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For the hydrogen atom we have a Coulomb force exerted by the proton
forcing the electron to orbit around it. Since the proton is 1837
heavier than the electron, we can ignore the reverse action. The potential
is thus
V(r) = - . |
(11.13) |
If we substitute this in the Schrödinger equation for u(r), we find
-  u(r) - u(r) = Eu(r). |
(11.14) |
The way to attack this problem is once again to combine
physical quantities to set the scale of length, and see what emerges. From
a dimensional analysis we find that the length scale is set by
the Bohr radius a0,
a0 =
= 0.53 x 10-10 m. |
(11.15) |
The scale of energy is set by these same parameters to be
= 2 Ry, |
(11.16) |
and one Ry (Rydberg) is
13.6 eV. Solutions can be found
by a complicated argument similar to the one for the Harmonic oscillator,
but (without proof) we have
En = -  

= - 13.6 eV. |
(11.17) |
and
| Rn = e-r/(na0)(c0 + c1r +...+ cn - 1rn - 1) |
(11.18) |
The explicit, and normalised, forms of a few of these states are
| R1(r) |
= |
2a0-3/2e-r/a0, |
(11.19) |
| R2(r) |
= |
2(2a0)-3/2 1 -
e-r/(2a0). |
(11.20) |
Remember these are normalised to
Rn(r)*Rm(r) dr = . |
(11.21) |
Notice that there are solution that do depend on
and
as well, and that we have not looked at such solutions here!
Next: 11.5 Now where does
Up: 11. 3D Schrödinger equation
Previous: 11.3 Solutions independent of