3.2 The Canonical Ensemble: Closed Systems
151
The lower limit is always z = 0, but the upper limit u depends upon the particle
energy: it takes values z = /(mg) for mgH > > and z = H for ≥ mgH . This
variable upper limit for the integration over z can be accommodated by utilizing the
Heaviside function H(mgH //), defined as
H(x) =
0, x > 1
1, 0 ≤ x ≤ 1
,
to give
(V , ,) =
4π A
3mg
2m
h 2
3
2 [
3
2 − (( − mgH )
3
2 H(mgH //)] .
Note that we may also write (V , ,) as
(V , ,) =
4πV
3mgH
2m
h 2
3
2 [
3
2 − (( − mgH )
3
2 H(mgH //)] ,
with the volume dependence V ≡ H A displayed explicitly.
3.2.2 Summary of Forms for the Canonical Partition Function
We have now obtained all relevant versions of the canonical partition function that
will be needed for our further employment. We may summarize these three versions
of the single-particle canonical partition function as
z(β, V ) =
⎧
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨
⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩
∞
0
(V , E)e −βE dE continuum energy states
r
e
−βE r
discrete energy states
i
i e
−βE i
discrete energy levels
(3.2.23)
Thus, the single-particle canonical partition function, or sum over (energy) states,
can be represented in two ways for discrete energies: either as a straightforward
sum over individual energy states or, equivalently, but often more conveniently, as a
sum over distinct energy levels weighted by their respective degeneracies (namely,
the number of states having the same energy). This procedure would be, to say the
least, tedious in the case of the very closely-spaced energy states associated with
the translational motion of any but the smallest-mass particles. For such cases, we