Single-Particle Electrodynamics - Assassination Science

wish of them—not least of which being simplicity.
2.7 Classical limit of quantum mechanics
A thorny question is the following: In what sense does classical mechanics
represent the “limit” of quantum mechanics?
The author will not add any new answers to this question. Most physicists
will be familiar with Ehrenfest’s theorem (see, e.g., [150, Ch. VI] and [69,
Sec. 31]): the equation of motion for the expectation value of an operator is
the expectation value of the corresponding classical Hamiltonian equation of
motion.
The rub is that, despite first appearances, the expectation values do not
follow the classical equations of motion, unless the functional dependence of
the Hamiltonian equation of motion is such that the expectation value of the
function is equivalent to the function of the relevant expectation values. This
is only exactly true of a few physical systems (where the Hamiltonian H is a
polynomial of the second degree in the q’s and b’s).
However, regardless of the system being considered, one can generally
construct wave-packets for which the classical equations of motion are a good
approximation—namely, those physical situations for which one may speak
with some validity of complementary variables taking on approximately welldefined
values (limited by the Heisenberg uncertainty principle) [69]. In such
cases, the difference between the taking the expectation of a function of
the conjugate variables, and taking the function of the expectations of the
conjugate variables, is small compared to the motion of the wave-packet as
a whole.
It is simple enough to review Ehrenfest’s theorem in theory, as we have
briefly done above. It is far more difficult to decide, for any given physical
application, just how well the classical equations equations of motion will
describe the gross motion of the particles under consideration. Clearly, the
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