Scattering amplitude
In quantum physics, the scattering amplitude is the probability amplitude of the outgoing spherical wave relative to the incoming plane wave in a stationary-state scattering process.
Formulation
Scattering in quantum mechanics begins with a physical model based on the Schrodinger wave equation for probability amplitude :where is the reduced mass of two scattering particles and is the energy of relative motion.
For scattering problems, a stationary wavefunction is sought with behavior at large distances in two parts. First a plane wave represents the incoming source and, second, a spherical wave emanating from the scattering center placed at the coordinate origin represents the scattered wave:
The scattering amplitude,, represents the amplitude that the target will scatter into the direction.
In general the scattering amplitude requires knowing the full scattering wavefunction:
For weak interactions a perturbation series can be applied; the lowest order is called the Born approximation.
For a spherically symmetric scattering center, the plane wave is described by the wavefunction
where is the position vector; ; is the incoming plane wave with the wavenumber along the axis; is the outgoing spherical wave; is the scattering angle ; and is the scattering amplitude.
The dimension of the scattering amplitude is length. The scattering amplitude is a probability amplitude; the differential cross-section as a function of scattering angle is given as its modulus squared,
Unitary condition
When conservation of number of particles holds true during scattering, it leads to a unitary condition for the scattering amplitude. In the general case, we haveOptical theorem follows from here by setting
In the centrally symmetric field, the unitary condition becomes
where and are the angles between and and some direction. This condition puts a constraint on the allowed form for, i.e., the real and imaginary part of the scattering amplitude are not independent in this case. For example, if in is known, then can be determined such that is uniquely determined within the alternative.
Partial wave expansion
In the partial wave expansion the scattering amplitude is represented as a sum over the partial waves,where is the partial scattering amplitude and are the Legendre polynomials. The partial amplitude can be expressed via the partial wave S-matrix element and the scattering phase shift as
Then the total cross section
can be expanded as
is the partial cross section. The total cross section is also equal to due to optical theorem.
For, we can write