130_notes.dvi

Higher order terms can be computed but its not recommended. Some atomic states, such as the2s state of Hydrogen, cannot decay by ...

Now suppose thewalls of the cavity are blackso that they emit and absorb photons at any energy. Then the result for the number o ...

34 Scattering of Photons In the scattering of photons, for example from an atom, aninitial state photonwith wave-number ~kand po ...

c(1)n;~k′ˆǫ(α′)(t) = 1 i ̄h e^2 2 mc^2 1 V ̄hc^2 √ ω′ω ǫ(μα)ǫ(α ′) μ δni ∫t 0 eiωnit ′ e−i(ω−ω ′)t dt′ = e^2 2 imV √ ω′ω ˆǫ(α)·ˆ ...

point. Diagrams (a) and (b) represent two second order terms. In diagram (a) the initial state photon is absorbed at timet 1 , l ...

× ∫t 0 dt 2 ei(ωni+ω ′−ω)t 2 We have calculated all the amplitudes. Thefirst order and second order amplitudes should be combine ...

Γ = ∫ dΩ ∣ ∣ ∣ ∣ ∣ ∣ δniǫˆ·ˆǫ′− 1 m ̄h ∑ j [ 〈n|ˆǫ′·~p|j〉〈j|ˆǫ·~p|i〉 ωji−ω + 〈n|ǫˆ·~p|j〉〈j|ˆǫ′·~p|i〉 ω′+ωji ] ∣ ∣ ∣ ∣ ∣ ∣ 2 × V ...

Also note that the formula yields an infinite result ifω=±ωji. This is not a physical result. In fact the cross section will be ...

Now we need to put the states in using an identity, then use the commutator withHto change~x to~p. 1 = ∑ j 〈i|j〉 〈j|i〉 i ̄hǫˆ·ˆǫ ...

34.3 Rayleigh Scattering Lord Rayleigh calculatedlow energy elastic scattering of lightfrom atoms using classical electromagneti ...

wave-vectors for the initial and final state. A look back at the calculation shows that we calculated the transition rate from a ...

34.5 Raman Effect The Kramers-Heisenberg formula clearly allows for theinitial and final state to be different. The atom changes ...

an electron is bound than they are if it is free. We will thereforecompute the amplitude from the second diagram. Hint = − e mc ...

i ̄h dcn dt = ∑ ~k,α ∑ j Hnje−iωtcjeiωnjt In the equations forcn, weexplicitly account for the fact that an intermediate state c ...

i ̄h dcn dt = ∆Ene−i∆Ent/ ̄h= ∑ ~k,α ∑ j HnjHjne−iωteiωnjt ei(−ωnj−∆ωn+ω)t− 1 ̄h(ωnj+ ∆ωn−ω) ∆En = ∑ ~k,α ∑ j |Hnj|^2 ei(ωnj+∆ωn ...

All photon energies contribute to the real part. Onlyphotons that satisfy the delta function constraint contribute to the imagin ...

= e^2 6 π^2 m^2 c^3 ∑ j ∫ ω|〈n|ei~k·~x~p|j〉|^2 (ωnj−ω) dω = 2 α ̄h 3 πm^2 c^2 ∑ j ∫ ω|〈n|ei ~k·~x ~p|j〉|^2 (ωnj−ω) dω Note that ...

= − 2 α 3 πm^2 c^2 |~p|^2 Ecut−off = −C|~p|^2 If we were hoping for little dependence on the cut-off we should be disappointed. ...

bound state and continuum state terms have the same fractionalchange, it is convenient to just use p^2 2 mfor all the correction ...

We now can compute the correction the same way he did. ∆E(nobs) = ∆En+C〈n|p^2 |n〉= ∆En+ 2 α 3 πm^2 c^2 Ecut−off〈n|p^2 |n〉 ∆En = ...