I've been studying the 1D schrodinger equation, and getting a feel for solutions in the harmonic oscillator, or potentials of inverse radius (atomic/hydrogen), and many versions of stair-step/ square potentials (square wells.) But, I've noticed that there are very few exact 1D potentials in the literature to study. I'm not satisfied with the conclusions I'm coming to based on these limited 1D cases. Is anyone aware of exact/analytical solutions for Schrodinger's equation in the case where energy ramps linearly, or where momentum ramps linearly with distance? For example, given a well with finite walls at +-a, E(particle) < V(wall) and a linear potential V(x)=x everywhere else, is there already a published exact solution? Or, perhaps instead of V(x)=x^2 - C , as in the harmonic well; is there a known solution for the case where V(x)=C - x^2 with finite walls at some arbitrary location +-a (so as to normalise the solution)? I'm interested in studying the effects on stationary states (purely real psi) and especially the phase of psi, where the change in momentum with distance is an easily expressed non-constant function. eg: I'm hoping for cases where either V or [itex] \hbar k [/itex] is linear in X over a region of the problem. ( I don't care much what the walls/ boundaries look like. ) I'm not seeing pre-worked problems on the internet or in my text books, and am wondering if someone has come across examples they could point me to as a time saver. Thanks!