9.4 Semiclassical Approximation
305
for t > t 0 . Here
−
∂ 2 R α (x 0 ,t 0 ;x,t)
∂x∂x 0
denotes a d × d matrix composed of derivatives
of R α . For example, for a system with two degrees of freedom,
−
∂ 2 R(x 0 , t 0 ; x, t)
∂x∂x 0
=
−
∂ 2 R
∂x 1 ∂x 01
−
∂ 2 R
∂x 1 ∂x 02
−
∂ 2 R
∂x 2 ∂x 01
−
∂ 2 R
∂x 2 ∂x 02
.
(9.49)
We can also write Eq. (9.48) in the form
G(x 0 , t 0 ; x, t) =
α
1
2πi ¯
h
d/2
Det
−
∂ 2 R α (x 0 , t 0 ; x, t)
∂x∂x 0
× exp
i
¯
h
R α (x 0 , t 0 ; x, t) − iκ α
π
2
(9.50)
for t > t 0 . In Eq. (9.50), κ α is the number of negative eigenvalues of the matrix
−
∂ 2 R α (x 0 ,t 0 ;x,t)
∂x∂x 0
−1
. For systems with one degree of freedom, it may also be taken
to be the number of conjugate points on the αth classical path in going from x 0 at
time t 0 to x at time t. We will discuss conjugate points in more detail below.
9.4.3 Conjugate Points
Let us now assume that Hamilton’s principal function is known in the neighborhood
of a given classical trajectory, x(p 0 , τ ). We want to know how the end point, x,
varies for fixed (x 0 , t 0 , t) as we vary p 0 , the initial momentum. From Appendix A,
we have
p 0 = −
∂R
∂x 0
t 0 ,x,t
.
(9.51)
If we change the initial momentum to p 0 + δp 0 , then we find
δp 0i = −
∂ 2 R
∂x 0i ∂x j
δx j
(9.52)
to first order in small quantities. Inverting Eq. (9.52), we find
δx j = J ji δp 0i ,
(9.53)
where the matrix, ¯
J , is defined as