= Relative-coordinate box spectrum of identical particles
{title2=$E_{r,s}=\frac{\pi^2\hbar^2}{8ma^2}(r^2+s^2)$}
For two equal-mass particles with orthonormal coordinates $y_\pm=(x_1\pm x_2)/\sqrt2$ confined to $|y_\pm|<a$, the <Hamiltonian> is the sum of two width-$2a$ <infinite square well> Hamiltonians. Exchange preserves $y_+$ and reverses $y_-$. The relative-coordinate box mode is even for odd index $s$ and odd for even index $s$. Pure spatial <bosonic exchange symmetry> therefore allows odd $s$, while pure spatial <fermionic exchange symmetry> allows even $s$.
The bosonic ground mode is $(r,s)=(1,1)$, while its first excited mode $(2,1)$ is degenerate in energy with the fermionic ground mode $(1,2)$. Angular spin degrees of freedom, if present, must be included in the full exchange symmetry rather than assigned the spatial rule automatically. The mean relative distance is $\langle|y_-|\rangle=a(1/2-2/\pi^2)$ in the lowest even mode and $a/2$ in the lowest odd mode, obtained by integrating the normalized squared cosine and sine. Multiplication by $\sqrt2$ gives the particle separation.
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