The position space wavefunctions can be recovered from the Fourier inversion
formula
ψ(q,t) =1
2 π∫
R^2eip·qψ ̃(p,t)d^2 pSince, 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) = 0ψ ̃(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 = 2mEp 1p 2p=√
2 mEθFigure 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 bydistributionsψ ̃(p) of the form
ψ ̃(p) =ψ ̃E(θ)δ(p^2 − 2 mE)