- #1
Petar Mali
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Homework Statement
How from
[tex](\hbar\omega-h)G_{f,f'}(\omega)=i2\langle\hat{S}_f^z\rangle\delta_{f,f'}+\langle\hat{S}^z\rangle\sum_gI(f-g)\{G_{f,f'}(\omega)-G_{g,f'}(\omega)\}[/tex]
get
[tex]G_{q}(\omega)=\frac{i\hbar}{2\pi}\frac{2\langle\hat{S}^z\rangle}{\hbar\omega-h-\epsilon(q)}[/tex]
where
[tex]G_{f,f'}(\omega)=\frac{1}{N}\sum_{\vec{q}} e^{i\vec{q}(\vec{f}-\vec{f'})}G_{\vec{q}}(\omega)[/tex]
[tex]\delta_{f,f'}=\frac{1}{N}\sum_{q}e^{i\vec{q}(\vec{f}-\vec{f'})}[/tex]
Homework Equations
The Attempt at a Solution
I really don't have a clue what to do. [tex]h[/tex] is constant.
If I use that spin don't depands of indices
[tex]\langle\hat{S}^z\rangle=\langle\hat{S}_g^z\rangle=S\sigma[/tex]
and use
[tex](\hbar\omega-h)\frac{1}{N}\sum_{\vec{q}} e^{i\vec{q}(\vec{f}-\vec{f'})}G_{\vec{q}}(\omega)=i2S\sigma\frac{1}{N}\sum_{q}e^{i\vec{q}(\vec{f}-\vec{f'})}+S\sigma\sum_gI(f-g)\{\frac{1}{N}\sum_{\vec{q}} e^{i\vec{q}(\vec{f}-\vec{f'})}G_{\vec{q}}(\omega)-\frac{1}{N}\sum_{\vec{q}} e^{i\vec{q}(\vec{g}-\vec{f'})}G_{\vec{q}}(\omega)\}[/tex]What now?
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