- #1
natugnaro
- 64
- 1
Homework Statement
Starting from the wave function of free electron and applying first order approximation of time independent perturbation theory find wave function of electron in periodic potential.
Homework Equations
Potential energy in reciprocal space
[tex]E_{p}(\vec{r}) = \sum_{\vec{g} }E_{g}\cdot e^{i\vec{g}\vec{r}}[/tex]
free electron wave function
[tex]\phi_{k}(\vec{r})=e^{i\vec{k}\vec{r}[/tex]
free electron wave function in first order approximation
[tex]\Psi_{k}(\vec{r})=\phi_{k}(\vec{r})+\sum_{\vec{k}^{'}\neq\vec{k}}\frac{\int\phi^{*}_{k'}(\vec{r})\cdot E_{p}(\vec{r})\cdot\phi_{k}(\vec{r})\cdot d^{3}r}{E(k)-E(k')}\cdot\phi_{k'}(\vec{r})[/tex]
The Attempt at a Solution
[tex]\int\phi^{*}_{k'}(\vec{r})\cdot E_{p}(\vec{r})\cdot\phi_{k}(\vec{r})\cdot d^{3}r=\sum_{\vec{g} }E_{g}\int e^{i(-\vec{k'}+\vec{g}+\vec{k})\cdot\vec{r}}d^{3}r=\sum_{\vec{g}}E_{g}\cdot\delta_{\vec{k'},\vec{k}+\vec{g}}[/tex]
I don't understan the last part , transition from complex exponential to Kronecker delta.
I have two options:
1.) When vectors satisfy k'=k+g then I have integral of [tex]\int d^{3}r[/tex] which must be 1, I am not clear about this.
2.) when vectors k' and k+g are not equal I have integral of [tex]\int [Cos(a) + i*Sin(a)] d^{3}r[/tex]
(a is some real number from the dot product in exponent) ,
this integral should be zero, but I don't see how ?