Quantcast
  • Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.

News

PO is now at the Physics Department of Bielefeld University!

New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback

Attributions

(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,047 questions , 2,200 unanswered
5,345 answers , 22,709 comments
1,470 users with positive rep
816 active unimported users
More ...

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

+ 0 like - 0 dislike
1096 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

-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.

asked Nov 27, 2019 in Theoretical Physics by Asaint (90 points) [ no revision ]

1 Answer

+ 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 Arnold Neumaier (15,787 points) [ no revision ]

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

Your answer

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):
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$\hbar$ysicsOv$\varnothing$rflow
Then 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).
Please complete the anti-spam verification




user contributions licensed under cc by-sa 3.0 with attribution required

Your rights
...