Chapter 21

Central Potentials and the

Hydrogen Atom

When the Hamiltonian function is invariant under rotations, we then expect
eigenspaces of the corresponding Hamiltonian operator to carry representations
ofSO(3). These spaces of eigenfunctions of a given energy break up into irre-
ducible representations ofSO(3), and we have seen that these are labeled by an
integerl= 0, 1 , 2 ,...and have dimension 2l+ 1. This can be used to find prop-
erties of the solutions of the Schr ̈odinger equation whenever one has a rotation
invariant potential energy. We will work out what happens for the case of the
Coulomb potential describing the hydrogen atom. This specific case is exactly
solvable because it has a second not-so-obviousSO(3) symmetry, in addition to
the one coming from rotations ofR^3.

21.1 Quantum particle in a central potential

In classical physics, to describe not free particles, but particles experiencing
some sort of force, one just needs to add a “potential energy” term to the
kinetic energy term in the expression for the energy (the Hamiltonian function).
In one dimension, for potential energies that just depend on position, one has

h=

p^2

2 m

+V(q)

for some functionV(q). In the physical case of three dimensions, this will be

h=

1

2 m

(p^21 +p^22 +p^23 ) +V(q 1 ,q 2 ,q 3 )

Quantizing and using the Schr ̈odinger representation, the Hamiltonian op-