Solving Two Bosons in a Box Problem

[Henk]

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.

 

[Meir Achuz]

Try to fix your latex.

 

[Henk]

I tried but the weird thing is that it even gives a mistake if I try something simpel as $$\frac{1}{2}$$

 

[Meir Achuz]

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