## La medida de la velocidad del electrón libre

Paul Dirac (1902-1984) gave an interesting general argument for a much stronger
version of Huygens' Principle in the context of quantum mechanics. In his
"Principles of Quantum Mechanics" he noted that a measurement of a component of
the instantaneous velocity of a free electron must give the value +-c, which
implies that electrons (and massive particles in general) always propagate along
null intervals, i.e., on the local light cone. At first this may seem to contradict
the fact that we observe
massive objects to move at speeds much less than the speed of light, but Dirac
points out that observed velocities are always average velocities over
appreciable time intervals, whereas the equations of motion of the particle
show that its velocity
oscillates between +c and -c in such a way that the mean value agrees with the
average value. He argues that this must be the case in any relativistic theory that incorporates the uncertainty principle, because in order to measure the velocity of a
particle we must measure its position at two different times, and then divide
the change in position by the elapsed time. To approximate as closely as
possible to the instantaneous velocity, the time interval must go to zero,
which implies that the
position measurements must approach infinite precision. However, according to
the uncertainty principle, the extreme precision of the position measurement
implies an approach to infinite indeterminacy in the momentum, which means that almost all values
of momentum - from zero to infinity - become equally probable. Hence the
momentum is almost certainly infinite, which corresponds to a speed of +-c.
This is obviously a very general argument, and applies to all massive particles
(not just fermions).

Tomado de
Huygens' Principle

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