The quantum statistical approach

[aaaa202]

Usually in the problems I have done, I found the partition function by simply summing exponential of the eigenenergies, but lately I have started to wonder why this approach is correct. Do we not want to sum over the energies of all possible states as we did in the classical case. In that case we need to take into account that there exists an infinite amount of superpositions of eigenstates, which are all valid states. Are these accounted for when I do calculate the partition function using only eigenenergies?
What got me wondering was that my teacher explained the fermi gas by what I thought was a very handwaving argument. He said that its heat capacity is so low because very few electrons can transition to higher energy states because usually the above states are all occupied. Well now, given the fact that there are an infinite amount of states a particle can be in, does this really provide a good explanation?

 

[Jano L.]

I agree with you, that's quite a logical defect in the quantum statistics. It really treats the density matrix $\rho_{mn}$ in the Hamiltonian basis as a basic object and in thermal equilibrium it assumes point-like distribution of probability over the eigenfunctions; the other wave functions are discarded. This saves it from divergences, but is quite hard to reconcile with the superposition principle.

We have discussed this in past here:

https://www.physicsforums.com/showthread.php?t=598233

but with no satisfactory resolution.

 

[Jazzdude]

thermal equilibrium it assumes point-like distribution of probability over the eigenfunctions; the other wave functions are discarded.

Maybe I'm misunderstanding what you're trying to say here, but you cannot tell which states went into the construction of a density matrix. So yes, there's a decomposition into eigenstates, but there are also arbitrarily many decomposition into any overcomplete family of states, which are not eigenstates of anything because they don't even have to be orthogonal.

So I don't see at what point the density matrix formalism in statistical quantum theory makes use of any assumption about the realization of the ensemble.

Cheers,

Jazz

 

[Jano L.]

The problem is that the thermal density matrix cannot be introduced via general definition as a sum over ensemble of $\phi_k$:

$$
\rho_{mn} = \sum_k p_k (\phi_m,\psi_k) (\psi_k,\phi_n),
$$

since there is uncountably many superpositions and the sum cannot be performed.


Instead, the thermal matrix is postulated as an infinite matrix that follows the Liouville - von Neumann equation derived from Schroedinger's equation for the above countable expansion.