152 Canonical ensemble
Thus, substituting the transformation into eqn. (4.5.24), we obtain
Q(N,V,T,r,r′) =1
h^3 N(
2 πm
β) 3 N/ 2 ∫
dNuexp[
−
1
2
βmω^2∑N
i=1i+ 1
iu^2]
. (4.5.47)
Now, each of the integrals overu 1 ,...,uNcan be performed independently and straight-
forwardly to give
Q(N,V,T,r,r′) =1
h^3 N(
2 πm
β) 3 N/ 2 (
2 π
βmω^2) 3 N/ 2
e−βmω(^2) (r−r′) (^2) /N+1
∏N
i=1(
i
i+ 1) 3 / 2
. (4.5.48)
Expanding the product, we find
∏Ni=1(
i
i+ 1) 3 / 2
=
(N
∏
i=1i
i+ 1) 3 / 2
=
(
1
2
2
3
3
4
···
N− 1
N
N
N+ 1
) 3 / 2
=
(
1
N+ 1
) 3 / 2
. (4.5.49)
Thus, substituting this result into eqn. (4.5.48) yields eqn. (4.5.33).
Finally, let us use the partition function expressions in eqns. (4.5.37)and (4.5.33)
to compute an observable, specifically, the expectation value〈|r−r′|^2 〉, known as the
mean-square end-to-end distanceof the polymer. From eqn. (4.5.35), we can set up
the expectation value as
〈|r−r′|^2 〉=1
Q(N,V,T)
1
h^6(
2 π
βhω) 3 N(
2 πm
β) 3
×
1
(N+ 1)^3 /^2
∫
drdr′|r−r′|^2 e−βmω(^2) (r−r′) (^2) /(N+1)
. (4.5.50)
Using the fact that 1/Q(N,V,T) = (λ^3 /V)(βhω/ 2 π)3(N+1), and transforming to
center-of-mass (R) and relative (s) coordinates yields
〈|r−r′|^2 〉=(
λ^3
V)(
βhω
2 π) 3 N+3
1
h^6(
2 π
βhω) 3 N(
2 πm
β) 3
×
1
(N+ 1)^3 /^2
∫
dRds s^2 e−βmω(^2) s (^2) /(N+1)
. (4.5.51)
The integration overRyields, again, a factor ofV, which cancels theVfactor in the
denominator. For thesintegration, we change to spherical polar coordinates, which
yields