Derivation of the Bloch theorem

[patric44]

Homework Statement

some questions about the derivation of Bloch theorem

Relevant Equations

in the attachments

hi guys
our solid state professor gave us a series of power point slides that contains the derivation of Bloch theorem , but some points is not clear to me , and when i asked him his answer was also not clear :

in the first part i understand the he represented both the potential energy and the electron plane wave as a Fourier series
but when he multiplied both together in the last equation he introduced k' why is that ! sinse k=k+G as it wil repreat in the next parabolic dispersion and subsequently he changed the index of Ck ⇒ C_k'-G isn't that also k ?
in the next page he set again k=k+g and took the exponential as a common factor but yet leaves the "C" coefficent as k-G
why he keep alternating between k'-G , k ...
and the jump from
$$ Ψ(r) ⇒Ψk(r) $$
is not very clear to me ?

- and how i suppose to solve this Schrodinger equation i mean its no longer a differential eq ?

 

[nrqed]

Homework Statement:: some questions about the derivation of Bloch theorem
Relevant Equations:: in the attachments

hi guys
our solid state professor gave us a series of power point slides that contains the derivation of Bloch theorem , but some points is not clear to me , and when i asked him his answer was also not clear :
View attachment 260161
in the first part i understand the he represented both the potential energy and the electron plane wave as a Fourier series
but when he multiplied both together in the last equation he introduced k' why is that ! sinse k=k+G as it wil repreat in the next parabolic dispersion and subsequently he changed the index of Ck ⇒ C_k'-G isn't that also k ?

He just introduced $\vec{k'} \equiv \vec{G} + \vec{k}$ so that the argument of the exponential would be $i \vec{k'} \cdot \vec{r}$ (he wanted that argument to be as simple as possible). So basically he replaces $\vec{k}$ by $\vec{k'} - \vec{G}$.

 

[patric44]

He just introduced $\vec{k'} \equiv \vec{G} + \vec{k}$ so that the argument of the exponential would be $i \vec{k'} \cdot \vec{r}$ (he wanted that argument to be as simple as possible). So basically he replaces $\vec{k}$ by $\vec{k'} - \vec{G}$.

i know that , why did he changed that into k in the next page as you can see when he took the e^ikr as a common factor .
and if he is considering k' = k+G = k then why didn't he drop it from the coefficient C ?
that is my question

 

[nrqed]

i know that , why did he changed that into k in the next page as you can see when he took the e^ikr as a common factor .
and if he is considering k' = k+G = k then why didn't he drop it from the coefficient C ?
that is my question

Ok, your questions were not very clear.

In the next page, he just renamed $\vec{k}' \rightarrow \vec{k}$. Since the sum is over the vectors, one can rename them, they are dummy indices.