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Homework Help: 2 Bosons in a box

  1. Apr 4, 2006 #1
    I'm having some difficulties with the following problem:

    Consider two (spinless)free bosons in a box of volume V with periodic boundary conditions. Let the momenta of the bosons be p and q.
    a) Write down the normalized wavefunction for p is not equal to q and p = q.

    \Psi_{pq}(r1,r2)

    I thought since they are bosons Y has to be symmetric thus:

    \Psi_{pq}(r1,r2) = \frac{1}{\sqrt{2}}(\varphi_{p}(r1)\varphi_{q}(r2)+ \varphi_{p}(r2)\varphi_{q}(r1))

    Where


    \varphi_{p}(r1)\varphi_{q}(r2) = \frac{1}{(2\pi)^3}(e^(i(p \cdot r1))(e^(i(q \cdot r2))


    and


    \varphi_{p}(r2)\varphi_{q}(r1) = \frac{1}{(2\pi)^3}(e^(i(p \cdot r2))(e^(i(q \cdot r1))


    For p=q this means:

    \Psi_{pq}(r1,r2) = \frac{1}{\sqrt{2}} \frac{1}{(2\pi)^3}(2e^(i(k \cdot (r1+r2)))

    b) Show that for p is not equal to q:

    \Psi_{pq}(r,r)|^2 > |\Psi_{pp}(r,r)|^2

    But I don't know how to do this.
     
    Last edited: Apr 4, 2006
  2. jcsd
  3. Apr 4, 2006 #2

    Meir Achuz

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    Try to fix your latex.
     
  4. Apr 4, 2006 #3
    I tried but the weird thing is that it even gives a mistake if I try something simpel as [tex]\frac{1}{2}[/tex]
     
  5. Apr 4, 2006 #4

    Meir Achuz

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    try writing it without the [tex] stuff. Then people who know latex can at least read it with a bit of difficulty.
     
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