scattering amplitude について

is the scattering amplitude. The dimension of the scattering amplitude is length.
The scattering amplitude is a probability amplitude and the differential cross-section as a function of scattering angle is given as its modulus squared
:

\frac = |f(\theta)|^2 \;.

In the low-energy regime the scattering amplitude is determined by the scattering length.
== Partial wave expansion ==

(詳細はMichael Fowler/ 1/17/08 Plane Waves and Partial Waves )〕
:

f=\sum_^\infty (2\ell+1) f_\ell P_\ell(\cos \theta)

,
where is the partial scattering amplitude and are the Legendre polynomials.
The partial amplitude can be expressed via the partial wave S-matrix element (

=e^

) and the scattering phase shift as
:

f_\ell = \frac = \frac = \frac = \frac \;.

Then the differential cross section is given by
:

\frac = |f(\theta)|^2 = \frac \left| \sum_^\infty (2\ell+1) e^ \sin \delta_\ell P_\ell(\cos \theta) \right|^2

,
and the total elastic cross section becomes
:

\sigma = 2 \pi \int_0^\pi \frac \sin \theta \, d \theta = \frac \operatorname f(0)

,
where is the imaginary part of .

抄文引用元・出典: フリー百科事典『ウィキペディア（Wikipedia）』
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