Quantum Mechanics for Mathematicians

whereU+(t,q−q 0 ) is the retarded propagator given by equations 12.12 and
12.13. Since

Dψ+(q,t) = (Dθ(t))ψ(q,t) +θ(t)Dψ(q,t) =iδ(t)ψ(q,t) =iδ(t)ψ(q,0)

ψ+(q,t) is a solution of 12.14 with

J(q,t) =iδ(t)ψ(q,0), Ĵ(ω,k) =

i

√

2 π

ψ ̃(k,0)

Using 12.16 to get an expression forψ+(q,t) in terms of the Green’s function
we have

ψ+(q,t) =

1

2 π

∫+∞

−∞

∫+∞

−∞

Ĝ(ω,k)√i

2 π

ψ ̃(k,0)e−iωteikqdωdk

=

(

1

2 π

) 2 ∫+∞

−∞

(∫+∞

−∞

∫+∞

−∞

iĜ(ω,k)e−iωteik(q−q

′)

dωdk

)

ψ(q′,0)dq′

Comparing this to equations 12.12 and 12.13, we find that the Green’s func-
tion that will give the retarded solutionψ+(q,t) is

Ĝ+(ω,k) = lim

→ 0 +

1

ω−k

2

2 m+i

and is related to the retarded propagator by

Û(ω,k) = i

2 π

Ĝ+(ω,k)

One can also define an “advanced” Green’s function by

Ĝ−= lim

→ 0 +

1

ω− 2 km^2 −i

and the inverse Fourier transform ofĜ−Ĵwill also be a solution to 12.14. Tak-
ing the difference between retarded and advanced Green’s functions gives an
operator

∆ =̂ i

2 π

(Ĝ+−Ĝ−)

with the property that, for any choice ofJ, ∆J will be a solution to the
Schr ̈odinger equation (since it is the difference between two solutions of the
inhomogeneous equation 12.14). The properties of ∆ can be understood by
using 12.11 to show that

∆ =̂ δ(ω−k

2

2 m

)