# Problem in Proof of Bloch theorem

In kittel's book on solid state physics a proof of bloch theorem is given . It says:

We consider N identical lattice points on a ring of length Na. the potential energy is periodic in a width $$U(x)=U(x+sa$$), where s is an integer. Let us be guided by the symmetry of the ring to look for solutions of the wave equation such that $$\psi(x+a)=C\psi(x)$$ where C is a constant . Then on going once around the ring $$\psi(x+Na)=\psi(x)=C^N\psi(x)$$ because $$\psi(x)$$ must be single valued. It follows that C is one of the N roots of unity or

$$C=exp(i2\pi s/N)$$ ; s=$$0,1,2,...N-1 .$$ $$(1)$$

We use $$(1)$$ to see that

$$\psi(x)=u_k(x)exp(i2\pi sx/Na)$$ $$(2)$$

$$(2)$$ is the bloch result

I couldnt follow the part where $$(1)$$ is used to arrive at the bloch result

To clarify, the Bloch result that you wish to show is that $$\psi(x) = e^{i k x}u_k(x)$$ where $$k = 2\pi s / Na$$ and $$u_k(x)$$ is a function which has the same periodicity as the potential $$U(x)$$, that is, $$u_k(x) = u_k(x+a)$$.
We can directly verify that (2) fulfills these conditions. From (2), we have $$u_k(x) = e^{-ikx}\psi(x)$$. To verify that this has the same periodicity as $$U(x)$$, show that $$u_k(x+a) = u_k(x)$$:
$$u_k(x+a) = e^{-ik(x+a)}\psi(x+a) = e^{-ik(x+a)}e^{i k a}\psi(x) = e^{-ikx}\psi(x) = u_k(x)$$
• I see that k can be written as $2\pi s/Na$ but how is C required in the proof. Commented Apr 11, 2021 at 17:54
• Sorry I didn't make that clearer. $C = e^{ika}$ is required in the proof: $u_k(x+a) = e^{-ik(x+a)}\psi(x+a) = e^{-ik(x+a)}C\psi(x) = e^{-ikx}e^{-ika}C\psi(x) = e^{-ikx}e^{-ika}e^{ika}\psi(x) = e^{-ikx}\psi(x) = u_k(x)$. We needed the relationship $\psi(x+a) = e^{ika}\psi(x)$ in the proof, given by how $C$ is defined in (1). Commented Apr 11, 2021 at 18:16
• Hi So are you already not assuming that $\psi(x)=u_k (x) e^{-ikx}$. We want to derive that right. I understand that using it we are able to satisfy the properties. Commented Apr 12, 2021 at 5:32
• Any $\psi(x)$ can be written as $\psi(x)=u_k (x) e^{-ikx}$ for some function $u_k (x)$. What we have shown here is that for any $\psi(x)$ satisfying $\psi(x+a) = C\psi(x)$, the corresponding $u_k(x)$ has the same periodicity as the potential. Commented Apr 12, 2021 at 19:34