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8.14: Appendix V- Kramers-Krönig Relations

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Suppose ˆχ(ω)ˆG(ω) is analytic in the UHP19. Then for all ν, we must have dν2πˆχ(ν)νω+iϵ=0 , where ϵ is a positive infinitesimal. The reason is simple: just close the contour in the UHP, assuming ˆχ(ω) vanishes sufficiently rapidly that Jordan’s lemma can be applied. Clearly this is an extremely weak restriction on ˆχ(ω), given the fact that the denominator already causes the integrand to vanish as |ω|1.

Let us examine the function 1νω+iϵ=νω(νω)2+ϵ2iϵ(νω)2+ϵ2 . which we have separated into real and imaginary parts. Under an integral sign, the first term, in the limit ϵ0, is equivalent to taking a principal part of the integral. That is, for any function F(ν) which is regular at ν=ω, limϵ0dν2πνω(νω)2+ϵ2F(ν)dν2πF(ν)νω . The principal part symbol means that the singularity at ν=ω is elided, either by smoothing out the function 1/(νϵ) as above, or by simply cutting out a region of integration of width ϵ on either side of ν=ω.

The imaginary part is more interesting. Let us write h(u)ϵu2+ϵ2 . For |u|ϵ, h(u)ϵ/u2, which vanishes as ϵ0. For u=0, h(0)=1/ϵ which diverges as ϵ0. Thus, h(u) has a huge peak at u=0 and rapidly decays to 0 as one moves off the peak in either direction a distance greater that ϵ. Finally, note that duh(u)=π , a result which itself is easy to show using contour integration. Putting it all together, this tells us that limϵ0ϵu2+ϵ2=πδ(u) . Thus, for positive infinitesimal ϵ, 1u±iϵ=uiπδ(u) , a most useful result.

We now return to our initial result [kka], and we separate ˆχ(ω) into real and imaginary parts: ˆχ(ω)=ˆχ(ω)+iˆχ(ω) . (In this equation, the primes do not indicate differentiation with respect to argument.) We therefore have, for every real value of ω, 0=dν2π[χ(ν)+iχ(ν)][νωiπδ(νω)] . Taking the real and imaginary parts of this equation, we derive the Kramers-Krönig relations: χ(ω)=+dνπˆχ(ν)νωχ(ω)=dνπˆχ(ν)νω .


This page titled 8.14: Appendix V- Kramers-Krönig Relations is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Daniel Arovas.

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