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Mathematical formulation [ edit ] The Schrödinger equation can be expressed like
\[\left[ {{\nabla ^2} + E} \right]\psi \left( {\bf{r}} \right) = V\left( {\bf{r}} \right)\psi \left( {\bf{r}} \right)\]
[
∇
2
+
E
]
ψ
(
r
)
=
V
(
r
)
ψ
(
r
)
{\displaystyle [\nabla ^{2}+E]\psi (\mathbf {r} )=V(\mathbf {r} )\psi (\mathbf {r} )}
where $V({\bf{r}})$
V
(
r
)
{\displaystyle V(\mathbf {r} )}
ψ
(
r
)
{\displaystyle \psi (\mathbf {r} )}
The unperturbed Green's function is defined as the solution of
$\left[ {{\nabla ^2} + E} \right]G\left( {{\bf{r}},{\bf{r'}}} \right) = \delta \left( {{\bf{r}} - {\bf{r'}}} \right)$
[
∇
2
+
E
]
G
(
r
,
r
′
)
=
δ
(
r
−
r
′
)
{\displaystyle [\nabla ^{2}+E]G(\mathbf {r} ,\mathbf {r} ')=\delta (\mathbf {r} -\mathbf {r} ')}
A plane wave can be expanded as
\[{e^{i{\bf{k}}{\bf{r}}}} = \sum\limits_{} {\left( {2l + 1} \right)} {i^l}{j_l}\left( {kr} \right){P_l}\left( {\cos \theta } \right)\]
e
i
k
⋅
r
=
∑
l
(
2
l
+
1
)
i
l
j
l
(
k
r
)
P
l
(
cos
θ
)
{\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=\sum _{l}(2l+1)i^{l}j_{l}(kr)P_{l}(\cos \theta )}
where ${j_l}\left( {kr} \right)$
j
l
(
k
r
)
{\displaystyle j_{l}(kr)}
P
l
(
cos
θ
)
{\displaystyle P_{l}(\cos \theta )}
Mathematical formulation [ edit ] The Schrödinger equation can be expressed like
[
∇
2
+
E
]
ψ
(
r
)
=
V
(
r
)
ψ
(
r
)
{\displaystyle \left[{{\nabla ^{2}}+E}\right]\psi \left({\bf {r}}\right)=V\left({\bf {r}}\right)\psi \left({\bf {r}}\right)}
[
∇
2
+
E
]
ψ
(
r
)
=
V
(
r
)
ψ
(
r
)
{\displaystyle [{{\nabla ^{2}}+E}]\psi ({\bf {r}})=V({\bf {r}})\psi ({\bf {r}})}
[
∇
2
+
E
]
ψ
(
r
)
=
V
(
r
)
ψ
(
r
)
{\displaystyle [\nabla ^{2}+E]\psi (\mathbf {r} )=V(\mathbf {r} )\psi (\mathbf {r} )}
where
V
(
r
)
{\displaystyle V({\bf {r}})}
V
(
r
)
{\displaystyle V(\mathbf {r} )}
ψ
(
r
)
{\displaystyle \psi ({\bf {r}})}
ψ
(
r
)
{\displaystyle \psi (\mathbf {r} )}
The unperturbed Green's function is defined as the solution of
[
∇
2
+
E
]
G
(
r
,
r
′
)
=
δ
(
r
−
r
′
)
{\displaystyle \left[{{\nabla ^{2}}+E}\right]G\left({{\bf {r}},{\bf {r'}}}\right)=\delta \left({{\bf {r}}-{\bf {r'}}}\right)}
[
∇
2
+
E
]
G
(
r
,
r
′
)
=
δ
(
r
−
r
′
)
{\displaystyle [\nabla ^{2}+E]G(\mathbf {r} ,\mathbf {r} ')=\delta (\mathbf {r} -\mathbf {r} ')}
A plane wave can be expanded as
e
i
k
⋅
r
=
∑
(
2
l
+
1
)
i
l
j
l
(
k
r
)
P
l
(
cos
θ
)
{\displaystyle {e^{i{\bf {k}}\cdot {\bf {r}}}}=\sum \limits _{}{\left({2l+1}\right)}{i^{l}}{j_{l}}\left({kr}\right){P_{l}}\left({\cos \theta }\right)}
e
i
k
⋅
r
=
∑
l
(
2
l
+
1
)
i
l
j
l
(
k
r
)
P
l
(
cos
θ
)
{\displaystyle e^{i\mathbf {k} \cdot \mathbf {r} }=\sum _{l}(2l+1)i^{l}j_{l}(kr)P_{l}(\cos \theta )}
where
j
l
(
k
r
)
{\displaystyle {j_{l}}\left({kr}\right)}
j
l
(
k
r
)
{\displaystyle j_{l}(kr)}
P
l
(
cos
θ
)
{\displaystyle {P_{l}}\left({\cos \theta }\right)}
P
l
(
cos
θ
)
{\displaystyle P_{l}(\cos \theta )}
References [ edit ]
External links [ edit ]
![{\displaystyle [\nabla ^{2}+E]\psi (\mathbf {r} )=V(\mathbf {r} )\psi (\mathbf {r} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3b9f9d846d1d004d539f9baf7c97364cb165d9f8)
![{\displaystyle [\nabla ^{2}+E]G(\mathbf {r} ,\mathbf {r} ')=\delta (\mathbf {r} -\mathbf {r} ')}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e41998bfb78b585e8651f14a80af5c03f3046a5)
![{\displaystyle \left[{{\nabla ^{2}}+E}\right]\psi \left({\bf {r}}\right)=V\left({\bf {r}}\right)\psi \left({\bf {r}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2591632ae387af9f2cd86077bd4c5a695d14c93)
![{\displaystyle [{{\nabla ^{2}}+E}]\psi ({\bf {r}})=V({\bf {r}})\psi ({\bf {r}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a89527c85f36c58a10b0defbfac802337eb3a481)
![{\displaystyle \left[{{\nabla ^{2}}+E}\right]G\left({{\bf {r}},{\bf {r'}}}\right)=\delta \left({{\bf {r}}-{\bf {r'}}}\right)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/de0655e5c0fed7f8e94b37d787c6b4e632878cc2)
