provides a unitary Lie algebra representation on the state spaceHof functions
of the position variablesq 1 ,q 2. This will be given by the operators
Γ′S(p 1 ) =−iP 1 =−∂
∂q 1
, Γ′S(p 2 ) =−iP 2 =−∂
∂q 2(19.1)
and
Γ′S(l) =−iL=−i(Q 1 P 2 −Q 2 P 1 ) =−(
q 1∂
∂q 2−q 2∂
∂q 1)
(19.2)
The Hamiltonian operator for the free particle isH=
1
2 m(P 12 +P 22 ) =−
1
2 m(
∂^2
∂q 12+
∂^2
∂q^22)
and solutions to the Schr ̈odinger equation can be found by solving the eigenvalue
equation
Hψ(q 1 ,q 2 ) =−1
2 m(
∂^2
∂q^21+
∂^2
∂q 22)
ψ(q 1 ,q 2 ) =Eψ(q 1 ,q 2 )The operatorsL,P 1 ,P 2 commute withHand so provide a representation of the
Lie algebra ofE(2) on the space of wavefunctions of energyE.
This construction of irreducible representations ofE(2) is similar in spirit
to the construction of irreducible representations ofSO(3) in section 8.4. There
the Casimir operatorL^2 commuted with theSO(3) action, and gave a differ-
ential operator on functions on the sphere whose eigenfunctions were spaces
of dimension 2l+ 1 with eigenvaluel(l+ 1), forlnon-negative and integral.
ForE(2) the quadratic functionp^21 +p^22 Poisson commutes withl,p 1 ,p 2. After
quantization,
|P|^2 =P 12 +P 22
is a second-order differential operator which commutes withL,P 1 ,P 2. This
operator has infinite dimensional eigenspaces that each carry an irreducible
representation ofE(2). They are characterized by a non-negative eigenvalue
that has physical interpretation as 2mEwherem,Eare the mass and energy
of a free quantum particle moving in two spatial dimensions.
From our discussion of the free particle in chapter 11 we see that, in mo-
mentum space, solutions of the Schr ̈odinger equation are given by
ψ ̃(p,t) =e−i^2 m^1 |p|^2 tψ ̃(p,0)and are parametrized by distributions
ψ ̃(p,0)≡ψ ̃(p)onR^2. These will have well-defined momentump 0 when
ψ ̃(p) =δ(p−p 0 )