(adsbygoogle = window.adsbygoogle || []).push({}); 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 E_{at}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 H_{0}and calculate the corresponding eigenvalues E_{0}(k) in the first Brillouin zone

2. Relevant equations

above

3. The attempt at a solution

We start by plugging H_{0}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 H_{0}.

Thank you

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# Homework Help: 1D atomic chain, Localized states

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