$$\require{cancel}$$

# 9.5: Optical Theorem

The differential scattering cross-section is simply the modulus squared of the scattering amplitude . The total cross-section is given by

 (980)

where . It follows that

 (981)

where use has been made of Equation (967). A comparison of this result with Equation (979) yields

 (982)

since . This result is known as the optical theorem. It is a reflection of the fact that the very existence of scattering requires scattering in the forward ( ) direction in order to interfere with the incident wave, and thereby reduce the probability current in this direction.

It is usual to write

 (983)

where

 (984)

is the th partial cross-section: i.e., the contribution to the total cross-section from the th partial wave. Note that the maximum value for the th partial cross-section occurs when the phase-shift takes the value .