Quantum Mechanics for Mathematicians

12.7 Green’s functions and solutions to the Schr ̈o-

dinger equations

The method of Green’s functions provides solutionsψto differential equations

Dψ=J (12.14)

whereDis a differential operator andJis a fixed function, by finding an inverse
D−^1 toDand then settingψ=D−^1 J. ForDa constant coefficient differential
operator, the Fourier transform will takeDto multiplication by a polynomial
D̂and we define the Green’s function ofDto be the function (or distribution)
with Fourier transform

Ĝ=^1

D̂

(12.15)

Since
D̂ĜĴ=Ĵ

the inverse Fourier transform ofĜĴwill be a solution to 12.14.
Note thatĜandGare not uniquely determined by the condition 12.15
sinceDmay have a kernel, and then solutions to 12.14 are only determined up
to a solutionψ 0 of the homogeneous equationDψ 0 = 0. In terms of Fourier
transforms,D̂may have zeros, and thenĜis ambiguous up to functions on the
zero set.
For the case of the Schr ̈odinger equation, we take

D=i

∂

∂t

+

1

2 m

∂^2

∂q^2

and then (Fourier transforming inqandtas above)

D̂=i(−iω) +^1

2 m

(ik)^2 =ω−

k^2

2 m

and

Ĝ=^1
ω−k
2
2 m
A solutionψof 12.14 will be given by computing the inverse Fourier transform
ofĜĴ

ψ(q,t) =

1

2 π

∫+∞

−∞

∫+∞

−∞

1

ω−k

2

2 m

Ĵ(ω,k)e−iωteikqdωdk (12.16)

HereD̂is zero on the setω= k

2
2 mand the non-uniqueness of the solution to
Dψ=Jis reflected in the ambiguity of how to treat the integration through
the pointsω=k

2
2 m.
For solutionsψ(q,t) of the Schr ̈odinger equation with initial dataψ(q,0) at
timet= 0, if we defineψ+(q,t) =θ(t)ψ(q,t) we get the “retarded” solution

ψ+(q,t) =

∫+∞

−∞

U+(t,q−q 0 )ψ(q 0 ,0)dq 0