Quantum Mechanics for Mathematicians

representation, where thePjare differentiation operators, this will be a second-
order differential operator, and the eigenvalue equation will be a second-order
differential equation (the time-independent Schr ̈odinger equation).
Using the Fourier transform, the space of solutions of the Schr ̈odinger equa-
tion of fixed energy becomes something much easier to analyze, the space of
functions (or, more generally, distributions) on momentum space supported
only on the subspace of momenta of a fixed length. In the case ofE(2) this
is just a circle, whereas forE(3) it is a sphere. In both cases, for each radius
one gets an irreducible representation in this manner.
In the case ofE(3) other classes of irreducible representations can be con-
structed. This can be done by introducing multi-component wavefunctions,
with a new action of the rotation groupSO(3). A second Casimir operator is
available in this case, and irreducible representations are eigenfunctions of this
operator in the space of wavefunctions of fixed energy. The eigenvalues of this
second Casimir operator turn out to be proportional to an integer, the “helicity”
of the representation.

19.1 The quantum free particle and representa-

tions ofE(2)

We’ll begin for simplicity with the case of two spatial dimensions. Recall from
chapter 18 that the Euclidean groupE(d) is a subgroup of the Jacobi group
GJ(d) =H 2 d+1oSp(2d,R). For the cased= 2, the translationsR^2 are a sub-
group of the Heisenberg groupH 5 (translations inq 1 ,q 2 ) and the rotations are
a subgroupSO(2)⊂Sp(4,R) (simultaneous rotations ofq 1 ,q 2 andp 1 ,p 2 ). The
Lie algebra ofE(2) is a sub-Lie algebra of the Lie algebragJ(2) of polynomials
inq 1 ,q 2 ,p 1 ,p 2 of degree at most two.
More specifically, a basis for the Lie algebra ofE(2) is given by the functions

l=q 1 p 2 −q 2 p 1 , p 1 , p 2

on thed= 2 phase spaceM=R^4 , wherelis a basis for the Lie algebraso(2)
of rotations,p 1 ,p 2 a basis for the Lie algebraR^2 of translations. The non-zero
Lie bracket relations are given by the Poisson brackets

{l,p 1 }=p 2 , {l,p 2 }=−p 1

which are the infinitesimal version of the rotation action ofSO(2) onR^2. There
is an isomorphism of this Lie algebra with a matrix Lie algebra of 3 by 3 matrices
given by

l↔





0 −1 0

1 0 0

0 0 0



, p 1 ↔





0 0 1

0 0 0

0 0 0



, p 2 ↔





0 0 0

0 0 1

0 0 0





Since we have realized the Lie algebra ofE(2) as a sub-Lie algebra of the
Jacobi Lie algebragJ(2), quantization via the Schr ̈odinger representation Γ′S