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

+ 0 like - 0 dislike
535 views

## 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

## 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. $$U = \langle E \rangle$$

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.

+ 3 like - 0 dislike

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.

answered Nov 29, 2019 by (14,537 points)

Thank you so much! Quite commendable that it's not behind a paywall. Hats off!

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysi$\varnothing$sOverflowThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.