The Hydrogen Atom

Finally, consider the hydrogen atom as a proton fixed at the origin, orbited by an electron of reduced mass $\mu$. The potential due to electrostatic attraction is

$$V(r) = - \frac{e^2}{4 \pi \epsilon_0 r}$$ (126)

in SI units. The kinetic energy term in the Hamiltonian is

$$\hat{T} = - \frac{\hbar^2}{2 \mu} \nabla^2$$ (127)

so we write out the Schrödinger equation in spherical polar coordinates as

$$- \frac{\hbar^2}{2 \mu} \left[ \frac{1}{r^2} \frac{\partial}... ...\epsilon_0 r} \psi(r, \theta, \phi) = E \psi(r, \theta, \phi)$$ (128)

It happens that we can factor $\psi(r, \theta, \phi)$ into $R(r)_{nl}Y_l^m(\theta, \phi)$, where $Y_l^m(\theta, \phi)$ are again the spherical harmonics. The radial part $R(r)$ then can be shown to obey the equation

$$- \frac{\hbar^2}{2 \mu r^2} \frac{d}{dr} \left( r^2 \frac{dR... ...[ \frac{\hbar^2 l(l+1)}{2 \mu r^2} + V(r) - E \right] R(r) = 0$$ (129)

which is called the radial equation for the hydrogen atom. Its (messy) solutions are

$$R_{nl}(r) = - \left[ \frac{(n - l - 1)!}{2n[(n+l)!]^3} \righ... ...^l e^{-r/na_0} L_{n+l}^{2l+1} \left( \frac{2r}{n a_0} \right)$$ (130)

where $0 \leq l \leq n - 1$, and $a_0$ is the Bohr radius, $\epsilon_0 h^2 / \pi \mu e^2$. The functions $L_{n+l}^{2l+1}(2r/na_0)$ are the associated Laguerre functions. The hydrogen atom eigenvalues are

$$E_n = - \frac{e^2}{8 \pi \epsilon_0 a_0 n^2} \hspace{1.0cm} n=1,2,\ldots$$ (131)

There are relatively few other interesting problems that can be solved analytically. For molecular systems, one must resort to approximate solutions.