Quantum Mechanics for Mathematicians

• The groupSO(3) acts on momentum vectors by rotation, with orbit of the
  group action the sphere of momentum vectors of fixed energyE >0. This
  is the sphere on which the Fourier transform of the wavefunctions in the
  representation is supported. Unlike the corresponding circle in theE(2)
  case, here there is a non-trivial subgroup of the rotation groupSO(3)
  which leaves a given momentum vector invariant. This is theSO(2)⊂
  SO(3) subgroup of rotations about the axis determined by the momentum
  vector, and it is different for different points in momentum space.

p 1 p 2

p 3

√

2 mE

SO(2)⊂SO(3)

Figure 19.2: Copy ofSO(2) leaving a given momentum vector invariant.

For single-component wavefunctions, a straightforward computation shows
that the second Casimir operatorL·Pacts as zero. By introducing wavefunc-
tions with several components, together with an action ofSO(3) that mixes the
components, it turns out that one can get new irreducible representations, with
a non-zero value of the second Casimir corresponding to a non-trivial weight of
the action of theSO(2) of rotations about the momentum vector.
Such multiple-component wavefunctions can be constructed as representa-
tions ofE(3) by taking the tensor product of our irreducible representation on
wavefunctions of energyE(call thisHE) and the finite dimensional irreducible
representationC^2 s+1
HE⊗C^2 s+1

The Lie algebra representation operators for the translation part ofE(3) act as
momentum operators onHEand as 0 onC^2 s+1. For theSO(3) part ofE(3),
we get angular momentum operators that can be written as

Jj=Lj+Sj≡Lj⊗ 1 + 1 ⊗Sj