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.

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.

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.

This is one of the failures of classical physics that appeared about 100 years ago.

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?

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

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.