Quantum Mechanics for Mathematicians

will span a 2l+ 1 dimensional (sincem=−l,−l+ 1,...,l− 1 ,l) space of energy
eigenfunctions forHof eigenvalueE.
For a general potential functionV(r), exact solutions for the eigenvaluesE
and corresponding functionsglE(r) cannot be found in closed form. One special
case where we can find such solutions is for the three dimensional harmonic os-
cillator, whereV(r) =^12 mω^2 r^2. These are much more easily found though using
the creation and annihilation operator techniques to be discussed in chapter 22.
The other well known and physically very important such case is the case
of a^1 rpotential, called the Coulomb potential. This describes a light charged
particle moving in the potential due to the electric field of a much heavier
charged particle, a situation that corresponds closely to that of a hydrogen
atom. In this case we have

V=−

e^2

r

whereeis the charge of the electron, so we are looking for solutions to

(

−~^2

2 m

(

d^2

dr^2

+

2

r

d

dr

−

l(l+ 1)

r^2

)

−

e^2

r

)

glE(r) =EglE(r) (21.1)

Since on functionsf(r)

d^2

dr^2

(rf) =r(

d^2

dr^2

+

2

r

d

dr

)f

multiplying both sides of equation 21.1 byrgives
(
−~^2
2 m

(

d^2

dr^2

−

l(l+ 1)

r^2

)

−

e^2

r

)

rglE(r) =ErglE(r)

The solutions to this equation can be found through a rather elaborate pro-
cess described in most quantum mechanics textbooks, which involves looking
for a power series solution. ForE≥0 there are non-normalizable solutions that
describe scattering phenomena that we won’t study here. ForE <0 solutions
correspond to an integern= 1, 2 , 3 ,..., withn≥l+ 1. So, for eachnwe getn
solutions, withl= 0, 1 , 2 ,...,n−1, all with the same energy

En=−

me^4

2 ~^2 n^2

A plot of the different energy eigenstates looks like this: