= Normal modes of two harmonically coupled identical particles
{title2=$y_\pm=(x_1\pm x_2)/\sqrt2$}
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.
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