The radial equation can be written in two different equivalent ways, using R(r) or u(r) = r R(r):
| d/dr (r2 dR/dr) - (2 m r2 / hbar2) [V(r) - E + hbar2 l(l+1)/(2 m r2] R = 0 | u(r) = r R(r) |
| -[hbar2 / (2 m)] d2u/dr2 +{V + [hbar2 / (2 m)] l (l+1) / r2 ]} u = Eu |
The solution, of course, depends on what potential we use. It will also depend on l, since l is in the equation, and there will be in general an infinite number of solutions for each l, not all necessarily bound. For the bound states, the functions must be oscillatory. In general, the functions will be complicated and must be evaluated numerically, perhaps in practice from a compiled table. There may be useful relations between functions. For the particular cases we will consider, the standard notation is
| Potential | R(r) | Energy |
|---|---|---|
|
V(r)=0, r<a V(r)=¥, r>a |
A jl(knlr) [nl(kr) rejected for finiteness at r=0] jl(bnl) = 0 and knl a = bnl | E = hbar2 knl2 /(2 m) |
| V(r) = - e2 /(4pe0r) |
R = A (2r)l
e -r
Ln - l - 12l + 1
(2r) L is the associated Laguerre polynomial |
En = - {m e4 /[2 hbar2
(4pe0)2]}/n2 = [hbar2 / (2 m)] [1 / a02] [1 / n2] = (1/2) [e2 / (4pe0 a0 )] [1/n2] Surprisingly, independent of l |
| Notes | k = sqrt(2mE)/hbar |
a0 = 4 p
e0 hbar2
/ (m e2) is the Bohr radius of hydrogen. r = r / (n a0) |
In particular,
| j0 = sin x / x | n0 = - cos x / x |
| j1 = sin x / x2 - cos x / x | n1 = cos x / x2 - sin x / x |
Explicit solution of hydrogen atom omitted in Fall, 2001
The spectrum of hydrogen is obtained by using the equation for the frequency and wavelength of a photon, E = h f = h / l together with the differences in the energies of various hydrogen levels [draw diagram]. The wavelengths are given by
This formula was discovered empirically before it was derived quantum-mechanically.
| Last Revised 01/10/22 |