How to extend the first postulate of statistical thermodynamics to reach Gibbs postulate?

Question

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.

Answer 1

The derivation of your mean formula and its formal context is extensively discussed in Part 2 (especially Chapter 9) of my online book 

A. Neumaier and D.Westra, Classical and Quantum Mechanics via Lie algebras, 2011. 

It does not apply to a single quantum harmonic oscillator, since the underlying assumption is that of a quantum system of macroscopic extent. Thus the expectations are calculated not from a wave function but from a density operator.

Comments

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