1.5 The S-Matrix, T-Matrix and Differential Cross Section
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energy conservation, we have
p
2
2μ
=
p
2
2μ
= E
(1.50)
Note that the T-matrix appearing in Eq. (1.49) also defines an off-shell T-matrix
for which
p
2
2μ
=
p
2
2μ
= E
(1.51)
The T-matrix
Since the cross section follows directly from the T-matrix, it becomes desirable to
have an integral equation for the T-matrix itself. Using Eq. (1.43) and inserting for
the resolvent from Eq. (1.18), we have
G(z) = G 0 (z) + G 0 (z)T (z)G 0 (z)
= G 0 (z) + G 0 (z)V G 0 (z) + G 0 (z)V G 0 (z)T (z)G 0 (z)
(1.52)
From this equation, it follows that
T (z) = V + V G 0 (z) T (z)
or
T (z) = V + T (z)G 0 (z)V
(1.53a, b)
These equations, written explicitly in representation space, are the integral equations for T-matrix. It may be noted that all these integral equations, like that of L–S
equation, have the same kernel.
1.6 The Unitarity of S-Matrix and Optical Theorem
There exists a conservation law for the probability flux if the given potential is
Hermitian. As a consequence, the S-matrix is required to be unitary, i.e.,
S
+ S =⇑
(1.54)
One can derive the consequence of this relation for the T-matrix starting from
Eq. (1.48).
However, one can also start using Eq. (1.53a) by multiplying by T
−1
(z) from the
right and by V
−1 from the left to get the equation: