We study "square" potentials, or piece-wise constant potentials, because they may crudely approximate real potentials. Let us concentrate on one-dimensional "square" potentials. Assume U(x) = U = constant in certain regions of space. In such regions the time-independent Schroedinger equation, HΦ(x) = EΦ(x,) can be written as

∂^{2}Φ(x)/∂x^{2} + (2m(E -
U)/ħ^{2})Φ(x) = 0.

We find the eigenfunction of H by solving HΦ_{E}(x) = EΦ_{E}(x).

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

Let E > U:

Then ∂^{2}Φ(x)/∂x^{2} + k^{2}Φ(x) =
0, k^{2} = 2m(E - U)/ħ^{2}.

The most general solution is Φ(x) = Ae^{ikx
}+ A'e^{-ikx}, with A
and A’ complex constants.

Let E < U:

Then ∂^{2}Φ(x)/∂x^{2} - ρ^{2}Φ(x) =
0, ρ^{2} = 2m(U - E)/ħ^{2}.

The most general solution is Φ(x) = Be^{iρx}^{
}+ B'e^{-iρx,} with B and B’
complex constants.

**Note: A solution exists in the classically forbidden region.**

Let E = U:

∂^{2}Φ(x)/∂x^{2} = 0

The most general solution is Φ(x) = Cx + C',
with C and C’ complex constants.

Approximate U(x) by U_{ε}(x), which is
equal to U(x) everywhere except in a small region x_{1 }± ε, where it does not have the step but varies continuously.
We can write the Schroedinger equation as a difference equation,

∂Φ_{ε}(x_{1 }+ ε)/∂x
+ ∂Φ_{ε}(x_{1 }- ε)/∂x
= (2m/ħ^{2})∫_{x1-ε}^{x1+ε }(U_{ε}(x) - E) Φ(x) dx

As ε --> 0, U_{ε}(x)
remains finite, and therefore the
integral remains finite and decreases as ε decreases.
For ε = 0 the integral is zero. Therefore ∂Φ_{ε}/∂x|_{ε-->0} is continuous.

At a finite step the **boundary
conditions** are that Φ(x) and ∂Φ(x)/∂x
are continuous.

(b) Assume U(x) does not remain finite on one side of the step.

Approximate U(x) by U_{ε}(x) such that U_{ε}(x)
has a step ∆U over a small interval 2ε = 2c/∆U.

As ε --> 0, ∆U
--> ∞, and ∫_{x1-ε}^{x1+ε }(U_{ε}(x) - E) Φ(x) dx
≈ Φ(x_{1})[c
- 2εE] --> cΦ(x_{1}).

At an infinite step ∂Φ_{ε}/∂x|_{ε-->0} is discontinuous, but it has a finite
discontinuity. Therefore Φ(x) remains
continuous as ε --> 0.

In a region of finite width ∆x, where |U| is
infinite, the wave function must be zero everywhere. Since the wave function
is
continuous, it also must be zero also at the boundary in the adjacent region.
If |U|
is infinite in a region with width ∆x = 0 (a δ-function,
for example), the
wave function can be finite there. It will have
the same value at the boundaries of the two adjacent regions and its derivative with
respect to x will be discontinuous.