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:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]H = \frac{p^2}{2m}+V_0(x)[/itex] where [itex]V_0(x)=V_0(x+a)[/itex]

is replaced with the following aproximate hamiltonian:

[itex]H\cong \sum_n H_{at}(n)+\sum_{n,n'}U(n,n')[/itex].

where, [itex]H_{at}(n)[/itex] - Atomic Hamiltonian for atom at site [itex]n[/itex],

[itex]U(n,n')[/itex] - Interaction Energy between atoms [itex]n[/itex] & [itex]n'[/itex],

in addition the interaction energy is non zero only for nearest neighbours that is

[itex]U(n,n') = 0[/itex] unless [itex]n' = n -1[/itex] or [itex]n' = n +1 [/itex] and also

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

We also assume that solutions to the atomic Schrödinger Equation are known:

[itex]H_{at}(n)ψ_n(x) = E_nψ_n(x) [/itex] where [itex]E_n=ε[/itex]

and last but not least:

[itex]ψ_k(x) = \sum_n e^{ikna} ψ_n(x)[/itex].

That is, the Bloch Functions are linear combinations of atomic orbitals.

Ok and now the equation in Driac natation is:

[itex]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> [/itex]

And the result we are expecting is:

[itex]E_k = ε-2αcos(ka)[/itex]

===============================================================

My attempt at solution

So what i do is rewriting the main equation in a form:

[itex]E_k = <ψ_{k'}|\sum_n H_{at}(n) |ψ_k> + <ψ_{k'}|[\sum_n U(n,n±1)]|ψ_k> [/itex]

and now using consequently all the relations writen above:

[itex]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> =[/itex]

[itex]<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> =[/itex]

[itex]<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> =[/itex]

[itex]\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> =[/itex]

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

[itex]\sum_n [ ε - α(e^{-ika} + e^{ika})] [/itex]

Finaly

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

My question is:What is wrong with my solution ?

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# Phonon Band Structure Quantum Atempt

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