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The animation shows position-space energy eigenfunctions for a finite square well. The half width, a, of the well can be changed with the slider as can the quantum number, n, by click-dragging in the energy spectrum on the left.

1. How does the energy spectrum depend on the well width?
2. Describe what happens to the wave function as the well width gets smaller.
3. Based on your observations, sketch the ground-state wave function for the bound state of an attractive Dirac delta function well.⁴

⁴If we have an attractive Dirac delta function well located at x = 0, −αδ(x), there is an infinitely-negative spike at x = 0. This is an example of a badly-behaved potential energy function and as a consequence we expect that there will be a discontinuity in the wave function. To find how much the slope of the wave function changes (kinks) across the Dirac delta function, we integrate the Schrödinger equation near the Dirac delta function (from -ε to ε) and then let the constant ε → 0 at the end of the calculation:

(dψ/dx)_> − (dψ/dx)_< = − (2mα/ħ²) ψ(0). (11.10)