Single-Particle Electrodynamics - Assassination Science

twice, then it clearly will not do to form a single expectation value using the
spinor |ψ〉; rather, we should expect two expectation values to appear. Since
the two cross-producted factors in (6.160) each involve µ once, it may be
reasonable to guess that the expectations should be placed around each of
these factors separately:
d t 〈ψ|σ|ψ〉 = ? 〈ψ|σ|ψ〉×〈ψ|Ω RR |ψ〉 . (6.163)
Using the expression (6.161) for Ω RR , and (6.150) for ˙σ, in (6.163), we then
have
d t 〈ψ|σ|ψ〉 = ? − 2 µ 2 ) µB 3
〈ψ|σ|ψ〉×
3 4πs( ( 〈ψ|σ|ψ〉×
s
ˆB ) . (6.164)
The guessed equation of motion (6.164) is, of course, equivalent to the classical
result (6.151), with the standard connection
σ classical ←→ 〈ψ|σ|ψ〉 ,
and will hence lead to the hyperbolic tangent dependence of the polarisation
equation (6.154).
So far, we have not changed the net results at all: by going to the
first-quantised theory, we still have the same classical radiation reaction behaviour.
We now take due note of the second quantisation of the photon
field. The most important feature that this introduces is that the interaction
leading to (6.164) must satisfy conservation of energy, momentum and
angular momentum. Since the photon has spin ¯h, the interaction must, to
satisfy conservation of angular momentum, flip the spin of the particle. As
it stands, however, the expression (6.164) assumes the radiation reaction to
be brought about by the continuous emission of radiation: it calculates the
matrix element from any given state to the same state (or one infinitesimally
close to it). Thus, to include the effects of the quantisation of the photon, we
need to insert spin-flipped states into (6.164); the resulting transition probability
is then given by the correct formula (6.139). The additional factor of
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