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Matalien l\"amp\"otilojen fysiikan teoria (Kyl-0.104)\hfil Autumn 2000\break
Electronic transport in mesoscopic systems\hfil Exercise set 8\break
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1. Derive the Kubo formula
$$G={2e^2\over h}{\hbar^2\over L^2}\sum_{m,n,k,k'}
v_mv_n\vert G^{\rm R}(m,k;n,k')\vert^2.$$
Instead of following the derivation by Datta (pages 159-160) use only continuum
expressions. (Note that $\Gamma_p(y,y')=\sum_m\chi_m(y)\hbar v_m\chi_m(y')$ does
not contain $a$ in our normalization and ${\rm Tr}A(y,y')=\int dyA(y,y)$, see first
exercise in set 7.) Also use the fact that $\bar T_{pq}$ should be independent
on the choice of $x_p$ and $x_q$ in the leads to justify the neglect of 
$G^{\rm R}(m,k_1;n,k_2)G^{\rm R*}(m,k_3;n,k_4)$ for $k_1\not= k_3$ etc.
Continue to solve exercise E.3.8.
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Exercises E.3.9 and E.4.1 - E.4.2 from Datta's book. 
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