The finite square well problem consists of a particle of mass m subject to the potential function in Figure 1.
Keep in mind that no physical potential could exhibit the
discontinuities which we see in this potential at the interface of
regions I and II (x=-a) or the interface between region II and III
(x=a). At some sufficiently small length scale, any
physical potential will look smooth and continuous. You should think of
the square well potential as an idealization of a smooth
potential which happens to vary continuously but very rapidly from
to V=0 in small regions near x=-a and x=a.
Mathematically, you should consider the potential
in Figure 1 as the result of a limiting process where the
steepness of the change in the potential at the interfaces of the
regions is taken to approach infinity.
The time independent Schrödinger equation (1) for the
that the pure energy states 
of energy E of a particle of mass m
may be rearranges into the following more convenient form,
We will begin by considering the possibility of a state ``bound''
within the well. This state will have energy E as sketched in the
figure. We must have that E>0, the minimum
potential in the system, but for a bound state we must also insist
that 
, so that particle has insufficient energy to travel to
large distances from the well.
The TISE holds in each of the three regions I, II, III of the figure. We shall treat each of these regions separately, proceeding from left to right beginning with region I.
