Question

4. A particle moves in a periodic one-dimensional potential, V(x a)-V(x); physically, this may represent the motion of non-interacting electrons in a crys- tal lattice. Let us call n), n - 0, +1, t2, particle located at site n, with (n'In) -Sn,Let H be the system Hamiltonian and U(a) the discrete translation operator: U(a)|n) - [n +1). In the tight- binding approximation, one neglects the overlap of electron states separated by a distance larger than a, so that where is the energy of a particle located in any site, and Д is the energy associated with the hopping between atomic orbitals centred on lattice sites. (a) Show that the linear combination is an eigenvector of both U(a), with eigenvalue e-10, and of H, with eigen- value E (0) = E0-2A cos θ. (b) Show that the wave function associated with 0) satisfies Bloch's Theorem

Answer 1

3) Lui na-N ine ei ene

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