Required math: calculus
Required physics: Schrödinger equation
Reference: Griffiths, David J. (2005), Introduction to Quantum Mechanics, 2nd Edition; Pearson Education – Problem 2.30.
The finite square well has the potential
where is a positive constant energy, and is a constant location on the axis.
We’ve seen that since the potential is an even function, we can look for solutions of the Schrödinger equation that are either even or odd. In the even function case, we’ve seen that the solution has the form
Imposing boundary conditions gives us the relations
from which we get the relation between and :
To complete the analysis, we need to normalize the wave function. Since it’s an even function, the integral from 0 to of the square modulus is half the integral from to , so we get:
Doing the integrals gives us
From the first boundary condition above, we have
Solving these last two equations, we get
Using the relation between and along with the trig identity we get
While we’re at it, we can also normalize the odd solution. Here, the solution has the form
The boundary conditions give
with the resulting relation between and :
The normalization integral is
The first boundary condition gives
Solving these two equations gives
Rewriting the relation between and we get
and inserting this into the solutions gives
The final result is then
where we’ve taken the negative root of in order to satisfy the original boundary conditions above (we could also have taken with a negative sign and with a positive sign).