- #1
Petar Mali
- 290
- 0
Homework Statement
How this would look in K space
[tex]-\sum_{\vec{n},\vec{m}}I_{\vec{n},\vec{m}}\hat{a}^+_{\vec{n}}\hat{a}_{\vec{n}}\langle \hat{a}^+_{\vec{n}}\hat{b}^+_{\vec{m}}\rangle[/tex]
I need to get
[tex]-\sum_{\vec{k}}\hat{a}^+_{\vec{n}}\hat{a}_{\vec{n}}\frac{1}{N}\sum_{\vec{q}}J(\vec{q})\langle \hat{a}^+_{\vec{q}}\hat{b}^+_{-\vec{q}}\rangle[/tex]
Homework Equations
[tex]\hat{a}, \hat{a}^+,\hat{b},\hat{b}^+[/tex] are Bose operators
[tex]I_{\vec{n},\vec{m}}=\frac{1}{N}\sum_{\vec{k}}J(\vec{k})e^{i\vec{k}(\vec{n}-\vec{m})[/tex]
[tex]\hat{a}^+_{\vec{n}}=\frac{1}{\sqrt{N}}\sum_{\vec{k}}\hat{a}^+_{\vec{k}}e^{-i\vec{k}\cdot\vec{n}}[/tex]
[tex]\hat{a}_{\vec{n}}=\frac{1}{\sqrt{N}}\sum_{\vec{k}}\hat{a}_{\vec{k}}e^{i\vec{k}\cdot\vec{n}}[/tex]
[tex]\hat{b}^+_{\vec{n}}=\frac{1}{\sqrt{N}}\sum_{\vec{k}}\hat{b}^+_{\vec{k}}e^{-i\vec{k}\cdot\vec{n}}[/tex]
[tex]\hat{b}_{\vec{n}}=\frac{1}{\sqrt{N}}\sum_{\vec{k}}\hat{b}_{\vec{k}}e^{i\vec{k}\cdot\vec{n}}[/tex]
The Attempt at a Solution
Homework Statement
[tex]-\sum_{\vec{n},\vec{m}}I_{\vec{n},\vec{m}}\hat{a}^+_{\vec{n}}\hat{a}_{\vec{n}}\langle \hat{a}^+_{\vec{n}}\hat{b}^+_{\vec{m}}\rangle=-\sum_{\vec{n},\vec{m}}\frac{1}{N^2}\sum_{\vec{k}}J(\vec{k})e^{i\vec{k}(\vec{n}-\vec{m})}\sum_{\vec{k}_1}\hat{a}^+_{\vec{k}_1}e^{-i\vec{k}_1\cdot\vec{n}}\sum_{\vec{k}_2}\hat{a}_{\vec{k}_2}e^{i\vec{k}_2\cdot\vec{n}}\frac{1}{N}\langle\sum_{\vec{k}_3}\hat{a}^+_{\vec{k}_3}e^{-i\vec{k}_3\cdot\vec{n}}\sum_{\vec{k}_4}\hat{b}^+_{\vec{k}_4}e^{-i\vec{k}_4\cdot\vec{n}}\rangle=[/tex]
[tex]=-\frac{1}{N^3}\sum_{\vec{k}}J(\vec{k})\sum_{\vec{k}_1,\vec{k}_2,\vec{k}_3,\vec{k}_4}\hat{a}^+_{\vec{k}_1}\hat{a}_{\vec{k}_2}\langle \hat{a}^+_{\vec{k}_3}\hat{b}^+_{\vec{k}_4}\rangle\sum_{\vec{n}}e^{i\vec{k}\cdot\vec{n}}e^{-i\vec{k}_1\cdot\vec{n}}e^{i\vec{k}_2\cdot\vec{n}}e^{-i\vec{k}_3\cdot\vec{n}}\sum_{\vec{m}}e^{-i\vec{k}\cdot\vec{m}}e^{-i\vec{k}_4\cdot\vec{m}[/tex]
So I get
[tex]=-\frac{1}{N^3}\sum_{\vec{k}}J(\vec{k})\sum_{\vec{k}_1,\vec{k}_2,\vec{k}_3,\vec{k}_4}\hat{a}^+_{\vec{k}_1}\hat{a}_{\vec{k}_2}\langle \hat{a}^+_{\vec{k}_3}\hat{b}^+_{\vec{k}_4}\rangle N\delta_{\vec{k}+\vec{k}_2,\vec{k}_1+\vec{k}_3}N\delta_{\vec{k}_3,-\vec{k}_4}[/tex]
And what now? I didn't get what I need to get!
[tex]=-\frac{1}{N}\sum_{\vec{k}}J(\vec{k})\sum_{\vec{k}_1,\vec{k}_2}\hat{a}^+_{\vec{k}_1}\hat{a}_{\vec{k}_2}\langle \hat{a}_{\vec{k}-\vec{k}_1+\vec{k}_2}\hat{a}_{-\vec{k}}\rangle[/tex]
Last edited: