Page 271 - Phase Space Optics Fundamentals and Applications

252 Chapter Eight

and choose for the time derivative of the parameterized position to be
proportional to the momentum, i.e.,

˙
P = m R (8.43)

where, in this section, the overdot denotes a temporal derivative (since
here t plays a role analogous to that of z in the rest of this chapter).
The equation for the evolution of the “rays” for this problem results
from considering the time derivative of Eq. (8.41) [in a step analogous
to that in Eq. (8.9)]:

∂ ∇ ∂ ∇ ·∇ ∂
˙ ˙

P =( R ·∇)∇ +∇ = ·∇ ∇ +∇ =∇ +
∂t m ∂t 2m ∂t
=−∇V( R, t) (8.44)
where Eq. (8.40) was used in the last step. This equation is clearly
Newton’s second law of motion, so the “rays” of the Schr¨odinger
equation are classical trajectories. In other words, the mathematical
relation between wave and ray optics is analogous to that between
quantum and classical mechanics. (In fact, it was the analogy between
ray optics and classical mechanics that inspired Schr¨odinger to postu-
late his wave equation for mechanics.) Like the eikonal, the action can
be parameterized as S = ( R, t), and its equation of evolution results
from considering
2
∂ |P|
˙ ˙
S = R ·∇ + = − V( R, t) (8.45)
∂t 2m
Notice that the right-hand side of this expression is the Lagrangian. 15
Let us now consider the rest of Eq. (8.39). As mentioned earlier, we
propose a Debye expansion for the amplitude of the form

∞
, j
A( r, t) = (i¯h) A j ( r, t) (8.46)
j=0
After substituting this into Eq. (8.39) (recalling that the first term has
been made to vanish), separating the different powers of ¯h, and rear-
ranging, we get the equations

∂ A 2 ∇
0 2
+∇ · A 0 = 0 (8.47a)
∂t m
2
∂ A j 2∇ A j ·∇ + A j ∇ 2
+ =−∇ A j−1 (8.47b)
∂t 2m