erator for a particle moving in a potentialV(q 1 ,q 2 ,q 3 ) will be
H=
1
2 m(P 12 +P 22 +P 33 ) +V(Q 1 ,Q 2 ,Q 3 )
=−
~^2
2 m(
∂^2
∂q^21+
∂^2
∂q 22+
∂^2
∂q^23)
+V(q 1 ,q 2 ,q 3 )=−
~^2
2 m∆ +V(q 1 ,q 2 ,q 3 )We will be interested in so-called “central potentials”, potential functions that
are functions only ofq 12 +q 22 +q 32 , and thus only depend uponr, the radial
distance to the origin. For suchV, both terms in the Hamiltonian will be
SO(3) invariant, and eigenspaces ofHwill be representations ofSO(3).
Using the expressions for the angular momentum operators in spherical co-
ordinates derived in chapter 8 (including equation 8.4 for the Casimir operator
L^2 ), one can show that the Laplacian has the following expression in spherical
coordinates
∆ =∂^2
∂r^2+
2
r∂
∂r−
1
r^2L^2
The Casimir operatorL^2 has eigenvaluesl(l+ 1) on irreducible representations
of dimension 2l+ 1 (integral spinl). So, restricted to such an irreducible repre-
sentation, we have
∆ =∂^2
∂r^2+
2
r∂
∂r−
l(l+ 1)
r^2
To solve the Schr ̈odinger equation, we want to find the eigenfunctions of
H. The space of eigenfunctions of energyEwill be a sum of irreducible repre-
sentations ofSO(3), with theSO(3) acting on the angular coordinates of the
wavefunctions, leaving the radial coordinate invariant. To find eigenfunctions
of the Hamiltonian
H=−~^2
2 m∆ +V(r)we can first look for functionsglE(r), depending onl= 0, 1 , 2 ,...and the energy
eigenvalueE, and satisfying
(
−~^2
2 m(
d^2
dr^2+
2
rd
dr−
l(l+ 1)
r^2)
+V(r))
glE(r) =EglE(r)Turning to the angular coordinates, we have seen in chapter 8 that represen-
tations ofSO(3) on functions of angular coordinates can be explicitly expressed
in terms of the spherical harmonic functionsYlm(θ,φ), on whichL^2 acts with
eigenvaluel(l+ 1). For each solutionglE(r) we will have the eigenvalue equation
HglE(r)Ylm(θ,φ) =EglE(r)Ylm(θ,φ)and the
ψ(r,θ,φ) =glE(r)Ylm(θ,φ)