Quantum Mechanics for Mathematicians

gnl(r) =glEn(r) the solutions are of the form

gnl(r)∝e−

r

na 0

(

2 r

na 0

)l

L^2 nl++1l

(

2 r

na 0

)

where theL^2 nl++1l are certain polynomials known as associated Laguerre polyno-
mials.
So, finally, we have found energy eigenfunctions

ψnlm(r,θ,φ) =gnl(r)Ylm(θ,φ)

for
n= 1, 2 ,...
l= 0, 1 ,...,n− 1
m=−l,−l+ 1,...,l− 1 ,l

The first few of these, properly normalized, are

ψ 100 =

1

√

πa^30

e−

ar

0

(called the 1Sstate, “S” meaningl= 0)

ψ 200 =

1

√

8 πa^30

(

1 −

r

2 a 0

)

e−

2 ar

0

(called the 2Sstate), and the three dimensionall= 1 (called 2P, “P” meaning
l= 1) states with basis elements

ψ 211 =−

1

8

√

πa^30

r

a 0

e−

2 ra

(^0) sinθeiφ
ψ 210 =−

1

4

√

2 πa^30

r

a 0

e−

2 ra

(^0) cosθ
ψ 21 − 1 =

1

8

√

πa^30

r

a 0

e−

2 ra

(^0) sinθe−iφ

21.2so(4) symmetry and the Coulomb potential

The Coulomb potential problem is very special in that it has an additional
symmetry, of a non-obvious kind. This symmetry appears even in the classi-
cal problem, where it is responsible for the relatively simple solution one can
find to the essentially identical Kepler problem. This is the problem of finding
the classical trajectories for bodies orbiting around a central object exerting a
gravitational force, which also has a^1 rpotential.