Quantum Mechanics for Engineers 

© Leon van Dommelen 

D.60 Checks on the expression for entropy
According to the microscopic definition, the differential of the entropy
should be
where the sum is over all system energy eigenfunctions
and is their probability. The differential can be simplified to
the latter equality since the sum of the probabilities is always one,
so 0.
This is to be compared with the macroscopic differential for the
entropy. Since the macroscopic expression requires thermal
equilibrium, in the microscopic expression above can be equated
to the canonical value where is the
energy of system eigenfunction . It simplifies the
microscopic differential of the entropy to

(D.38) 
the second inequality since is a constant in the summation and
0.
The macroscopic expression for the differential of entropy is given by
(11.18),
Substituting in the differential first law (11.11),
and plugging into that the definitions of and ,
and differentiating out the product in the first term, one part drops
out versus the second term and what is left is the differential for
according to the microscopic definition (D.38). So, the
macroscopic and microscopic definitions agree to within a constant on
the entropy. That means that they agree completely, because the
macroscopic definition has no clue about the constant.
Now consider the case of a system with zero indeterminacy in energy.
According to the fundamental assumption, all the eigenfunctions with
the correct energy should have the same probability in thermal
equilibrium. From the entropy’s point of view, thermal
equilibrium should be the stable most messy state, having the maximum
entropy. For the two views to agree, the maximum of the microscopic
expression for the entropy should occur when all eigenfunctions of the
given energy have the same probability. Restricting attention to only
the energy eigenfunctions with the correct energy, the
maximum entropy occurs when the derivatives of
with respect to the are zero. Note that the constraint that the
sum of the probabilities must be one has been added as a penalty term
with a Lagrangian multiplier, {D.48}. Taking
derivatives produces
showing that, yes, all the have the same value at the maximum
entropy. (Note that the minima in entropy, all zero except one,
do not show up in the derivation; is zero when 0,
but its derivative does not exist there. In fact, the infinite
derivative can be used to verify that no maxima exist with any of the
equal to zero if you are worried about that.)
If the energy is uncertain, and only the expectation energy is known,
the penalized function becomes
and the derivatives become
which can be solved to show that
with and constants. The requirement to conform with the
given definition of temperature identifies as and the
fact that the probabilities must sum to one identifies as
1/.
For two systems and in thermal contact, the probabilities of
the combined system energy eigenfunctions are found as the products of
the probabilities of those of the individual systems. The maximum of
the combined entropy, constrained by the given total energy ,
is then found by differentiating
can be simplified by taking apart the logarithm and noting that
the probabilities and sum to one to give
Differentiation now produces
which produces and
and the common constant then
implies that the two systems have the same temperature.