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Homework Statement
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|\leqx_{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.
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.