Quantum Mechanics for Mathematicians

Taking the derivative, the Lie algebra representation on functions is given
by

ρ′(X)f=

d

dt

ρ(etX)f|t=0=

d

dt

f(etX·x 1 ,etX·x 2 ,etX·x 3 )|t=0

whereX∈so(3). Recall that a basis forso(3) is given by

l 1 =





0 0 0

0 0 − 1

0 1 0



 l 2 =





0 0 1

0 0 0

−1 0 0



 l 3 =





0 −1 0

1 0 0

0 0 0





which satisfy the commutation relations

[l 1 ,l 2 ] =l 3 ,[l 2 ,l 3 ] =l 1 ,[l 3 ,l 1 ] =l 2

Digression.A Note on Conventions
We’re using the notationljfor the real basis of the Lie algebraso(3) =su(2).
For a unitary representationρ, theρ′(lj)will be skew-adjoint linear operators.
For consistency with the physics literature, we’ll use the notationLj=iρ′(lj)
for the self-adjoint version of the linear operator corresponding tolj in this
representation on functions. TheLjsatisfy the commutation relations

[L 1 ,L 2 ] =iL 3 ,[L 2 ,L 3 ] =iL 1 ,[L 3 ,L 1 ] =iL 2

We’ll also use elementsl±=l 1 ±il 2 of the complexified Lie algebra to create
raising and lowering operatorsL±=iρ′(l±).
As with theSU(2)case, we won’t include a factor of~as is usual in physics
(the usual convention isLj=i~ρ′(lj)), since for considerations of the action of

the rotation group it would cancel out (physicists define rotations usinge

i~θLj
).
The factor of~is only of significance whenLj is expressed in terms of the
momentum operator, a topic discussed in chapter 19.
In theSU(2)case, theπ′(Sj)had half-integral eigenvalues, with the eigen-
values ofπ′(2S 3 )the integral weights of the representation. Here theLjwill
have integer eigenvalues, the weights will be the eigenvalues of 2 L 3 , which will
be even integers.

Computingρ′(l 1 ) we find