- #1

Petar Mali

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If I have cubic structure where plane is define by vector [tex]\rho[/tex] and in [tex]z[/tex] direction I have planes [tex]...m-2,m-1,m,m+1,m+2...[/tex]

and if I have for example

[tex]\sum_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}\hat{B}_{m+1,\vec{\rho}}[/tex]

how to go with that in K-space?

If I had

[tex]\sum_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}[/tex]

I will say

[tex](m,\vec{\rho})=\vec{n}[/tex]

and then I will have

[tex]\sum_{\vec{n}}\hat{B}_{\vec{n}}\hat{B}_{\vec{n}}=\sum_{\vec{n}}\frac{1}{\sqrt{N}}\sum_{\vec{k}}\hat{B}_{\vec{k}}e^{i\vec{k}\cdot\vec{n}}\frac{1}{\sqrt{N}}\sum_{\vec{q}}\hat{B}_{\vec{q}}e^{i\vec{q}\cdot\vec{n}}[/tex]

[tex]=\frac{1}{N}\sum_{\vec{k},\vec{q}}\hat{B}_{\vec{k}}\hat{B}_{\vec{q}}N\delta_{\vec{k},-\vec{q}}=\sum_{\vec{k}}\hat{B}_{\vec{k}}\hat{B}_{-\vec{k}}[/tex]

But what to do in case with m+1. Thanks for your answer!

and if I have for example

[tex]\sum_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}\hat{B}_{m+1,\vec{\rho}}[/tex]

how to go with that in K-space?

If I had

[tex]\sum_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}\hat{B}_{m,\vec{\rho}}[/tex]

I will say

[tex](m,\vec{\rho})=\vec{n}[/tex]

and then I will have

[tex]\sum_{\vec{n}}\hat{B}_{\vec{n}}\hat{B}_{\vec{n}}=\sum_{\vec{n}}\frac{1}{\sqrt{N}}\sum_{\vec{k}}\hat{B}_{\vec{k}}e^{i\vec{k}\cdot\vec{n}}\frac{1}{\sqrt{N}}\sum_{\vec{q}}\hat{B}_{\vec{q}}e^{i\vec{q}\cdot\vec{n}}[/tex]

[tex]=\frac{1}{N}\sum_{\vec{k},\vec{q}}\hat{B}_{\vec{k}}\hat{B}_{\vec{q}}N\delta_{\vec{k},-\vec{q}}=\sum_{\vec{k}}\hat{B}_{\vec{k}}\hat{B}_{-\vec{k}}[/tex]

But what to do in case with m+1. Thanks for your answer!

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