= Double-well oscillator scale in a linear magnetic field
{title2=$\omega_d=\sqrt{2\hbar q\alpha k}/m$}
For $q\alpha k>0$, let $x_0=\sqrt{2\hbar k/(q\alpha)}$. Expansion of the <magnetic-gradient quartic confinement> at either minimum gives $V(x_0+\delta x)=(q\alpha)^2x_0^2(\delta x)^2/(2m)+\cdots$. The local oscillator frequency is therefore $|q\alpha|x_0/m$. Both lowest parity states have leading energy $\hbar\omega_d/2$ if $x_0$ greatly exceeds the oscillator width. Local anharmonicity changes their common energy, while <quantum tunnelling> gives their much smaller splitting.
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