Figure 69: Responseg(t)of a system to aδ(t)driving force splitted into an even and an odd component
Since the even component of the Green’s function is related to the odd component also the real part
of the susceptibility is related to the imaginary part of the susceptibility. For example if the real part
χ′is known, the inverse Fourier transform gives the even componentE(t)of the Green’s function.
By knowingE(t)the odd component can be determined withO(t) = sgn(t)E(t). The Fourier
transformation ofO(t)leads toχ′′. The relationship between the real part and the imaginary part of
the susceptibility is called theKramers-Kronig relation:
χ′′(ω) =1
π∫∞−∞χ′(ω′)
ω′−ωdω′ (102)χ′(ω) = −1
π∫∞−∞χ′′(ω′)
ω′−ω
dω′ (103)This relation is very useful because if you measure e.g. the response of a solid to a light wave you
can either measure the depth which the light wave is penetrating into the material or the polarisation
caused by the light wave and calculate the other one. Or you measure both and check with the
Kramers-Kronig relationship if it is right.
”If you calculated a linear response and want to tell everybody else about it, put it in the Kramers-
Kronig relation to see if it is right. Because if you publish a response and then somebody else says:
’Oh, but it violates causality’ - it looks stupid.”
10.2 Ballistic Transport
The formulas for ballistic transport are the same as for electrons in vacuum, because ballistic means
there are no obstacles and no scattering. Newton’s law with the forceF=−eEis
F = ma = −eE = mdv
dt