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Final Exam/4/ Synchrotron radiation (Section 23.5; lectures 2301 and 2302)/3/ Rayleigh scattering and implications (Section 21.4; lecture 2101)/2/ Retardation (Section 20.3; lecture 2001)/1/ E. M. waves in a plasma and implications; Drude model (Sections 18.5, 18.6; lecture 1802)You can bring one page of handwritten notes.
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Synchrotron RadiationThe frequency spectrum. A single charge moving relativistically on a circular arc emits synchrotron radiation over a wide range of frequencies.Zangwill gives an approximate formula:
dI(ω)/dΩ = factor × (1+γ2θ2) ×[ (1+γ2θ2) K2/3
2(ξ) + γ2θ2 K1/32(ξ) ]
wherefactor = 3q2γ2ω2 / (16 π3 ε0 ω*2)
andξ = ( 1+γ2θ2 )3/2 (ω/2ω*) ; ω*=3γ3ω0 /2 .
(and modified Bessel functions)
Two comments. If γ >>1 then .../1/ … the intensity is significant in a narrow beam; θ ~ 1/ γ ; i.e., negligible for θ >> 1/ γ ; that’s because K(ξ) decreases rapidly for ξ>1. /2/ … the spectrum is peaked at ω = ω* ; that’s because K(ξ) decreases rapidly for ξ>1.
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Larmor’s formula, nonrelativistic
Larmor’s formula contributed to the downfall of classical physics.
A classical (planetary) model of the H atom. But classical physics implies …
The electron spirals down to the origin as the electron loses energy by radiation.
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This is one of the failures of classical physics that appeared about 100 years ago.
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The quantum theory of the atomA hydrogen atom is not like this:
Because the electron is a wave; like this:
The energy is
The electron cannot emit E.M. waves, because the wave cannot have a lower energy than -13.6 eV.
The quantum theory of the E. M. fields
E and B are quantized.
In Q.E.D, E and B are non-commuting quantum operators that create and annihilate photons.
Q.E.D. is an example of “second quantization” (Dirac)
A “classical E.M. wave” i.e., that can be accurately described by the classical theory, is a stream of photons in which the number of photons is indefinite.
Analogy: think of a classical mechanical oscillator
What is the quantum state of this motion?
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Larmor’s formula, relativisticThe power emitted by a charged particle is
Now integrate dP/dΩ to get the total power.
By hand, this is a long calculation; Zangwill calls it “tedious” ...
… so do it by Mathematica ...
Result
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Final Exam/4/ Synchrotron radiation (Section 23.5; lectures 2301 and 2302)/3/ Rayleigh scattering and implications (Section 21.4; lecture 2101)/2/ Retardation (Section 20.3; lecture 2001)/1/ E. M. waves in a plasma and implications; Drude model (Sections 18.5, 18.6; lecture 1802)You can bring one page of handwritten notes.