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Infinite Potential Well
Consider a particle of mass and energy moving in the following simple potential:

(302) 
It follows from Eq. (301) that if (and, hence, ) is
to remain finite then must go to zero in regions where the potential
is infinite. Hence, in the regions and .
Evidently, the problem is equivalent to that of a particle trapped in a
onedimensional box of length .
The boundary conditions on in
the region are

(303) 
Furthermore, it follows from Eq. (301) that satisfies

(304) 
in this region, where

(305) 
Here, we are assuming that . It is easily demonstrated that there are
no solutions with which are capable of satisfying the boundary conditions (303).
The solution to Eq. (304), subject to the boundary conditions
(303), is

(306) 
where the are arbitrary (real) constants, and

(307) 
for
. Now, it can be seen from Eqs. (305) and (307)
that the energy is only allowed to take certain discrete values:
i.e.,

(308) 
In other words, the eigenvalues of the energy operator are discrete. This
is a general feature of bounded solutions: i.e., solutions in which
as
. According to the discussion in Sect. 4.12,
we expect the stationary eigenfunctions to satisfy
the orthonormality constraint

(309) 
It is easily demonstrated that this is the case, provided
.
Hence,

(310) 
for
.
Finally, again from Sect. 4.12, the general timedependent solution can be written as a linear superposition of stationary solutions:

(311) 
where

(312) 
Next: Square Potential Barrier
Up: OneDimensional Potentials
Previous: Introduction
Richard Fitzpatrick
20100720