(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Consider two noninteracting particles p and q each with mass m in a cubical box od size a. Assume the energy of the particles is

[tex] E = \frac{3 \hbar^2 \pi^2}{2ma^2} + \frac{6\hbar^2 pi^2}{2ma^2} [/tex]

Using the eigenfunctions

[tex] \psi_{n_{x},n_{y},n_{z}} (x_{p},y_{p},z_{p}) [/tex]

and

[tex] \psi_{n_{x},n_{y},n_{z}} (x_{q},y_{q},z_{q}) [/tex]

write down the two particle wave functions which could describe the system when the particles are

a) distinguishable, spinless bosons

b) identical, spinless bosons

c) identical spin-half fermions in a symmetric spin state

d) identical spin half fermions in an antisymmetric spin state

2. Relevant equations

For a cube the wavefunction is given by

[tex] \psi_{n_{x},n_{y},n_{z}} = N \sin\left(\frac{n_{x}\pi x}{a}\right)\sin\left(\frac{n_{y}\pi y}{a}\right)\sin\left(\frac{n_{z}\pi z}{a}\right) [/tex]

[tex] E = \frac{\hbar^2 \pi^2}{2ma^2} (n_{x}^2 + n_{y}^2 +n_{z}^2) [/tex]

3. The attempt at a solution

for the fermions the wavefunction mus be antisymmetric under exhange

c) [tex] \Phi^{(A)}(p,q) = \psi^{(A)} (r_{p},r_{q}) \chi^{(S)}_{S,M_{s}}(p,q) [/tex]

where chi is the spin state

so since the energy is 3 E0 for the first particle there possible value nx,ny,nz are n=(1,1,1) and the second particle n'=(1,1,2).

we could select

[tex] \Psi^{(A)} (x_{p},x_{q},t) = \frac{1}{\sqrt{2}} (\psi_{n}(x_{p})\psi_{n'}(x_{q} - \psi_{n}(x_{q})\psi_{n'}(x_{p}) \exp[-\frac{i(E_{n} + E_{n'})t}{\hbar} [/tex]

A means it is unsymmetric

the spin state chi could be

[tex] \chi^{(S)}_{1,1}(p,q) = \chi_{+}(p) \chi_{-}(q) [/tex]

S means it is symmetric

For d it is similar but switched around

for a) and b) i have doubts though

For a) the bosons must be distinguishable so we could have WF like this

[tex] \Psi_{1} (r_{p},r_{q},t) = \psi_{n}(r_{p})\psi_{n'}(r_{q}) \exp[-\frac{i(E_{n} + E_{n'})t}{\hbar} [/tex]

for one of the particles. Under exchange this would be symmetric.

b) if the bosons are identical then we simply have to construct a wavefunction that is smmetric like we did in part c for the fermions.

Is this right? Please help!

thanks for any and all help!

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# Wavefunctions of fermions and bosons

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