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.The latter is described by the wavefunction
ψ
(
r
)
=
e
i
k
z
+
f
(
θ
)
e
i
k
r
r
,
{\displaystyle \psi (\mathbf {r} )=e^{ikz}+f(\theta ){\frac {e^{ikr}}{r}}\;,}
where
r
≡
(
x
,
y
,
z
)
{\displaystyle \mathbf {r} \equiv (x,y,z)}
is the position vector;
r
≡
|
r
|
{\displaystyle r\equiv |\mathbf {r} |}
;
e
i
k
z
{\displaystyle e^{ikz}}
is the incoming plane wave with the wavenumber k along the z axis;
e
i
k
r
/
r
{\displaystyle e^{ikr}/r}
is the outgoing spherical wave; θ is the scattering angle; and
f
(
θ
)
{\displaystyle f(\theta )}
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,