Quantum Mechanics for Mathematicians

operators

L−L+=(L 1 −iL 2 )(L 1 +iL 2 )

=L^21 +L^22 +i[L 1 ,L 2 ]

=L^21 +L^22 −L 3

so
L^2 =L^21 +L^22 +L^23 =L−L++L 3 +L^23
For the representationρofSO(3) on functions onS^2 constructed above,
we know that on a highest weight vector of the irreducible representationρl
(restriction ofρto the 2l+ 1 dimensional irreducible subspace of functions that
are linear combinations of theYlm(θ,φ)), we have the two eigenvalue equations

L+f= 0, L 3 f=lf

with solution the functions proportional toYll(θ,φ). Just from these conditions
and our expression forL^2 we can immediately find the scalar eigenvalue ofL^2
since
L^2 f=L−L+f+ (L 3 +L^23 )f= (0 +l+l^2 )f=l(l+ 1)f

We have thus shown that our irreducible representationρlcan be characterized
as the representation on whichL^2 acts by the scalarl(l+ 1).
In summary, we have two different sets of partial differential equations whose
solutions provide a highest weight vector for and thus determine the irreducible
representationρl:

• L+f= 0, L 3 f=lf
  which are first-order equations, with the first using complexification and
  something like a Cauchy-Riemann equation, and


• 
  

  L^2 f=l(l+ 1)f, L 3 f=lf
  where the first equation is a second-order equation, something like a
  Laplace equation.

  
  

That a solution of the first set of equations gives a solution of the second set
is obvious. Much harder to show is that a solution of the second set gives a
solution of the first set. The space of solutions to

L^2 f=l(l+ 1)f

forla non-negative integer includes as we have seen the 2l+ 1 dimensional
vector space of linear combinations of theYlm(θ,φ) (there are no other solu-
tions, although we will not show that). Since the action ofSO(3) on functions
commutes with the operatorL^2 , this 2l+ 1 dimensional space will provide a
representation, the irreducible one of spinl.