Quantum Mechanics for Mathematicians

If we now restrict attention to the subspaceHE⊂ Hof energy eigenstates
of energyE, on this space we can define rescaled operators

K=

√

−m

2 E

W

On this subspace, equation 21.2 becomes the relation

2 H(K^2 +L^2 +~^21 ) =−me^41

and we will be able to use this to find the eigenvalues ofHin terms of those of
L^2 andK^2.
We will assume thatE <0, in which case we have the following commutation
relations
[Lj,Lk] =i~jklLl
[Lj,Kk] =i~jklKl
[Kj,Kk] =i~jklLl

Defining

M=

1

2

(L+K), N=

1

2

(L−K)

one has
[Mj,Mk] =i~jklMl
[Nj,Nk] =i~jklNl
[Mj,Nk] = 0

This shows that we have two commuting copies ofso(3) acting on states, spanned
respectively by theMjandNj, with two corresponding Casimir operatorsM^2
andN^2.
Using the fact that
L·K=K·L= 0

one finds that
M^2 =N^2
Recall from our discussion of rotations in three dimensions that representa-
tions ofso(3) =su(2) correspond to representations ofSpin(3) =SU(2), the
double cover ofSO(3) and the irreducible ones have dimension 2l+ 1, withl
half-integral. Only forlintegral does one get representations ofSO(3), and it
is these that occur in theSO(3) representation on functions onR^3. For four di-
mensions, we found thatSpin(4), the double cover ofSO(4), isSU(2)×SU(2),
and one thus hasspin(4) =so(4) =su(2)×su(2) =so(3)×so(3). This is exactly
the Lie algebra we have found here, so one can think of the Coulomb problem
at a fixed negative value ofEas having anso(4) symmetry. The representa-
tions that will occur can include the half-integral ones, since neither of the two
so(3) factors is theso(3) of physical rotations in 3-space (the physical angular
momentum operators are theL=M+N).