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Finally, consider the hydrogen atom as a proton fixed at the origin,
orbited by an electron of reduced mass
. The potential due to
electrostatic attraction is
 |
(126) |
in SI units. The kinetic energy term in the Hamiltonian is
 |
(127) |
so we write out the Schrödinger equation in spherical polar coordinates as
 |
(128) |
It happens that we can factor
into
, where
are again the spherical harmonics. The radial part
then can
be shown to obey the equation
![\begin{displaymath}
- \frac{\hbar^2}{2 \mu r^2} \frac{d}{dr} \left( r^2 \frac{dR...
...[ \frac{\hbar^2 l(l+1)}{2 \mu r^2} + V(r) - E \right] R(r) = 0
\end{displaymath}](img325.png) |
(129) |
which is called the radial equation for the hydrogen atom.
Its (messy) solutions are
![\begin{displaymath}
R_{nl}(r) = - \left[ \frac{(n - l - 1)!}{2n[(n+l)!]^3} \righ...
...^l e^{-r/na_0} L_{n+l}^{2l+1}
\left( \frac{2r}{n a_0} \right)
\end{displaymath}](img326.png) |
(130) |
where
, and
is the Bohr radius,
. The functions
are the associated
Laguerre functions. The hydrogen atom eigenvalues are
 |
(131) |
There are relatively few other interesting problems that can be solved
analytically. For molecular systems, one must resort to approximate
solutions.
Next: Approximate Methods
Up: Some Analytically Soluble Problems
Previous: The Rigid Rotor
Contents
David Sherrill
2006-08-15