## Question

I'm searching for the derivation which enables me to say:

> The first postulate of statistical thermodynamics can be extended to

> arrive at the Gibbs postulate, a postulate which relates the energy of

> said microstates to internal energy of a system as calculated by

> classical thermodynamics

-wikibooks

Motivation

> Gibbs's Postulate is one which relates the internal energy, $U$, of a

> system as determined by thermodynamics to the average ensemble energy,

> $E$, as determined by statistical mechanics. $$ U = \langle E \rangle $$

-wikibooks

What is this average exactly? For say, the quantum harmonic oscillator is it:

$$ U = \langle E \rangle= \langle \psi |H| \psi \rangle $$

Where $H$ is the Hamiltonian and $\psi$ is the wavefunction. The partition function on the other hand makes no explicit reference to the wavefunction of the system as it uses trace.

Note: using the trace method to determine the average ensemble energy one does not need any knowledge of the eigenkets involved. Hence, the search for a derivation which enables this transition.

P.S: Feel free to limit to an example to that of the quantum harmonic oscillator.