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stunner5000pt
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Homework Statement
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
Homework 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]
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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