# Phonon Band Structure Quantum Atempt

1. Sep 6, 2012

### kanonear

I found a lecture on the internet where a LCAO assumptions are used to calculate the phonon band structure. In this lecture (Here) we can find that the true Hamiltonian:

$H = \frac{p^2}{2m}+V_0(x)$ where $V_0(x)=V_0(x+a)$

is replaced with the following aproximate hamiltonian:

$H\cong \sum_n H_{at}(n)+\sum_{n,n'}U(n,n')$.

where, $H_{at}(n)$ - Atomic Hamiltonian for atom at site $n$,

$U(n,n')$ - Interaction Energy between atoms $n$ & $n'$,
in addition the interaction energy is non zero only for nearest neighbours that is

$U(n,n') = 0$ unless $n' = n -1$ or $n' = n +1$ and also

$<ψ_{n''}|U(n,n ± 1)|ψ_{n'}> = - α$ for $n'' = n$ & $n' = n ± 1$ otherwise it's zero.

We also assume that solutions to the atomic Schrödinger Equation are known:
$H_{at}(n)ψ_n(x) = E_nψ_n(x)$ where $E_n=ε$
and last but not least:
$ψ_k(x) = \sum_n e^{ikna} ψ_n(x)$.
That is, the Bloch Functions are linear combinations of atomic orbitals.

Ok and now the equation in Driac natation is:
$E_k = <ψ_k|\sum_n H_{at}(n) |ψ_k> + <ψ_k|[\sum_n U(n,n-1) + U(n,n+1)]|ψ_k>$

And the result we are expecting is:
$E_k = ε-2αcos(ka)$

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My attempt at solution

So what i do is rewriting the main equation in a form:
$E_k = <ψ_{k'}|\sum_n H_{at}(n) |ψ_k> + <ψ_{k'}|[\sum_n U(n,n±1)]|ψ_k>$

and now using consequently all the relations writen above:
$E_k = <ψ_{n'}|\sum_n e^{-ikn'a}\sum_n H_{at}(n) \sum_n e^{ikna}|ψ_n> + <ψ_{n'}|\sum_n e^{-ikn'a} [ \sum_n U(n,n±1)] \sum_n e^{ikna}|ψ_n> =$

$<ψ_{n'}|\sum_n e^{-ikn'a} H_{at}(n) e^{ikna}|ψ_n> + <ψ_{n'}|\sum_n e^{-ikn'a} U(n,n±1) e^{ikna}|ψ_n> =$

$<ψ_{n'}|\sum_n e^{ik(n-n')a} H_{at}(n) |ψ_n> + <ψ_{n'}|\sum_n e^{ik(n-n')a} U(n,n±1) |ψ_n> =$

$\sum_n e^{ik(n-n')a} ε <ψ_{n'}|ψ_n> + <ψ_{n+1}|\sum_n e^{-ika} U(n,n+1)|ψ_n> + <ψ_{n-1}|\sum_n e^{ika} U(n,n-1)|ψ_n> =$

$\sum_n e^{ik(n-n')a} ε δ_{n,n'} + \sum_n e^{-ika}(-α) + \sum_n e^{ika} (-α) =$

$\sum_n [ ε - α(e^{-ika} + e^{ika})]$

Finaly

$E_k = \sum_n [ ε - 2αcos(ka)]=n[ ε - 2αcos(ka)]$

My question is: What is wrong with my solution ?

Last edited: Sep 6, 2012