The Square Well Potential

Problem:

Consider the square potential well shown in the figure below.

(a)  Find the most general solution Φ(x) of the eigenvalue equation HΦ(x) = EΦ(x), (E < 0), in regions 1, 2, and 3 and apply boundary conditions.
(b)  Solve the equation that results from part (a) graphically, and find the conditions under which even and odd solutions exist.

Solution:


Summary
The operator representing the energy of a system is H.  The eigenvalues of H are E.  If the potential U(x) is independent of time, then separation of variables is possible, and we can write ψ(x,t) = Φ(x)Χ(t).  If the wave function is of this form, then Φ(x) = ΦE(x) is an eigenfunction of the operator H, and the energy of the system is certain.  
We find the eigenfunction of H by solving HΦE(x) = EΦE(x).

We can solve this equation in regions of piecewise constant potentials.

Confinement leads to energy quantization.