Next: The Harmonic Oscillator
Up: Some Analytically Soluble Problems
Previous: Some Analytically Soluble Problems
Contents
Consider a particle constrained to move in a single dimension, under
the influence of a potential
which is zero for
and
infinite elsewhere. Since the wavefunction is not allowed to become
infinite, it must have a value of zero where
is infinite, so
is nonzero only within
. The Schrödinger equation
is thus
![\begin{displaymath}
- \frac{\hbar^2}{2m} \frac{d^2\psi}{dx^2} = E \psi(x)
\hspace{0.5cm} 0 \leq x \leq a
\end{displaymath}](img292.png) |
(115) |
It is easy to show that
the eigenvectors and eigenvalues of this problem are
![\begin{displaymath}
\psi_n(x) = \sqrt{\frac{2}{a}} sin \left( \frac{n \pi x}{a} ...
...\hspace{1.0cm} 0 \leq x \leq a \hspace{1.0cm} n = 1,2,3,\ldots
\end{displaymath}](img293.png) |
(116) |
![\begin{displaymath}
E_n = \frac{h^2 n^2}{8 m a^2} \hspace{1.0cm} n=1,2,\ldots
\end{displaymath}](img294.png) |
(117) |
Extending the problem to three dimensions is rather straightforward;
see McQuarrie [1], section 6.1.
David Sherrill
2006-08-15