The position space wavefunctions can be recovered from the Fourier inversion
formula
ψ(q,t) =
1
2 π
∫
R^2
eip·qψ ̃(p,t)d^2 p
Since, in the momentum space representation, the momentum operator is
the multiplication operator
Pψ ̃(p) =pψ ̃(p)
an eigenfunction for the Hamiltonian with eigenvalueEwill satisfy
(
|p|^2
2 m
−E
)
ψ ̃(p) = 0
ψ ̃(p) can only be non-zero ifE=|p|
2
2 m, so free particle solutions of energyEwill
thus be parametrized by distributions that are supported on the circle
|p|^2 = 2mE
p 1
p 2
p=
√
2 mE
θ
Figure 19.1: Parametrizing free particle solutions of Schr ̈odinger’s equation via
distributions supported on a circle in momentum space.
Going to polar coordinatesp= (pcosθ,psinθ), such solutions are given by
distributionsψ ̃(p) of the form
ψ ̃(p) =ψ ̃E(θ)δ(p^2 − 2 mE)