ

+ ∇ · (ρu) = 0. (3.97)

‚t

This is a continuity equation for ρ (see (3.83))!

Equation (3.92) becomes

‚u

= ’∇( 1 mu2 + V + Q).

m (3.98)

2

‚t

The gradient of u2 can be rewritten as

2 ∇(u ) = (u · ∇)u,

1 2 (3.99)

as will be shown below; therefore

‚u Du

+ (u · ∇)u = ’∇(V + Q),

m =m (3.100)

‚t Dt

which is the Lagrangian equation of motion, similar to (3.86). The local force

per unit volume equals ’ρ∇(V + Q). This force depends on position, but

not on velocities, and thus the ¬‚uid motion is frictionless, with an external

force due to the potential V and an “internal force” due to the quantum

potential Q.

3.4 Quantum hydrodynamics 69

Proof We prove (3.99). Since u is the gradient of S, the u-¬eld is irro-

tational: curl u = 0, for all regions of space where ρ = 0. Consider the

x-component of the gradient of u2 :

‚uy

1 ‚ux ‚uz ‚ux ‚ux ‚ux

(∇u2 )x = ux + uy + uz = ux + uy + uz

2 ‚x ‚x ‚x ‚x ‚y ‚z

= (u · ∇)ux , (3.101)

because curl u = 0 implies that

‚uy ‚ux ‚uz ‚ux

= and = .

‚x ‚y ‚x ‚z

The quantum potential Q, de¬ned by (3.93), can also be expressed in

derivatives of ln ρ:

2

Q=’ ∇2 ln ρ + 1 (∇ ln ρ)2 , (3.102)

2

4m

which may be more convenient for some applications. The quantum po-

tential is some kind of internal potential, related to the density distribution,

as in real ¬‚uids. One may wonder if a simple de¬nition for the stress tensor

(3.88) exists. It is indeed possible to de¬ne such a tensor (Takabayasi, 1952),

for which

f int = ’ρ∇Q = ∇σ, (3.103)

when we de¬ne

2

def

σ= ρ∇∇ ln ρ. (3.104)

4m

This equation is to be read in cartesian coordinates (indexed by ±, β, . . .) as

2

‚ 2 ln ρ

σ±β = ρ . (3.105)

4m ‚x± ‚xβ

3.4.2 The classical limit

In the absence of the quantum force Q the ¬‚uid behaves entirely classically;

each ¬‚uid element moves according to the classical laws in a potential ¬eld

V (r), without any interaction with neighboring ¬‚uid elements belonging to

the same particle. If the ¬‚uid is interpreted as a probability density, and the

initial distribution is a delta-function, representing a point particle, then in

the absence of Q the distribution will remain a delta function and follow

a classical path. Only under the in¬‚uence of the quantum force will the

70 From quantum to classical mechanics: when and how

distribution change with time. So the classical limit is obtained when the

quantum force (which is proportional to 2 ) is negligible compared to the

interaction force. Note, however, that the quantum force will never be small

for a point particle, and even near the classical limit particles will have a

non-zero quantum width.

3.5 Quantum corrections to classical behavior

For molecular systems at normal temperatures that do not contain very fast

motions of light particles, and in which electronically excited states play no

role, classical simulations will usually su¬ce to obtain relevant dynamic and

thermodynamic behavior. Such simulations are in the realm of molecular dy-

namics (MD),which is the subject of Chapter 6. Still, when high-frequency

motions do occur or when lighter particles and lower temperatures are in-

volved and the quantum wavelength of the particles is not quite negligible

compared to the spatial changes of interatomic potentials, it is useful to in-

troduce quantum e¬ects as a perturbation to the classical limit and evaluate

the ¬rst-order quantum corrections to classical quantities. This can be done

most simply as a posterior correction to quantities computed from unmodi-

¬ed classical simulations, but it can be done more accurately by modifying

the equations of motions to include quantum corrections. In general we shall

be interested to preserve equilibrium and long-term dynamical properties of

the real system by the classical approximation. This means that correct-

ness of thermodynamic properties has priority over correctness of dynamic

properties.

As a starting point we may either take the quantum corrections to the

partition function, as described in Chapter 17, Section 17.6 on page 472, or

the imaginary-time path-integral approach, where each particle is replaced

by a closed string of n harmonically interacting beads (see Section 3.3 on

page 44). The latter produces the correct quantum partition function. In

the next section we shall start with the Feynman“Hibbs quantum-corrected