Quantum Mechanics for Mathematicians

and so the constant characterizing an irreducible will be the energy 2mE. Our ir-
reducible representation will be on the space of solutions of the time-independent
Schr ̈odinger equation

1

2 m

(P 12 +P 22 +P 32 )ψ(q) =−

1

2 m

(

∂^2

∂q^21

+

∂^2

∂q 22

+

∂^2

∂q 32

)

ψ(q) =Eψ(q)

Using the Fourier transform

ψ(q) =

1

(2π)

(^32)

∫

R^3

eip·qψ ̃(p)d^3 p

the time-independent Schr ̈odinger equation becomes
(
|p|^2
2 m

−E

)

ψ ̃(p) = 0

and we have distributional solutions

ψ ̃(p) =ψ ̃E(p)δ(|p|^2 − 2 mE)

characterized by distributionsψ ̃E(p) defined on the sphere|p|^2 = 2mE.
Such complex-valued distributions on the sphere of radius

√

2 mEprovide a
Fourier-transformed versionu ̃of the irreducible representation ofE(3). Here
the action of the groupE(3) is by

̃u(a, 1 )ψ ̃E(p) =e−i(a·p)ψ ̃E(p)

for translations, by
̃u( 0 ,R)ψ ̃E(p) =ψ ̃E(R−^1 p)

for rotations, and by

̃u(a,R)ψ ̃E(p) =u ̃(a, 1 ) ̃u( 0 ,R)ψ ̃E(p) =e−ia·R

− (^1) p ̃
ψE(R−^1 p)
for a general element.

19.3 Other representations ofE(3)

For the case ofE(3), besides the representations parametrized byE >0 con-
structed above, as in theE(2) case there are finite dimensional representations
where the translation subgroup ofE(3) acts trivially. Such irreducible represen-
tations are just the spin-srepresentations (ρs,C^2 s+1) ofSO(3) fors= 0, 1 , 2 ,....
E(3) has some structure not seen in theE(2) case, which can be used to
construct new classes of infinite dimensional irreducible representations. This
can be seen from two different points of view:

• There is a second Casimir operator which one can show commutes with
  theE(3) action, given by

L·P=L 1 P 1 +L 2 P 2 +L 3 P 3