358
7 Classical Statistical Mechanics
As we have seen, z(T , V ) is defined quantum mechanically via z ≡
r e −ββ r ,
with the summation over r a summation over all quantum states r associated with
an atom, and hence we might logically expect that the classical expression for z
should have a similar form, i.e., z(T , V ) ∝
· · ·
e −βH(p,q) dpdq, in which p, q
collectively denote all the momenta and coordinates upon which the Hamiltonian
(energy) depends. We could write dp ≡ dp 1 · · · dp s and dq ≡ dq 1 · · · dq s , with s
giving the number of degrees of freedom of the molecule. For example, a closedshell atom in its ground electronic state will have s = 3 (corresponding to q 1 =
x, q 2 = y, q 3 = z; p 1 = p x , p 2 = p y , p 3 = p z ). This specific example uses
Cartesian coordinates and conjugate momenta: more generally, the momenta and
coordinates do not need to be Cartesian. See Appendix G for additional background
information on classical mechanics.
The classical Hamiltonian for a single atom of mass m is just its kinetic energy,
namely
H trans (r, p) =
1
2m
p
2 .
If we now follow our conjecture in the paragraph above that
z(T , V ) ∝
e
−βH(r,p) dpdr ,
we see that
z(T , V ) ≡ z trans (T , V ) ∝
e
−βp 2 /(2m) dpdr
= C trans V
∞
0
π
0
2π
0
e
−βp 2 /(2m) p
2 sin ϑ dϕdϑdp
= C trans V 4π
∞
0
e
−βp 2 /(2m) p
2 dp ,
with C trans a constant. Upon evaluation of the definite integral, we obtain
z trans (T , V ) = C trans V
2πm
β
3
2 = C trans (2πmk B T )
3
2 V .
(7.1.2)
Comparison between the classical result (7.1.2) and Eq. (7.1.1) indicates that C trans
must equal h −3 in order for the two results to be consistent.
In the classical phase-space description of the motion of a (structureless) particle,
each point (r, p) in the six-dimensional phase space represents a possible state of
motion for the particle. This leads to the statement that the number of states available
to the particle in a specific region of its phase space must be proportional to the
(six-dimensional) volume of that region. Thus, for a structureless classical particle
having position r −
1
2 dr ≤ r ≤ r +
1
2 dr and possessing momentum p −
1
2 dp ≤
7 Classical Statistical Mechanics
As we have seen, z(T , V ) is defined quantum mechanically via z ≡
r e −ββ r ,
with the summation over r a summation over all quantum states r associated with
an atom, and hence we might logically expect that the classical expression for z
should have a similar form, i.e., z(T , V ) ∝
· · ·
e −βH(p,q) dpdq, in which p, q
collectively denote all the momenta and coordinates upon which the Hamiltonian
(energy) depends. We could write dp ≡ dp 1 · · · dp s and dq ≡ dq 1 · · · dq s , with s
giving the number of degrees of freedom of the molecule. For example, a closedshell atom in its ground electronic state will have s = 3 (corresponding to q 1 =
x, q 2 = y, q 3 = z; p 1 = p x , p 2 = p y , p 3 = p z ). This specific example uses
Cartesian coordinates and conjugate momenta: more generally, the momenta and
coordinates do not need to be Cartesian. See Appendix G for additional background
information on classical mechanics.
The classical Hamiltonian for a single atom of mass m is just its kinetic energy,
namely
H trans (r, p) =
1
2m
p
2 .
If we now follow our conjecture in the paragraph above that
z(T , V ) ∝
e
−βH(r,p) dpdr ,
we see that
z(T , V ) ≡ z trans (T , V ) ∝
e
−βp 2 /(2m) dpdr
= C trans V
∞
0
π
0
2π
0
e
−βp 2 /(2m) p
2 sin ϑ dϕdϑdp
= C trans V 4π
∞
0
e
−βp 2 /(2m) p
2 dp ,
with C trans a constant. Upon evaluation of the definite integral, we obtain
z trans (T , V ) = C trans V
2πm
β
3
2 = C trans (2πmk B T )
3
2 V .
(7.1.2)
Comparison between the classical result (7.1.2) and Eq. (7.1.1) indicates that C trans
must equal h −3 in order for the two results to be consistent.
In the classical phase-space description of the motion of a (structureless) particle,
each point (r, p) in the six-dimensional phase space represents a possible state of
motion for the particle. This leads to the statement that the number of states available
to the particle in a specific region of its phase space must be proportional to the
(six-dimensional) volume of that region. Thus, for a structureless classical particle
having position r −
1
2 dr ≤ r ≤ r +
1
2 dr and possessing momentum p −
1
2 dp ≤