Editing Old quantum theory (section)

=== Hydrogen atom ===
The angular part of the hydrogen atom is just the rotator, and gives the quantum numbers ''l'' and ''m''. The only remaining variable is the radial coordinate, which executes a periodic one-dimensional potential motion, which can be solved.

For a fixed value of the total angular momentum ''L'', the Hamiltonian for a classical Kepler problem is (the unit of mass and unit of energy redefined to absorb two constants):
: <math>
H= { p_r^2 \over 2 } + {l^2 \over 2 r^2 } - {1\over r}.
</math>

Fixing the energy to be (a negative) constant and solving for the radial momentum <math>p_r</math>, the quantum condition integral is:
: <math>
\oint \sqrt{2E - {l^2\over r^2} + { 2\over r}}\ dr=  k h
</math>
which can be solved with the method of residues,<ref name="Sommerfeld"/> and gives a new quantum number <math>k</math> which determines the energy in combination with <math>l</math>. The energy is:
: <math>
 E= -{1 \over 2 (k + l)^2}
</math>
and it only depends on the sum of ''k'' and ''l'', which is the ''principal quantum number'' ''n''. Since ''k'' is positive, the allowed values of ''l'' for any given ''n'' are no bigger than ''n''. The energies reproduce those in the Bohr model, except with the correct quantum mechanical multiplicities, with some ambiguity at the extreme values.