How Does Bloch's Theorem Apply in Solid State Physics?

[silverwhale]

Hi Everybody,

I am learning solid state physics and read today through Kittel. I am still stuck at the proof of Blochs theorem i.e. the proof of Blochs function.

For the Schrodinger equation of a periodic potential the general ansatz:

$$\psi = \sum_k C_k e^{ikr}$$

is made.

Then, by looking at the Schrodinger equation in the reciprocal lattice, we see that the $C_k$ are a linear combination of $C_{k-G}$, where $G$ is the reciprocal lattice vector.

Finally, the wave function is indexed by an index $k$ and rewritten as:

$$\psi_k = \sum_G C_{k-G} e^{i(k-G)r},$$

where the sum goes now over G.

I have two questions:
1) How can we go from the sum over k to a sum over G?
2) Is $\psi_k$ the same function as $\psi$? If so, why do we index it with k?

I hope this is clear enough, and I would be glad for ANY answer! :)
I am searching for an ansatz for nearly a week now!

 

[eljose79]

Thanks!


Hi there,

First of all, congratulations on delving into the world of solid state physics! It can definitely be a challenging subject, but it is also incredibly fascinating.

To answer your first question, the transition from a sum over k to a sum over G is possible because of the periodicity of the potential. This means that the wave function must also be periodic, and thus can be represented as a Fourier series. The k values represent the different frequencies in the Fourier series, while the G values represent the different harmonics. So when we sum over G, we are essentially summing over all the different harmonics that make up the periodic wave function.

As for your second question, yes, \psi_k and \psi are essentially the same function. However, the index k is used to differentiate between different solutions of the Schrodinger equation. This is because for a periodic potential, there are multiple solutions, each with a different value of k. So by indexing the wave function with k, we can specify which solution we are referring to.

I hope this helps clarify things for you. Keep up the great work in your studies! If you have any further questions, don't hesitate to ask. Science is all about curiosity and asking questions. Good luck!