 Magnetic-gradient quartic confinement

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 $\pm \sqrt{2\hslash k/(q\alpha )}$; 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.