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Phonon Band Structure Quantum Atempt

  1. Sep 6, 2012 #1
    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:

    [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 ?
     
    Last edited: Sep 6, 2012
  2. jcsd
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