 Double-well oscillator scale in a linear magnetic field

For $q\alpha k>0$, let $x_0=\sqrt{2\hslash k/(q\alpha )}$. Expansion of the magnetic-gradient quartic confinement at either minimum gives $V(x_0+\delta x)=(q\alpha {)}^2{x}_0^2(\delta x{)}^2/(2m)+\cdots$. The local oscillator frequency is therefore $|q\alpha |x_0/m$. Both lowest parity states have leading energy $\hslash {\omega }_d/2$ if $x_0$ greatly exceeds the oscillator width. Local anharmonicity changes their common energy, while quantum tunnelling gives their much smaller splitting.