??? question "calculate $E_0$ and the velocity $v$"
The edge of the Brilloin zone has $k = \pi/a$. Substituting this in the free electron dispersion $E = \hbar^2 k^2/2m$ we get $E_0 = \hbar^2 \pi^2/2 m a^2$, and $v=\hbar k/m=\hbar \pi/ma$.
Note that this Hamiltonian is diagonal so the eigenenergies are on the diagonal and the eigenfunctions are the $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$ plane waves.
Note that this Hamiltonian is diagonal so the eigenenergies are on the diagonal and the eigenfunctions are the $|k\rangle$ and $|k'\rangle$ plane waves.
As we will see below, the lattice potential $V(x)$ may couple the states $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$. The coupling strength is given by the matrix element $W=\langle \mathbf{k} |V| |\mathbf{k'}\rangle$. Including the coupling into the Hamiltonian, we get
As we will see below, the lattice potential $V(x)$ may couple the states $|\mathbf{k}\rangle$ and $|\mathbf{k'}\rangle$. The coupling strength is given by the matrix element $W=\langle \mathbf{k} |V| |\mathbf{k'}\rangle$. Including the coupling into our Hamiltonian, we get
$$
H\begin{pmatrix}\alpha \\\beta \end{pmatrix} =
...
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@@ -114,7 +114,7 @@ Check out section 15.1.1 of the book for the details of this calculation.
#### Physical meaning of $W$
Now we expand the definition of $W$:
Now we expand the definition of $W$, concentrating on 1D:
