Page 270 - Phase Space Optics Fundamentals and Applications

Rays and Waves 251

write the wavefunction in terms of a slowly varying amplitude and a
rapidly oscillating phase, i.e.,

i
( r, t) = A( r, t) exp ( r, t) (8.38)
¯ h

Notice that the phase was chosen to be proportional to the inverse
of ¯h. Therefore, like k for the Helmholtz case, ¯h −1 plays the role of
the large asymptotic parameter. The resulting approximate results are
then valid within the so-called semiclassical regime, i.e., when ¯h is
small compared to all other quantities (or variations of quantities) in
theproblemunderstudythatpresentthesameunits(action=length×
mass × speed).
After substitution of Eq. (8.38) and reordering, Eq. (8.37) can be
written as

2
∂ |∇ | 2 ∂ A 2∇ A·∇ + A∇ 2 2
+ + V − i¯h + − ¯h ∇ A = 0
∂t 2m ∂t 2m
(8.39)
Again, we face a decision between two approaches:
1. Assume that ¯h is very small, and expand A in a Debye series.
2. Assume that both A and are real, and separate the real and
imaginary parts of Eq. (8.39).
As before, let us consider approach 1 first.

8.5.1 Semiclassical Mechanics
We start by setting the leading term in Eq. (8.39) to zero. This leads to
the equation

∂ |∇ | 2
+ + V = 0 (8.40)
∂t 2m
This is the equation for the action in classical mechanics. Like the
eikonal equation, it can be solved parameterically. We start by pa-
rameterizing the position as = R(t, 1 , 2 , 3 ). We then define the
r
momentum P and the Hamiltonian H, respectively, as the spatial and

(minus the) temporal derivatives of the action
P =∇ ( R, t) (8.41)

∂
H =− ( R, t) (8.42)
∂t