12.3 The free particle propagator
In almost all cases,^11 H 0 will be the Hamiltonian for a free probe of massm,
H 0 =
P^2
2 m(12.10)
wherePis the momentum operator. The eigenstates are|φk〉=|k〉with energyE= ̄h^2 k^2 / 2 m,
with wave function,
φk(x) =〈x|k〉=
eik·x
(2π)^3 /^2(12.11)
The normalization has been chosen so that〈k′|k〉=δ(3)(k′−k), which may be verified by inserting
a complete set of position eigenstates
∫
d^3 x〈k′|x〉〈x|k〉=δ(3)(k′−k) (12.12)In the position basis the Lippmann-Schwinger equation becomes,
〈x|ψ±k〉=〈x|φk〉+λ∫
d^3 y〈x|1
E−H 0 ±iε|y〉〈y|H 1 |ψk±〉 (12.13)One defined thepropagatoror Green functions for the free non-relativistic particle as follows,
G±(x,y;k^2 ) =
̄h^2
2 m〈x|1
E−H 0 ±iε|y〉 (12.14)The factor of ̄h^2 / 2 mhas been pulled out for later convenience and to make the propagator depend
only onk^2 , but not onm. SinceH 0 is diagonal in the momentum basis|ℓ〉, the propagator may be
computed by inserting a complete set of momentum eigenstates,
G±(x,y;k^2 ) =∫
d^3 l〈x|1
k^2 −l^2 ±iε
|l〉〈l|y〉=∫ d (^3) l
(2π)^3
eil·(x−y)
k^2 −l^2 ±iε
(12.15)
One may either evaluate this Fourier integral directly, or use the fact thatG±satsifies,
(∆ +k^2 )G±(x,y;k^2 ) =δ(3)(x−y) (12.16)Fork= 0,G±is just the Coulomb equation for aδ-function charge with solution
G±(x,y; 0) =−1
4 πr
r=|x−y| (12.17)(^11) One notable exception is the Coulomb problem for which the exact solution is known, abeit complicated.