Quantum Mechanics for Mathematicians

scattering states

(E >0)

bound states

(E <0)

n= 1

n= 2

n= 3

n= 4

l= 0 l= 1 l= 2 l= 3

1 s

2 s

3 s

4 s

2 p

3 p

4 p

3 d

4 d 4 f

Figure 21.1: Energy eigenstates in the Coulomb potential.

The degeneracy in the energy values leads one to suspect that there is some
extra group action in the problem commuting with the Hamiltonian. If so, the
eigenspaces of energy eigenfunctions will come in irreducible representations
of some larger group thanSO(3). If the representation of the larger group
is reducible when one restricts to theSO(3) subgroup, givingncopies of the
SO(3) representation of spinl, that would explain the pattern observed here.
In the next section we will see that this is the case, and there use representation
theory to derive the above formula forEn.
We won’t go through the process of showing how to explicitly find the func-
tionsglEn(r) but just quote the result. Setting

a 0 =

~^2

me^2

(this has dimensions of length and is known as the “Bohr radius”), and defining