The Square Well Potential


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.


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.