1.4 Resolvent Equation and Lippmann–Schwinger Equation
5
g(z) = [z − H ]
−1
, where z = E ± iε
(1.14)
which operating on the plane wave state |
p gives the scattering states | p
(±) , viz.
| p
(±)
= lim
ε→0
±iεg(E ± iε)|
p
(1.15)
This is how we arrive at a time-independent theory starting from time-dependent
approach.
Now using the relation (1.14), we write
z − z
= g
−1
(z) − g
−1
(z
)
and multiplying both sides by g(z) g(z
) we get
g
z
− g(z) =
z − z
g(z) g
z
(1.16)
With the definition of the resolvent for a free particle, i.e.,
g 0 (z) = (z − H 0 )
−1
(1.17)
we write
V = H − H 0 = g
−1
0 (z) − g
−1
(z)
On multiplication with g(z) g 0 (z), we get the resolvent equation:
g(z) = g 0 (z) + g 0 (z)V g(z),
= g 0 (z) + g(z)V g 0 (z)
(1.18a, b)
Let us now derive, using this operator identity, the integral equation for the
scattering states. Thus, from Eq. (1.15), we have
| p
(±)
= lim
ε→0
[±iε g(E ± iε)|
p]
= lim
ε→0
±iε[g 0 (E ± iε) + g 0 (E + iε)V g(E + iε)]|
p
(1.19)
For the first term on the right-hand side of the equation, we use the relation
±iε g 0 (E ± iε) |
p = |
p
(1.20)
and, using Eq. (1.15), for the second term, we write
|
p
(±) = |
p + g 0 (E ± i0) V |
p
±
(1.21)