Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.31.
The finite square well has the potential
where is a positive constant energy, and is a constant location on the axis.
The delta function potential can be thought of as the limit of the finite square well as and in such a way that the area of the rectangle in the well is a constant. That is, the integral of the potential is the same in both cases, so that
The energies of the bound states for the even solution of the finite square well are given by
Substituting for we get
As , , so becomes very small. In this limit, , so we can approximate equation 3 by
If we retain only the term in (discarding higher powers of ), we get
This is the energy we found earlier when analyzing the delta function well.
We can do a similar analysis for the scattering states. The transmission coefficient for the finite square well is
For small we can use the approximation and we get
Substituting for gives
This is the same formula we obtained when analyzing the delta function potential directly.