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A formula may be derived mathematically for the rate of scattering when a beam of electrons passes through a material.
The interaction picture [ edit ] Define the unperturbed Hamiltonian by
H
0
{\displaystyle H_{0}}
H
1
{\displaystyle H_{1}}
H
{\displaystyle H}
The eigenstates of the unperturbed Hamiltonian are assumed to be
H
=
H
0
+
H
1
{\displaystyle H=H_{0}+H_{1}\ }
H
0
|
k
⟩
=
E
(
k
)
|
k
⟩
{\displaystyle H_{0}|k\rangle =E(k)|k\rangle }
In the interaction picture , the state ket is defined by
|
k
(
t
)
⟩
I
=
e
i
H
0
t
/
ℏ
|
k
(
t
)
⟩
S
=
∑
k
′
c
k
′
(
t
)
|
k
′
⟩
{\displaystyle |k(t)\rangle _{I}=e^{iH_{0}t/\hbar }|k(t)\rangle _{S}=\sum _{k'}c_{k'}(t)|k'\rangle }
By a Schrödinger equation , we see
i
ℏ
∂
∂
t
|
k
(
t
)
⟩
I
=
H
1
I
|
k
(
t
)
⟩
I
{\displaystyle i\hbar {\frac {\partial }{\partial t}}|k(t)\rangle _{I}=H_{1I}|k(t)\rangle _{I}}
which is a Schrödinger-like equation with the total
H
{\displaystyle H}
H
1
I
{\displaystyle H_{1I}}
Solving the differential equation , we can find the coefficient of n-state.
c
k
′
(
t
)
=
δ
k
,
k
′
−
i
ℏ
∫
0
t
d
t
′
⟨
k
′
|
H
1
(
t
′
)
|
k
⟩
e
−
i
(
E
k
−
E
k
′
)
t
′
/
ℏ
{\displaystyle c_{k'}(t)=\delta _{k,k'}-{\frac {i}{\hbar }}\int _{0}^{t}dt'\;\langle k'|H_{1}(t')|k\rangle \,e^{-i(E_{k}-E_{k'})t'/\hbar }}
where, the zeroth-order term and first-order term are
c
k
′
(
0
)
=
δ
k
,
k
′
{\displaystyle c_{k'}^{(0)}=\delta _{k,k'}}
c
k
′
(
1
)
=
−
i
ℏ
∫
0
t
d
t
′
⟨
k
′
|
H
1
(
t
′
)
|
k
⟩
e
−
i
(
E
k
−
E
k
′
)
t
′
/
ℏ
{\displaystyle c_{k'}^{(1)}=-{\frac {i}{\hbar }}\int _{0}^{t}dt'\;\langle k'|H_{1}(t')|k\rangle \,e^{-i(E_{k}-E_{k'})t'/\hbar }}
The transition rate [ edit ] The probability of finding
|
k
′
⟩
{\displaystyle |k'\rangle }
|
c
k
′
(
t
)
|
2
{\displaystyle |c_{k'}(t)|^{2}}
In case of constant perturbation,
c
k
′
(
1
)
{\displaystyle c_{k'}^{(1)}}
c
k
′
(
1
)
=
⟨
k
′
|
H
1
|
k
⟩
E
k
′
−
E
k
(
1
−
e
i
(
E
k
′
−
E
k
)
t
/
ℏ
)
{\displaystyle c_{k'}^{(1)}={\frac {\langle \ k'|H_{1}|k\rangle }{E_{k'}-E_{k}}}(1-e^{i(E_{k'}-E_{k})t/\hbar })}
|
c
k
′
(
t
)
|
2
=
|
⟨
k
′
|
H
1
|
k
⟩
|
2
s
i
n
2
(
E
k
′
−
E
k
2
ℏ
t
)
(
E
k
′
−
E
k
2
ℏ
)
2
1
ℏ
2
{\displaystyle |c_{k'}(t)|^{2}=|\langle \ k'|H_{1}|k\rangle |^{2}{\frac {sin^{2}({\frac {E_{k'}-E_{k}}{2\hbar }}t)}{({\frac {E_{k'}-E_{k}}{2\hbar }})^{2}}}{\frac {1}{\hbar ^{2}}}}
Using the equation which is
lim
α
→
∞
1
π
s
i
n
2
(
α
x
)
α
x
2
=
δ
(
x
)
{\displaystyle \lim _{\alpha \rightarrow \infty }{\frac {1}{\pi }}{\frac {sin^{2}(\alpha x)}{\alpha x^{2}}}=\delta (x)}
The transition rate of an electron from the initial state
k
{\displaystyle k}
k
′
{\displaystyle k'}
P
(
k
,
k
′
)
=
2
π
ℏ
|
⟨
k
′
|
H
1
|
k
⟩
|
2
δ
(
E
k
′
−
E
k
)
{\displaystyle P(k,k')={\frac {2\pi }{\hbar }}|\langle \ k'|H_{1}|k\rangle |^{2}\delta (E_{k'}-E_{k})}
where
E
k
{\displaystyle E_{k}}
E
k
′
{\displaystyle E_{k'}}
δ
{\displaystyle \delta }
The scattering rate [ edit ] The scattering rate w(k) is determined by summing all the possible finite states k' of electron scattering from an initial state k to a final state k', and is defined by
w
(
k
)
=
∑
k
′
P
(
k
,
k
′
)
=
2
π
ℏ
∑
k
′
|
⟨
k
′
|
H
1
|
k
⟩
|
2
δ
(
E
k
′
−
E
k
)
{\displaystyle w(k)=\sum _{k'}P(k,k')={\frac {2\pi }{\hbar }}\sum _{k'}|\langle \ k'|H_{1}|k\rangle |^{2}\delta (E_{k'}-E_{k})}
The integral form is
w
(
k
)
=
2
π
ℏ
L
3
(
2
π
)
3
∫
d
3
k
′
|
⟨
k
′
|
H
1
|
k
⟩
|
2
δ
(
E
k
′
−
E
k
)
{\displaystyle w(k)={\frac {2\pi }{\hbar }}{\frac {L^{3}}{(2\pi )^{3}}}\int d^{3}k'|\langle \ k'|H_{1}|k\rangle |^{2}\delta (E_{k'}-E_{k})}
C. Hamaguchi (2001). Basic Semiconductor Physics . Springer. pp. 196–253. J.J. Sakurai. Modern Quantum Mechanics 
