Partition function of classical oscillator with small anharmonic factor

[castlemaster]

Homework Statement

Having a unidemsional array of N oscillators with same frequency w and with an anharmonic factor [tex]ax^4[/tex] where 0 < a << 1

Calculate, up to the first order of a, the partition function.

Homework Equations

For one oscillator

[tex]Z=\frac{1}{h}\int{e^{-\beta(p^2/2m+1/2mw^2x^2+ax^4)}dpdx}[/tex]

The Attempt at a Solution

Tried to

[tex]Z=\frac{K'}{h}\int{e^{\frac{-\beta*mw^2x^2}{2}(1+\frac{2a}{mw^2}x^2))}dpdx}[/tex]

and I guess I can approximate [tex](1+bx^2)[/tex] to something ... but I know more or less the solution and I can't figure out how to reach it.

 

[qbert]

[tex]
Z=\frac{1}{h}\int{e^{-\beta(p^2/2m+1/2mw^2x^2+ax^4)}dpdx}
[/tex]
[tex]
=\frac{1}{h}\int{e^{-\beta(p^2/2m+1/2mw^2x^2)}e^{-\beta ax^4}dpdx}
[/tex]
[tex]
=\frac{1}{h}\int{e^{-\beta(p^2/2m+1/2mw^2x^2)} (1 -\beta ax^4 + \cdots) }dpdx}
[/tex]

you should probably know how to calculate the last.
but here are some useful integrals

[tex] \int e^{-ax^2} = \sqrt{\pi/a} [/tex]
[tex] \int x^2 e^{-ax^2} = \frac{1}{2} \sqrt{\frac{\pi}{a^3}} [/tex]
[tex] \int x^4 e^{-ax^2} = \frac{3}{4} \sqrt{\frac{\pi}{a^5}} [/tex]

 

[Dickfore]

good luck.

 

[castlemaster]

That's what I thought but part b of the problem say:

show that [tex]C_{v}=Nk(1-\frac{6*\alfa*k}{m^2w^4}T)[/tex]

but
[tex]E=-\frac{d lnZ}{d\beta}[/tex]
and
[tex]C_{v}=\frac{d E}{dT}[/tex]

if I do all the integrals I get something like

[tex]ln Z=Nln(K_{1}\beta^{n})[/tex]

and for the properties of the ln

[tex]ln Z=Nln(K_{1}) +Nnln(\beta)[/tex]

and making the derivate over B will never give the Cv mentioned.

What I'm doing wrong?

Thanks for the answer

 

[castlemaster]

if I do

[tex]u=\frac{\beta mw^2x^2}{2}[/tex]

then I get something like

[tex]\frac{1}{\beta mw^2}\int{e^{-u}e^{-\frac{4\alpha u^2}{\beta m^2w^4}}}[/tex]

I think it approches what I need to end with, at least the variables are similar,

someone has a clue? maybe using the fact that the derivate of exp(ax^n) is nax^(n-1)exp(ax^n)

I'll appreciate any clue

thanks

 

[qbert]

i already told you how to do this.

step 1. do the math.
you end up with [itex] z = c_1 \beta^{-1} - a c_2 \beta^{-2} [/itex]
for constants c₁, and c₂.

step 2. still do the math.
find [itex] E = - \frac{d \log z}{d\beta} [/itex], this will still have an a in
the denominator, drop it (using small a approximation)

step 3. do even more math.
find [itex] C_v = \frac{dE}{dT} [/itex]

step 4. ?

step 5. profit.

 

[castlemaster]

You are right qbert,
I was just doing a stupid mistake everytime.

ln(a*b) = lna+lnb RIGHT
ln(a+b) = lna+lnb STUPID
ln(1+ax) aprox.= ax for a small

thanks for the patience