Tight binding - evaluate integral

1. Mar 13, 2012

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1. The problem statement, all variables and given/known data

The energy of an electron within a band as a function of its wavevector is given by the
tight-binding expression (in one dimension),

E(k)=-$\alpha$-$\gamma$$\Sigma$$_{m}$ exp (-ik$\rho$$_{m}$)

(a)What are typical expressions for integrals $\alpha$ and $\gamma$?
(b) Evaluate the integral $\gamma$ for the following wavefunction, assuming it is an eigenstate of the Hamiltonian, being careful to distinguish the cases 2x$_{0}$$\leq$$\rho$ and 2x$_{0}$>$\rho$:

$\phi$(x)=$\sqrt{\frac{1}{2x_{0}}}$ |x|$\leq$x$_{0}$

$\phi$(x)=0 |x|>x$_{0}$

(c) Hence evaluate the energy of an electron in a linear chain of these atoms with a
spacing a and make a graph of the result for the two cases 2x0 a and 2x0 > a.

3. The attempt at a solution

$\alpha$=-<$\phi$$_{n}$|H|$\phi$$_{n}$>
$\gamma$=-<$\phi$$_{m}$|H|$\phi$$_{n}$>

But I cannot do part b) because I do not know what $\phi$$_{m}$ and $\phi$$_{n}$ are. All that I know is that sometimes n=m and sometimes it does not. Please help.

2. Mar 13, 2012

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What is $\rho$? Why does it matter whether $\rho$ is bigger or smaller than x$_{0}$?