Ramped Finite Wells

Restart Finite Well

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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 = 2m = 1).  Using the slider, you can change the ramping potential, Vr, 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.