Stationary scattering wave
edit
The (elastic) scattering stationary state (azimuthally symmetric) is described by a wave-function with the following asymptotic,
ψ
(
r
→
)
⟶
r
→
∞
e
i
k
→
r
→
+
f
(
θ
)
e
i
k
r
r
,
{\displaystyle \psi ({\vec {r}}){\underset {r\to \infty }{\longrightarrow }}e^{i{\vec {k}}{\vec {r}}}+f(\theta ){\frac {e^{ikr}}{r}}\;,}
(1
where
e
i
k
→
r
→
{\displaystyle e^{i{\vec {k}}{\vec {r}}}}
k
→
{\displaystyle {\vec {k}}}
f
(
θ
)
e
i
k
r
r
{\displaystyle f(\theta ){\frac {e^{ikr}}{r}}}
f
(
θ
)
{\displaystyle f(\theta )}
Cross-section
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Consider a detector with the window
d
Ω
{\displaystyle d\Omega }
θ
{\displaystyle \theta }
r
{\displaystyle r}
d
N
/
d
t
{\displaystyle dN/dt}
j
r
{\displaystyle j_{r}}
d
N
d
t
=
j
r
r
2
d
Ω
.
{\displaystyle {\frac {dN}{dt}}=j_{r}r^{2}d\Omega \,.}
The radial flux from the stationary wave (1
j
r
=
ℏ
2
m
i
(
ψ
∗
∂
ψ
∂
r
−
∂
ψ
∗
∂
r
ψ
)
=
|
f
(
θ
)
|
2
1
r
2
ℏ
k
m
.
{\displaystyle j_{r}={\frac {\hbar }{2mi}}\left(\psi ^{*}{\frac {\partial \psi }{\partial r}}-{\frac {\partial \psi ^{*}}{\partial r}}\psi \right)=|f(\theta )|^{2}{\frac {1}{r^{2}}}{\frac {\hbar k}{m}}\,.}
(2
The cross-section
d
σ
{\displaystyle d\sigma }
d
σ
=
1
|
j
i
|
d
N
d
t
.
{\displaystyle d\sigma ={\frac {1}{|j_{i}|}}{\frac {dN}{dt}}\,.}
(3
