Page 77 - STATISTICAL MECHANICS: From First Principles to Macroscopic Phenomena

Molecular ideal gas 63

We assume that the centers of mass R i are far enough apart so that interactions

(P{q} i )) are negli-
between particles in separate molecules (represented by the φ n i
gible. Then, when the Hamiltonian acts on this term in the wave function, the only
terms in the Hamiltonian which contribute significantly are those parts V 1 (P{q} i )
which describe the interactions between the particles in each molecule. Then the
eigenvalue can be shown to be

2 2
¯ h k
i
E ν = +
n i (5.16)
2M
i
where

2
p /2m + V 1 (Pq i ) φ n (Pq i ) =
n φ n (Pq i ) (5.17)
Pi
Pi
Notice that for terms in the wave function in which particles have been interchanged
by permutation between molecules, different terms in the potential energy are sig-
nificant. Here k i is the momentum associated with the center of mass of the ith

molecule and p Pi , P{q} i are the remaining degrees of freedom associated with that
molecule, which must be treated quantum mechanically.
In (5.15), P permutes labels associated with all the indistinguishable particles,
including ones on different molecules, in principle. However, here the permutations
which interchange identical particles on different molecules may be neglected. To
see this consider a particle labelled α on the ith molecule and an identical particle
labelled α on another molecule i . In the term in the partition function associated

with the permutation which interchanges these two particles and does nothing else,

the factors depending on k i are
V −¯h 2 k 2 i m α ·( r α − r α )
e 2M e k i M dk i (5.18)

2π 3
where m α is the mass of the particles being interchanged and M is the mass of the
molecule as before. The factor m α /M arises from the definition of the center of

mass of the molecule R i = 1 m α r α . The integral is done as in Chapter 4 and
M α
we have
1 −π r 2 (m α /M) 2
e λ 2 (5.19)
λ 3
where r =| r α − r α |. This will be small if the density is low enough but the

condition is slightly more stringent than (5.13) might imply because of the factor