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Homework Help: Partition function of classical oscillator with small anharmonic factor

  1. Aug 30, 2010 #1
    1. The problem statement, all variables and given/known data

    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.

    2. Relevant equations

    For one oscillator

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

    3. 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.
     
  2. jcsd
  3. Aug 30, 2010 #2
    [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]
     
  4. Aug 30, 2010 #3
    good luck.
     
  5. Aug 30, 2010 #4
    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
     
  6. Sep 2, 2010 #5
    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
     
  7. Sep 2, 2010 #6
    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.
     
  8. Sep 4, 2010 #7
    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
     
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