12
1 Essentials of Non-relativistic Two-Body Scattering Theory
T
−1
(z) = V
−1
− G 0 (z)
(1.55)
As G 0 (z) has a cut along the positive real energy axis so has T(z) and T
−1
(z).
Writing Eq. (1.55) for z = E + iε,
T
−1
(E + iε) = V
−1
− G 0 (E + iε)
(1.56a)
and for z = E − iε,
T
−1
(E − iε) = V
−1
− G 0 (E − iε)
(1.56b)
Subtracting these two equations, we have
T
−1
(E + iε) − T
−1
(E − iε) = −G 0 (E + iε) + G 0 (E − iε)
And multiplying this equation by T (E + iε) T (E − iε) on both sides, we get
T (E − iε) − T (E + iε) = T (E + iε)[−G 0 (E + iε) + G 0 (E − iε)] T (E − iε)
(1.57)
Using the first resolvent equation, the difference in parentheses reduces to:
−G 0 (E + iε) + G 0 (E − iε) = 2iεG 0 (E + iε)G 0 (E − iε)
= 2iε[(E − H 0 )
2
+ ε
2
]
(1.58)
In the limit ε → 0, we get the unitarity relation:
T (E − i0) − T (E + i0) = 2πi T (E + i0)δ(E − H 0 )T (E − i0),
or
T (E + i0) − T
+
(E + i0) = −2πi T (E + i0)δ(E − H 0 )T
+
(E + i0) (1.59)
Expressed in momentum representation, this expression for scattering in the
forward direction reduces to:
Im
p|T (E + i0)|
p = −π
d
p
δ(E − p
2
/(2μ))
p|T (E + i0)
p
2
= −π μ p
d ˆ
p
p|T (E + i0)
p
2 , |
p| =
p
=
2μE
= −
p
16π 3 μ
σ tot
(1.60)
This is the well-known optical theorem which relates the imaginary part of the
forward amplitude to the total cross section.