- #1

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## Homework Statement

Question attached:

Hi,

To me this looks like a classical, continuous system, as a pose to a quantum, discrete system, so I am confused as to how to work the system in the grand canonical ensemble since , in my notes it has only been introduced as a quantum thing,(consistent with what I found only after a quick google to check I hadn't missed anything here: https://ps.uci.edu/~cyu/p115A/LectureNotes/Lecture9/html_version/lecture9.html) .

## Homework Equations

I do have the relation that:

## Z(T,\mu,V)= \sum\limits_{N=0}^{N=\infty}e^{\beta \mu N} z(\beta,N,V)## [2]

(where I have used a capital Z to denote the grand ensemble and a small z to denote the canonical ensemble)

However the derivation of this is from a quantum canonical ensemble as a pose to the classical canonical ensemble.

But even so, this seems like my best bet at attempted the question, although I will now be substituing in the classical canonical ensemble and not the quantum canonical ensemble, which from the derivation I think the formula should really be used for.

## The Attempt at a Solution

So my plan is to integrate over phase-space with this Hamiltonian in the canonial ensemble and apply the formula above.

Doing the integral I get:

##z_1 = \frac{\sqrt{2\pi}}{h\beta c}V\frac{\Gamma(d)}{\Gamma(\frac{d}{2})} ##

(Please ask me for details of my working on this if needed, but from what I see, that's not the issue here)

I then use Gibb's formula that ##z_N=\frac{(z_1)^N}{N!} ##

##\implies Z=\sum\limits_{N=0}^{N=\infty}\frac{e^{\beta \mu N}}{N!}( \frac{\sqrt{2\pi}}{h\beta c}V\frac{\Gamma(d)}{\Gamma(\frac{d}{2})})^N ## [1]

I know that ## \Phi(V,T,\mu)=\frac{1}{\beta} In(Z) = -P(T,\mu) V ##

and so from this I need the log of [1], and using the above its easy enough to compute ##p##, however I can't see how to simplify the sum.

Is the use of [2] wrong here?

Is there a quantum way to look at this system, or a classical way to apply the grand canonical ensemble ?

Many thanks in advance.