Normal modes of two harmonically coupled identical particles ($y_{\pm }=(x_1\pm x_2)/\sqrt{2}$)

For equal masses in identical quadratic traps with an interaction depending quadratically on $x_1-x_2$, the displayed orthogonal coordinates of the normal modes separate the Hamiltonian operator into a collective and a relative oscillator. orthogonality preserves both the kinetic Laplacian and the absolute coordinate Jacobian. Particle exchange fixes $y_{+}$ and negates $y_{-}$. Hence bosonic exchange symmetry selects even relative oscillator levels, and fermionic exchange symmetry selects odd ones. For a trap frequency $\omega$ and interaction $-m{\omega }^2{\sin }^2\alpha (x_1-x_2{)}^2/4$, the two frequencies are $\omega$ and $\omega \cos \alpha$; the relative oscillator length diverges when the latter approaches zero.