## Ramped Finite Wells

*Please wait for the animation to completely load.*

Ramped wells consist of a potential energy function that is
proportional to *x* added to either a finite well or an infinite
well. The result is a finite or an infinite well with a ramped
bottom. For an infinite well, these solutions must also satisfy
the boundary condition (ψ = 0) at the infinite wells, while in
the case of a finite well such as the one shown, we must match the solution in the classically-forbidden regions.

Such a spatially-varying potential energy function means that for
a given energy eigenstate, *E *−* V*(*x*) will also change
over the extent of the well. One set of potential energy
function is shown in the animation
**(***ħ *= 2*m* = 1). Using the slider, you can change the ramping potential,
*V*_{r}, to
see the effect on the energy eigenfunctions and the energy levels. To see the other bound states, simply click-drag in the
energy level diagram on the left to select a level. The selected level will turn red.

In particular, where
the well is deeper, the difference between *E* and *V* is greater. This means that the
curviness*
*of the energy eigenfunction is greater there. In addition, where the well is
deeper we would expect a smaller energy eigenfunction amplitude.