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1D atomic chain, Localized states

  1. Nov 15, 2017 #1
    1. The problem statement, all variables and given/known data

    1D atomic chain with one atom in the primitive cell and the lattice constant a. The system in described within the tight binding model and contains N-->∞ primitive cells indexed by the integer n. The electronic Hamiltonian is $$H_{0} = \sum_{n} (|n \rangle E_{at} \langle n | -|n+1 \rangle \beta \langle n| - |n \rangle \beta \langle n+1 | )$$

    with Eat being the energy on one electron in the state ##|n \rangle ##at site n and## \beta >0 ## represents the energy overlap integral responsable for the interaction between first neighbors. We assume that the atomic orbitals | n \rangle are orthonormalized and neglect the overlap of atomic orbitals on different sites, thus ## \langle n|n' \rangle = \delta_{nn'} ##

    First off, show that the electronic states described by :

    $$ | k \rangle = \frac{1}{\sqrt{N!}}\sum_{n} e^{ikna} |n \rangle $$

    are eigenstates of H0 and calculate the corresponding eigenvalues E0(k) in the first Brillouin zone
    2. Relevant equations

    3. The attempt at a solution

    We start by plugging H0 into the equation $$ H_{0}|k \rangle = E_{0} | k \rangle $$ and thus obtaining

    $$(\sum_{n} (|n \rangle E_{at} \langle n | -|n+1 \rangle \beta \langle n| - |n \rangle \beta \langle n+1 | ))| k \rangle$$

    and now replacing the form of ##|k \rangle"## one gets

    $$(\sum_{n} (|n \rangle E_{at} \langle n | -|n+1 \rangle \beta \langle n| - |n \rangle \beta \langle n+1 | ))(\frac{1}{\sqrt{N!}}\sum_{n} e^{ikna} |n \rangle)$$

    moving on with

    $$ H_{0}|k \rangle = \frac{1}{\sqrt(N!)}\sum_{n}|n \rangle E_{at} \langle n |e^{ikna}|n \rangle - |n+1 \rangle \beta \langle n|e^{ikna}|n \rangle - |n \rangle \beta \langle n+1 | e^{ikna}|n \rangle $$

    and here is where I get stuck. I don t know how to evaluate ##\langle n |e^{ikna}|n \rangle## and ## |n \rangle \beta \langle n+1 | e^{ikna}|n \rangle ##

    From my intuition I think that after solving the LHS of the Schrodinger equation like I started to do above, at one point I should get that the expression above is of form ## Y|k \rangle ## with Y being a number and thus showing that the ket k is a eigenstate of H0.

    Thank you
  2. jcsd
  3. Nov 20, 2017 #2
    Thanks for the thread! This is an automated courtesy bump. Sorry you aren't generating responses at the moment. Do you have any further information, come to any new conclusions or is it possible to reword the post? The more details the better.
  4. Nov 21, 2017 #3
    You should start by using a different summation index in the Hamiltonian and the state. You used n as the index of sites in the Hamiltonian. Use some other index, say m, when you write |k>. Then use the orthogonality of states at different sites. Also notice that a matrix element like <n| eikna |n> = eikna.
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