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Mathematical formulation

The Schrödinger equation can be expressed like

\[\left[ {{\nabla ^2} + E} \right]\psi \left( {\bf{r}} \right) = V\left( {\bf{r}} \right)\psi \left( {\bf{r}} \right)\]

$[\nabla ^{2}+E]\psi (\mathbf {r} )=V(\mathbf {r} )\psi (\mathbf {r} )$

where $V({\bf{r}})$ $V(\mathbf {r} )$ is the potential of the solid and $\psi ({\bf{r}})$$\psi (\mathbf {r} )$ is the wave function of the electron that has to be calculated.

The unperturbed Green's function is defined as the solution of

$\left[ {{\nabla ^2} + E} \right]G\left( {{\bf{r}},{\bf{r'}}} \right) = \delta \left( {{\bf{r}} - {\bf{r'}}} \right)$

$[\nabla ^{2}+E]G(\mathbf {r} ,\mathbf {r} ')=\delta (\mathbf {r} -\mathbf {r} ')$

A plane wave can be expanded as

\[{e^{i{\bf{k}}{\bf{r}}}} = \sum\limits_{} {\left( {2l + 1} \right)} {i^l}{j_l}\left( {kr} \right){P_l}\left( {\cos \theta } \right)\]

$e^{i\mathbf {k} \cdot \mathbf {r} }=\sum _{l}(2l+1)i^{l}j_{l}(kr)P_{l}(\cos \theta )$

where ${j_l}\left( {kr} \right)$ $j_{l}(kr)$ are spherical Bessel functions and ${P_l}\left( {\cos \theta } \right)$ $P_{l}(\cos \theta )$ are Legendre polynomials.

Mathematical formulation

The Schrödinger equation can be expressed like

$\left[{{\nabla ^{2}}+E}\right]\psi \left({\bf {r}}\right)=V\left({\bf {r}}\right)\psi \left({\bf {r}}\right)$

$[{{\nabla ^{2}}+E}]\psi ({\bf {r}})=V({\bf {r}})\psi ({\bf {r}})$

$[\nabla ^{2}+E]\psi (\mathbf {r} )=V(\mathbf {r} )\psi (\mathbf {r} )$

where $V({\bf {r}})$ $V(\mathbf {r} )$ is the potential of the solid and $\psi ({\bf {r}})$ $\psi (\mathbf {r} )$ is the wave function of the electron that has to be calculated.

The unperturbed Green's function is defined as the solution of

$\left[{{\nabla ^{2}}+E}\right]G\left({{\bf {r}},{\bf {r'}}}\right)=\delta \left({{\bf {r}}-{\bf {r'}}}\right)$

$[\nabla ^{2}+E]G(\mathbf {r} ,\mathbf {r} ')=\delta (\mathbf {r} -\mathbf {r} ')$

A plane wave can be expanded as

${e^{i{\bf {k}}\cdot {\bf {r}}}}=\sum \limits _{}{\left({2l+1}\right)}{i^{l}}{j_{l}}\left({kr}\right){P_{l}}\left({\cos \theta }\right)$

$e^{i\mathbf {k} \cdot \mathbf {r} }=\sum _{l}(2l+1)i^{l}j_{l}(kr)P_{l}(\cos \theta )$

where ${j_{l}}\left({kr}\right)$ $j_{l}(kr)$ are spherical Bessel functions and ${P_{l}}\left({\cos \theta }\right)$ $P_{l}(\cos \theta )$ are Legendre polynomials.

External links

• www.example.com