Phonon Band Structure Quantum Atempt

[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:
$E_k = ε-2αcos(ka)$

===============================================================
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 written 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 ?

 

[eljose79]

There are a few potential issues with your solution:

1. Your use of the δ_{n,n'} term may not be correct. In this context, δ_{n,n'} represents the Kronecker delta function, which takes the value 1 when n=n' and 0 otherwise. However, in your solution, you are using it as a summation term, which may not be appropriate.

2. You seem to have mixed up the terms in the summations. In the original equation, the first summation is over the atomic Hamiltonian for each atom (H_{at}(n)), while the second summation is over the interaction energy between nearest neighbor atoms (U(n,n±1)). In your solution, you have combined both terms into a single summation, which may not be correct.

3. Your final result does not match the expected result of E_k = ε-2αcos(ka). This could be due to the issues mentioned above, or there may be other errors in your calculations.

It is important to carefully check your algebra and ensure that you are using the correct mathematical notation and concepts when solving equations. It may also be helpful to double check your solution with a colleague or a textbook to identify any potential errors.