302
9 Semiclassical Theory: Path Integrals
The subscript α indicates that the quantity is evaluated along the αth classical path.
Terms of cubic order or higher in Eq. (9.30) give contributions at least of order
√ ¯
h and smaller. There will be a contribution to the Green’s function from each
classical path. If we neglect cubic and higher-order terms in Eq. (9.30) and substitute
Eq. (9.30) into Eq. (9.23), we find
G N (x 0 , t 0 ; x, t) ≈ θ(t − t 0 )
α
mN
2πi ¯
h(t − t 0 )
N/2
× exp
i
¯
h
R N (x 0 , {x i } α , x)
∞
−∞
dy 1 . . .
∞
−∞
dy N −1
× exp
⎡
⎣ i
2 ¯
h
N −1
i=1
N −1
j =1
∂ 2 R N
∂x i ∂x j
α
y i y j
⎤
⎦ ,
(9.31)
where the summation
α is over all classical paths and we have dropped the
subscript α on y i .
Let us now introduce a matrix ¯
M (N −1) (α) such that its (ij )th element is
¯
M
(N −1)
ij
(α) =
t
m
∂ 2 R N
∂x i ∂x j
α
.
(9.32)
The matrix ¯
M (N −1) (α) can be written
¯
M
(N −1) (α) =
⎛
⎜
⎜
⎜
⎜
⎜
⎝
a 1 −1 0 . . . 0
−1 a 2 −1 . . . 0
0 −1 a 3 . . . 0
. . .
. . .
. . .
. . .
. . .
0 0 0 . . . a N −1
⎞
⎟
⎟
⎟
⎟
⎟
⎠
(9.33)
with
a i = 2 −
((t) 2
m
∂ 2 V
∂x 2
i
α
.
(9.34)
We next note that for an N × N real symmetric matrix, ¯
M, we can write
∞
−∞
dy 1 . . .
∞
−∞
dy N exp
⎡
⎣ −
1
2k
N
i=1
N
j =1
M i,j y i y j
⎤
⎦ =
(2πk) N
Det[ ¯
M]
1
2
.
(9.35)