lecture notes2

# At very high frequencies h binding energy of

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: time t=0, then the polarization at any other time is: ∞ Px ( t ) = ∫−∞ χ (ω )e− iωt dω . Since P is real, we may write this as: € ∞ Px ( t ) = ∫ [Re χ (ω ) cosωt + Im χ (ω ) sin ωt ]dω . −∞ This has to be zero for t<0. Therefore, € ∞ ∞ ∫ Re χ (ω ) cosωtdω = − ∫ Im χ (ω ' ) sinω ' tdω ' for t<0. −∞ −∞ For t>0, we may write the same equation but with −t: € ∞ ∞ ∫ Re χ (ω ) cosωtdω = ∫ Im χ (ω ' ) sinω ' tdω ' for t>0. In either case, € € −∞ ∫ −∞ ∞ −∞ ∞ Re χ (ω ) cos ωtdω = sgn t ∫ Im χ (ω ' ) sin ω ' tdω ' . −∞ 6 Now we note that since P and E are real, the susceptibility function satisfies χ(−ω)=χ*(ω), so Re χ is even and Im χ is odd. Therefore, the left ­hand side is the Fourier transform of Re χ. We may thus use the inverse transform, ∞ 1∞ Re χ (ω ) = ∫−∞ sgn t ∫−∞ Im χ (ω ' ) sinω ' tdω ' cosωtdt , 2π To go further, we introduce σ0 as an infintesimal parameter, € ∞ 1 ∞ −σ | t | Re χ (ω ) = ∫−∞ e sgn t ∫−∞ Im χ (ω ' ) sinω ' tdω ' cosωtdt , 2π so that we are justified in performing the integral over t: € ∞ ω ' +ω ω ' −ω ∫−∞ e−σ | t | sgn t sinω ' t cosωtdt = (ω ' +ω )2 + σ 2 + (ω ' −ω )2 + σ 2 . This gives (using the oddness of Im χ to combine the two terms) € 1∞ ω ' −ω Re χ (ω ) = ∫−∞ Im χ (ω ' ) dω ' . π (ω ' −ω ) 2 + σ 2 If we take the limit as σ0, the integral becomes a principal part: € ∞ Im χ (ω ' ) 1 Re χ (ω ) = PP ∫−∞ dω ' , π ω ' −ω where PP indicates that one is to exclude a region |ω’ ­ω|<ς from the integration region, and take the limit as ς0. An analogous relation (swapping Re with Im and € introducing a − sign) allows one to go from Re χ to Im χ. It is possible to express the same equation in terms of only the positive frequencies using the oddness of Im χ: ∞ ω ' Im χ (ω ' ) 2 Re χ (ω ) = PP ∫ 0 dω ' π ω '2 −ω 2 . ∞ Re χ (ω ' ) 2ω Im χ (ω ) = − PP ∫ dω ' 0 ω ' 2 −ω 2 π These are called the Kramers Kronig relations. With the help of the Kramers ­Kronig relations, we need only specify Im χ(ω). € An interesting law can be obtained for the behavior of Im...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online