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Rays and Waves 253

for j ≥ 1. Notice that Eq. (8.47a) is a continuity equation. As in the
optical case, it can be solved through integration to give

9
−1
[ R(t 0 , )] [ R(t, )]
A 0 [ R(t, ),t] = A 0 [ R(t 0 , ),t 0 ] (8.48)
( ) ( )
where = ( 1 , 2 , 3 ). These results are the basis of many semiclassical
techniques used to understand and model quantum dynamics based
on classical mechanics. These techniques include the WKB (or JWKB)
16
method and the Van Vleck-Gutzwiller propagator. 17,18 However, the
amplitude estimate in Eq. (8.48) diverges when classical trajectories
cross. This problem is analogous to the caustic problem in optics.

8.5.2 Bohmian Mechanics and the
Hydrodynamic Model
Now let us consider approach 2, where both A and are assumed to
be real. After simple manipulation, the real and imaginary parts of
Eq. (8.39) can be written as

2
∂ |∇ | 2 2 ∇ A
+ + V − ¯h = 0 (8.49a)
∂t 2m A

∂ A 2 2 ∇
+∇ · A = 0 (8.49b)
∂t m
This form of separating Schr¨odinger’s equation is the basis of Louis
deBroglie’s and David Bohm’s pilot wave interpretation for quantum
mechanics. 19 Notice that Eq. (8.49a) is almost identical to the classical
2
2
equation for the action, except for the extra term −¯h ∇ A/A. This
term is referred to as the quantum potential, and like the last term in
Eq. (8.35a), it has the effect of steering the trajectories away from the
classical ones in order to keep them from crossing. The interpretation
of deBroglie and Bohm is that there is a directly undetectable “pilot
wave” whose behavior is ruled by Schr¨odinger’s equation and which
guides the motion of the detectable particle.
Besides the philosophical interpretation of these results, Eqs. (8.49a)
and (8.49b) serve as the basis for computational methods. This formal-
ism is referred to as the hydrodynamic model 20 since, as seen from
Eq. (8.49b), the square modulus of the wave function satisfies a conti-
nuity equation akin to that of a fluid. However, as in the optical case,
the fact that Eqs. (8.49a) and (8.49b) are coupled makes their solution
difficult, both algebraically and computationally, especially when the
wave function presents zeros.