310
9 Semiclassical Theory: Path Integrals
∂
∂x 0
∂R
∂x
x 0 ,τ
x,τ
=
∂
∂x 0
∂S
∂x
x 0 ,e
x,τ
=
∂
∂x 0
∂S
∂x
x 0 ,e
x,e
+
∂
∂e
∂S
∂x
x 0 ,e
x 0 ,x
∂e
∂x 0
x,τ
.
(9.73)
If we use the chain rule for partial derivatives,
∂e
∂x 0
x,τ
∂x 0
∂τ
x,e
∂τ
∂e
x 0 ,x
= −1,
(9.74)
we can combine Eqs. (9.72), (9.73), and (9.74) to obtain
∂
∂x 0
∂R
∂x
x 0 ,τ
x,τ
=
∂
∂x 0
∂S
∂x
x 0 ,e
x,e
−
∂
∂e
∂S
∂x
x 0 ,e
x 0 ,x
∂
∂x 0
∂S
∂e
x 0 ,x
x,e
∂ 2 S
∂e 2
−1
x 0 ,x
.
(9.75)
Let us also note that
∂ 2 R
∂τ 2
x 0 ,x
= −
∂e
∂τ
x 0 ,x
= −
∂ 2 S
∂e 2
−1
x 0 ,x
,
(9.76)
where we have again used Eq. (9.72). Thus we finally obtain
Det[ ¯
D 1 ] ≡
∂
∂x 0
∂R
∂x
x 0 ,τ
x,τ
∂ 2 R
∂τ 2
x 0 ,x
= −
∂ 2 S
∂e 2
x 0 ,x
×
∂
∂x 0
∂S
∂x
x 0 ,e
x,e
+
∂
∂e
∂S
∂x
x 0 ,e
x 0 ,x
×
∂
∂x 0
∂S
∂e
x 0 ,x
x,e
= Det
∂ 2 S
∂x 0 ∂x
∂ 2 S
∂e∂x
∂ 2 S
∂x 0 ∂e
∂ 2 S
∂e 2
,
(9.77)
where ¯
D 1 is defined by Eq. (9.77) and the subscript 1 means the system has one degree of
freedom. We have expressed partial derivatives of Hamilton’s principal function in terms of
partial derivatives of the action integral.
If we now combine the results above, we obtain for the energy Green’s function
for a system with one degree of freedom
G osc (x 0 , x; e) =
1
i ¯
h
β
Det( ¯
D 1,β ) exp
i
¯
h
S β (x 0 , x; e)
,
(9.78)
where ¯
D 1,β is computed for the βth path.