depending on two variablesθ,p. To put this delta-function in a more useful form,
recall the discussion leading to equation 11.9 and note that forp≈
√
2 mEone
has the linear approximation
p^2 − 2 mE≈ 2√
2 mE(p−√
2 mE)so one has the equality of distributions
δ(p^2 − 2 mE) =1
2
√
2 mEδ(p−√
2 mE)In the one dimensional case (see equation 11.10) we found that the space of
solutions of energyEwas parametrized by two complex numbers, corresponding
to the two possible momenta±
√
2 mE. In this two dimensional case, the space of
such solutions will be infinite dimensional, parametrized by distributionsψ ̃E(θ)
on the circle.
It is this space of distributionsψ ̃E(θ) on the circle of radius
√
2 mEthat will
provide an infinite dimensional representation of the groupE(2), one that turns
out to be irreducible, although we will not show that here. The position space
wavefunction corresponding toψ ̃E(θ) will be
ψ(q) =1
2 π∫∫
eip·qψ ̃E(θ)δ(p^2 − 2 mE)pdpdθ=
1
2 π∫∫
eip·qψ ̃E(θ)1
2
√
2 mEδ(p−√
2 mE)pdpdθ=
1
4 π∫ 2 π0ei√
2 mE(q 1 cosθ+q 2 sinθ)ψ ̃E(θ)dθFunctionsψ ̃E(θ) with simple behavior inθwill correspond to wavefunctions
with more complicated behavior in position space. For instance, takingψ ̃E(θ) =
e−inθone finds that the wavefunction along theq 2 direction is given by
ψ(0,q) =1
4 π∫ 2 π0ei√ 2 mE(qsinθ)
e−inθdθ=
1
2
Jn(√
2 mEq)whereJnis then’th Bessel function.
Equations 19.1 and 19.2 give the representation of the Lie algebra ofE(2)
on wavefunctionsψ(q). The representation of this Lie algebra on theψ ̃E(θ) is
given by the Fourier transform, and we’ll denote this byΓ ̃′S. Using the formula
for the Fourier transform we find that
̃Γ′S(p 1 ) =−̃∂
∂q 1=−ip 1 =−i√
2 mEcosθΓ ̃′S(p 2 ) =−̃∂
∂q 2=−ip 2 =−i√
2 mEsinθ