From Wikipedia, the free encyclopedia
This is not a Wikipedia article : It is an individual user's work-in-progress page, and may be incomplete and/or unreliable. For guidance on developing this draft, see Wikipedia:So you made a userspace draft . Finished writing a draft article? Are you ready to request an experienced editor review it for possible inclusion in Wikipedia? Submit your draft for review!
Draft mathtest
new article content ...
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 )}
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 )}
![{\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)
