Statistical Mechanics - Physics at Oregon State University

6.3. MAXIMUM ENTROPY PRINCIPLE. 131
for a closed system we get the same answers, which shows that the procedure is
OK.
How to deal with discrete eigenvalues.
So far we have described our state vectors in a Hilbert space with states
at a given energy, volume, and particle number. Since the Hamiltonian is the
operator that governs all quantum mechanics, volume and particle number are
ingredients of the Hamiltonian. But energy is the outcome! Therefore, it is
much better to work in the more natural Hilbert space of state vectors that
correspond to the Hamiltonian. The density operator in this case is
ρ = Cδ(U − H) (6.73)
where C is some constant, determined by 1 = CTr δ(U − H).
This form of the density operator in the microcanonical ensemble implicitly
assumes that we take the thermodynamic limit. For a finite system the energy
eigenvalues are discrete and the δ -function will yield a number of infinitely high
peaks. Therefore we are not able to take the derivative
∂S
∂U which is needed
to define the temperature!
We can, however, rewrite the delta function. Using a standard property we
find:
ρ = C U
δ(
V V
H
− ) (6.74)
V
and in the thermodynamic limit the spacing in the energy density U
V goes to
zero. But that is the wrong order of limits, which can lead to problems!
Mathematical details about delta function limits.
In order to be more precise mathematically, we need a limiting procedure
for the delta-function. For example, one can choose a Gaussian approximation
ρ = C 1
ɛ √ (U−H)2
e− ɛ
π 2 (6.75)
which gives the correct result in the limit ɛ → 0. In order to evaluate the trace
of ρlogρ we now have to sum over states with all possible values of the energy.
Clearly we will get a value for the entropy which is defined for all values of U,
and the partial derivative of the entropy with respect to U will give a good value
for the inverse temperature. But now we have two limits to deal with: N → ∞
and ɛ → 0. The thermodynamic limit N → ∞ has to be taken after all physical
state variables have been set equal to the values needed in the problem. For
example, if needed the limit T → 0 has to be taken first. The limit ɛ → 0
is not related to a physical state variable. This limiting procedure is a purely
mathematical trick. Therefore, the order of taking limits is first to assign all