Scattering amplitudes Definition and 21 Threads

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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