# What is Scattering amplitudes: Definition and 21 Discussions

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,

d
σ

d
Ω

=

|

f
(
θ
)

|

2

.

{\displaystyle {\frac {d\sigma }{d\Omega }}=|f(\theta )|^{2}\;.}

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