Consider a one-dimensional chain of sites available for electrons. Let N be the number of sites in the chain. The probabilities of finding the electron on any of the sites is given by 2, where yj, j = 1, ..., N is a set of complex wave function values fully describing the electron state. The dynamical system of equations describing the time dependence of the wave function V¡(t) reads i \hbar \frac{d \Psi_{j}}{d t}=\left\{\begin{array}{cc}

-\tau\left(\Psi_{j+1}+\Psi_{j-1}\right), & 1

-\tau \Psi_{j-1}, & j=N \\

-\tau \Psi_{j+1}, & j=1

\end{array}\right. Here is the parameter describing coupling of nearest neighbouring sites.

(b) Consider the cases N = 2, 3, 4 and, for each case, write down the system (1) withexcluded time in the form of the matrix equation \mathrm{H}_{N} \Psi=E \Psi

c) For the cases N = 2, 3, 4 divide Eq. (2) by the coupling parameter 7 and replace-E/T = 28 to obtain the dimensionless form of the equation: (f) In the general case, prove the following recurrent relation for polynomials PN(E): P_{N}(\varepsilon)=2 \varepsilon P_{N-1}(\varepsilon)-P_{N-2}(\varepsilon) by expressing the matrix determinant in Eq. (4) in terms of its minors. The recurrent relation (6) is common for both types of Chebyshev polynomials TNand UN. Identify which of the two types corresponds to the polynomial PN(=) bychecking the cases N =V = 2, 3.[3 marks] 1) Using the right the expression for one of these polynomials T_{N}(\cos (\theta))=\cos (n \theta), U_{N}(\cos (\theta)) \sin (\theta)=\sin ((N+1) \theta) solve the equation PÂ(ɛ) = 0 for values of . Recover the allowed electron energies E using the relation E = 2ET. ) Addressing the limit N » 1, identify the upper and the lower limits for allowedenergies in terms of the coupling parameter 7. Express the width of the energyband as the difference between the two limits.[2 marks] Prove that the zero energy state E = 0 is allowed for any odd value of N