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Infinite Potential Well
Consider a particle of mass
and energy
moving in the following simple potential:
![\begin{displaymath}
V(x) = \left\{\begin{array}{lcl}
0&\mbox{\hspace{1cm}}&\mbox...
...leq a$}\ [0.5ex]
\infty&&\mbox{otherwise}
\end{array}\right..
\end{displaymath}](img788.png) |
(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
one-dimensional box of length
.
The boundary conditions on
in
the region
are
![\begin{displaymath}
\psi(0) = \psi(a) = 0.
\end{displaymath}](img794.png) |
(303) |
Furthermore, it follows from Eq. (301) that
satisfies
![\begin{displaymath}
\frac{d^2 \psi}{d x^2} = - k^2 \psi
\end{displaymath}](img795.png) |
(304) |
in this region, where
![\begin{displaymath}
k^2 = \frac{2 m E}{\hbar^2}.
\end{displaymath}](img796.png) |
(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
![\begin{displaymath}
\psi_n(x) = A_n \sin(k_n x),
\end{displaymath}](img799.png) |
(306) |
where the
are arbitrary (real) constants, and
![\begin{displaymath}
k_n = \frac{n \pi}{a},
\end{displaymath}](img801.png) |
(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.,
![\begin{displaymath}
E_n = \frac{n^2 \pi^2 \hbar^2}{2 m a^2}.
\end{displaymath}](img803.png) |
(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
![\begin{displaymath}
\int_0^a \psi_n(x) \psi_m(x) dx = \delta_{nm}.
\end{displaymath}](img807.png) |
(309) |
It is easily demonstrated that this is the case, provided
.
Hence,
![\begin{displaymath}
\psi_n(x) = \sqrt{\frac{2}{a}} \sin\left(n \pi \frac{x}{a}\right)
\end{displaymath}](img809.png) |
(310) |
for
.
Finally, again from Sect. 4.12, the general time-dependent solution can be written as a linear superposition of stationary solutions:
![\begin{displaymath}
\psi(x,t) = \sum_{n=0,\infty} c_n \psi_n(x) {\rm e}^{-{\rm i} E_n t/\hbar},
\end{displaymath}](img810.png) |
(311) |
where
![\begin{displaymath}
c_n = \int_0^a\psi_n(x) \psi(x,0) dx.
\end{displaymath}](img811.png) |
(312) |
Next: Square Potential Barrier
Up: One-Dimensional Potentials
Previous: Introduction
Richard Fitzpatrick
2010-07-20