Microstates and oscillators help

[karnten07]

Homework Statement

The number of microstates of a system of N oscillators containing Q quanta of energy homework is given by

W(N,Q) = (N+Q-1)!/[(N-1)!Q!]

Show that when one further quantum is added to the system the number of microstates increases by a factor of approximately (1+N/Q), provided that N,Q>>1.

Homework Equations

The Attempt at a Solution

So W(N,Q+1) = (N+Q)!/[(N-1)!(Q+1)!]

My problem class leader showed us how to do the question but I am unsure of how he did it, so this is what he did:

W(N,Q+1)/W(N,Q)= [(N+Q)!(N-1)!Q!]/[(N-1)!(Q+1)!(N+Q-1)!]

= [(N+Q)!Q!]/[(Q+1)!(N+Q-1)!] by cancelling

= N+Q/Q+1 = 1+N/Q

I think he got to the last step by approxiamtion since N,Q>>1 but i don't see how that works. Also i don't know why there aren't factorial signs there, whether he or i forgot to write them in or whether they disappear for some reason.

Any explanations would be so helpful, thanks

 

[kdv]

Homework Statement

The number of microstates of a system of N oscillators containing Q quanta of energy homework is given by

W(N,Q) = (N+Q-1)!/[(N-1)!Q!]

Show that when one further quantum is added to the system the number of microstates increases by a factor of approximately (1+N/Q), provided that N,Q>>1.

Homework Equations

The Attempt at a Solution

So W(N,Q+1) = (N+Q)!/[(N-1)!(Q+1)!]

My problem class leader showed us how to do the question but I am unsure of how he did it, so this is what he did:

W(N,Q+1)/W(N,Q)= [(N+Q)!(N-1)!Q!]/[(N-1)!(Q+1)!(N+Q-1)!]

= [(N+Q)!Q!]/[(Q+1)!(N+Q-1)!] by cancelling

= N+Q/Q+1 = 1+N/Q

I think he got to the last step by approxiamtion since N,Q>>1 but i don't see how that works. Also i don't know why there aren't factorial signs there, whether he or i forgot to write them in or whether they disappear for some reason.

Any explanations would be so helpful, thanks

First, notice that (Q+1)!/Q! = Q+1 Do you see why?

In general, if you have (X+1)!/ X! , this is equal to X+1 (where X can be anything)

Now, note that $$\frac{N+Q}{Q+1} = \frac{1 + N/Q}{1+1/Q} \approx 1+N/Q - 1/Q - N/Q^2 \ldots$$ where I have used the binomial expansion and have neglected the corrections of order $1/Q^2, N/Q^2$ and higher .

 

[karnten07]

First, notice that (Q+1)!/Q! = Q+! Do you see why?

In general, if you have (X+1)!/ X! , this is equal to X (where X can be anything)

Now, note that $$\frac{N+Q}{Q+1} = \frac{1 + N/Q}{1+1/Q} \approx 1+N/Q - 1/Q - N/Q^2 \ldots$$ where I have used the binomial expansion and have neglected the corrections of order $1/Q^2, N/Q^2$ and higher .

Yes i see it now, thankyou so much.

 

[kdv]

Yes i see it now, thankyou so much.

I made a few typos in my post (now corrected)

I meant to say (Q+1)!/Q! = Q+1

and (X+1)!/X! = X+1

Sorry for the typos.

And you are welcome.