= Magnetic-gradient quartic confinement
{title2=$V_k(x)=[\hbar k-q\alpha x^2/2]^2/(2m)$}
A spinless charged particle in $B_z=\alpha x$ admits <vector potential> $A_y=\alpha x^2/2$. Separating a plane wave along y produces a one-dimensional quartic <potential energy>. For nonzero $q\alpha$, $k/(q\alpha)>0$ gives two minima at $\pm\sqrt{2\hbar k/(q\alpha)}$; the opposite sign gives one minimum at zero. The field gradient creates confinement even though there is no <scalar potential>. The quartic potential is even, so its nondegenerate <eigenstates> have definite spatial <parity>.
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