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Tight binding - evaluate integral

  1. Mar 13, 2012 #1
    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)=-[itex]\alpha[/itex]-[itex]\gamma[/itex][itex]\Sigma[/itex][itex]_{m}[/itex] exp (-ik[itex]\rho[/itex][itex]_{m}[/itex])

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

    [itex]\phi[/itex](x)=[itex]\sqrt{\frac{1}{2x_{0}}}[/itex] |x|[itex]\leq[/itex]x[itex]_{0}[/itex]

    [itex]\phi[/itex](x)=0 |x|>x[itex]_{0}[/itex]

    (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

    [itex]\alpha[/itex]=-<[itex]\phi[/itex][itex]_{n}[/itex]|H|[itex]\phi[/itex][itex]_{n}[/itex]>
    [itex]\gamma[/itex]=-<[itex]\phi[/itex][itex]_{m}[/itex]|H|[itex]\phi[/itex][itex]_{n}[/itex]>

    But I cannot do part b) because I do not know what [itex]\phi[/itex][itex]_{m}[/itex] and [itex]\phi[/itex][itex]_{n}[/itex] are. All that I know is that sometimes n=m and sometimes it does not. Please help.
     
  2. jcsd
  3. Mar 13, 2012 #2
    What is [itex]\rho[/itex]? Why does it matter whether [itex]\rho[/itex] is bigger or smaller than x[itex]_{0}[/itex]?
     
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