Relative-coordinate box spectrum of identical particles ($E_{r,s}=\frac{{\pi }^2{\hslash }^2}{8m{a}^2}(r^2+s^2)$)

For two equal-mass particles with orthonormal coordinates $y_{\pm }=(x_1\pm x_2)/\sqrt{2}$ 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 $⟨|y_{-}|⟩=a(1/2-2/{\pi }^2)$ in the lowest even mode and $a/2$ in the lowest odd mode, obtained by integrating the normalized squared cosine and sine. Multiplication by $\sqrt{2}$ gives the particle separation.