Jump to content

User:Ierakaris/sandbox/Draft mathtest

From Wikipedia, the free encyclopedia

Draft mathtest

new article content ...

Mathematical formulation

[edit]

The Schrödinger equation can be expressed like

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

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

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

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

A plane wave can be expanded as

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

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

Mathematical formulation

[edit]

The Schrödinger equation can be expressed like

where is the potential of the solid and is the wave function of the electron that has to be calculated.

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

A plane wave can be expanded as

where are spherical Bessel functions and are Legendre polynomials.


References

[edit]
[edit]