Quantum Mechanics for Mathematicians

or, in components

l 1 =q 2 p 3 −q 3 p 2 , l 2 =q 3 p 1 −q 1 p 3 , l 3 =q 1 p 2 −q 2 p 1

The Euclidean groupE(3) is a subgroup of the Jacobi groupGJ(3) in the
same way as in two dimensions, and, just as in theE(2) case, exponentiating
the Schr ̈odinger representation Γ′S

Γ′S(l 1 ) =−iL 1 =−i(Q 2 P 3 −Q 3 P 2 ) =−

(

q 2

∂

∂q 3

−q 3

∂

∂q 2

)

Γ′S(l 2 ) =−iL 2 =−i(Q 3 P 1 −Q 1 P 3 ) =−

(

q 3

∂

∂q 1

−q 1

∂

∂q 3

)

Γ′S(l 3 ) =−iL 3 =−i(Q 1 P 2 −Q 2 P 1 ) =−

(

q 1

∂

∂q 2

−q 2

∂

∂q 1

)

Γ′S(pj) =−iPj=−

∂

∂qj

provides a representation ofE(3).
As in theE(2) case, the above Lie algebra representation is just the in-
finitesimal version of the action ofE(3) on functions induced from its action
on position spaceR^3. Given an elementg= (a,R)∈E(3) we have a unitary
transformation on wavefunctions

u(a,R)ψ(q) =ψ(g−^1 ·q) =ψ(R−^1 (q−a))

Such group elementsgwill be a product of a translation and a rotation, and
treating these separately, the unitary transformationsuare exponentials of the
Lie algebra actions above, with

u(a, 1 )ψ(q) =e−i(a^1 P^1 +a^2 P^2 +a^3 P^3 )ψ(q) =ψ(q−a)

for a translation bya, and

u( 0 ,R(φ,ej))ψ(q) =e−iφLjψ(q) =ψ(R(−φ,ej)q)

forR(φ,ej) a rotation about thej-axis by angleφ.
This representation ofE(3) on wavefunctions is reducible, since in terms of
momentum eigenstates, rotations will only take eigenstates with one value of the
momentum to those with another value of the same norm-squared. We can get
an irreducible representation by using the Casimir operatorP 12 +P 22 +P 32 , which
commutes with all elements in the Lie algebra ofE(3). The Casimir operator
will act on an irreducible representation as a scalar, and the representation will
be characterized by that scalar. The Casimir operator is just 2mtimes the
Hamiltonian

H=

1

2 m

(P 12 +P 22 +P 32 )