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Section 12.5: Ramped Infinite and Finite Wells
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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. Solutions to such ramped wells obey a time-independent Schrödinger equation of the following form
[−(ħ2/2m)(d2/dx2) + αx] ψ(x) = Eψ(x) ,
where α refers to the strength of the ramping function. We can now put this equation into a more standard form
[d2/dx2 − 2mαx/ħ2 + 2mE/ħ2] ψ(x) = 0 ,
which has as its solution, Airy functions. For an infinite well, such as shown in Animation 1: Infinite, these solutions must also satisfy the boundary condition (ψ = 0) at the infinite wells, while in the case of a finite well, we must match the Airy functions with exponentials in the classically-forbidden regions.
Such a spatially-varying potential energy function means that for a given energy eigenfunction, E − V(x) will also change over the extent of the well. Two such potential energy functions (one infinite, one finite) are 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.