Diploma thesis - Fachbereich Physik

2.5. EXAMPLE: ORDINARY INTEGRAL 19
The aim is now to expand Z(j) until the first order in ¯h. For this purpose, it is useful
to recall the following result [22, p. 337]:
∫
1 ∞
(
√ dx x n exp − 1 )
(n − 1)!! ¯hn/2
2π¯h 2¯h λx2 = √ [λ > 0] , (2.78)
λ
n+1
−∞
where n is an arbitrary even integer. For n being odd the integral vanishes. Inserting the
expansion (2.77) into (2.73) and expanding the exponential function into a Taylor series
yields
Z(j) =
1
√ exp
2π¯h
×
[
− 1¯h ]∫ ∞
[
A(x cl, j) dδx exp − 1 ]
−∞ 2¯h A′′ (x cl )δx 2
[
1 − 1
6¯h A′′′ (x cl )δx 3 − 1
24¯h A(4) (x cl )δx 4 + 1
72¯h 2 (A′′′ (x cl )) 2 δx 6 + . . .
]
(2.79)
where terms contributing to Z(j) in an order higher than linear in ¯h according to the result
(2.78) have been omitted. Applying (2.78) then yields
[
Z(j) = exp − 1¯h ] [ ]
A(x 1
cl, j) √
A
′′
(x cl ) − ¯h A (4) (x cl )
√
8 (A
′′
(x cl )) + 5¯h (A ′′′ (x cl )) 2
√ 5 24 (A
′′
(x cl )) + 7 O(¯h2 ) .
,
(2.80)
This expression can be converted, using the Taylor expansion
)
ln(1 − x) = −
(x + x2
2 + x3
3 + . . . + xn
n + . . . , (2.81)
into the form
Z(j) = exp
Defining
one obtains
[
− 1¯h A(x cl, j) − 1 2 ln A′′ (x cl ) + ¯h
{ 5 [A ′′′ (x cl )] 2
24 [A ′′ (x cl )] − 1 }
A (4) (x cl )
3 8 [A ′′ (x cl )] 2
]
+ O(¯h 2 )
.
(2.82)
F(j) := − 1 ln Z(j) , (2.83)
β
F(j) = 1
¯hβ A(x cl, j) + ¯h
{
2¯hβ ln A′′ (x cl ) − ¯h2 5 [A ′′′ (x cl )] 2
¯hβ 24 [A ′′ (x cl )] − 1 3 8
}
A (4) (x cl )
+ O (¯h 3) .
[A ′′ (x cl )] 2
(2.84)
In the saddle-point approximation, the reduced Planck constant ¯h is considered a small
quantity. Nevertheless, in order to include the zero-temperature limit, i.e. β → ∞, in the
calculation, a factor ¯hβ cannot be considered a small, but rather an arbitrary quantity.
Moreover, a factor 1/β has to be rewritten as ¯h/¯hβ. The average X is now defined by
X := X(j) := 1 ∫
1 ∞
√ dx x exp
[− 1¯h ]
Z(j)
A(x, j) . (2.85)
2π¯h
−∞