Understanding Quantum Physics: An Advanced Guide for the Perplexed

example is also an ideal
situation. However, it seems difficult to explain why the electron speeds up at
the node and where the infinite energy required for the acceleration comes
from. Moreover, the sudden acceleration of the electron near the node may also
result in large radiation (Aharonov, Anandan and Vaidman 1993), which is
inconsistent with the predictions of quantum mechanics. Again, it seems very difficult
to explain why the accelerating electron does not radiate here.
    Let’s further
consider an electron in a superposition of two energy eigenstates in two boxes
ψ 1 (x) + ψ 2 (x). In this example, even if one assumes that
the electron can move with infinite velocity (e.g. at the nodes), it cannot
continuously move from one box to another due to the restriction of box walls.
Therefore, any sort of continuous motion cannot generate the effective charge
density e|ψ 1 (x) + ψ 2 (x)| 2 . One may still
object that this is merely an artifact of the idealization of infinite
potential. However, even in this ideal situation, the model should also be able
to generate the effective charge density by means of some sort of ergodic
motion of the electron; otherwise it will be inconsistent with quantum
mechanics. On the other hand, it is very common in quantum optics experiments
that a single-photon wave packet is split into two branches moving along two
well separated paths in space. The wave function of the photon disappears
outside the two paths for all practical purposes. Moreover, the experimental
results are not influenced by the environment and setup between the two paths
of the photon. Thus it is very difficult to imagine that the photon performs a
continuous ergodic motion back and forth in the space between its two paths.
    In view of these
serious drawbacks of the classical ergodic models and their inconsistency with
quantum mechanics, we conclude that the ergodic motion of particles cannot be
continuous. If
the motion of a particle is discontinuous, then the particle can readily move
throughout all regions where the wave function is nonzero during an arbitrarily
short time interval at a given instant. Furthermore, if the probability density
of the particle appearing in each position is proportional to the modulus
square of its wave function there at every instant, the discontinuous motion
can also generate the right effective mass and charge density. This will solve
the above problems plagued by the classical ergodic models. The discontinuous
ergodic motion requires no existence of a finite ergodic time. Moreover, a
particle undergoing discontinuous motion can also move from one region to
another spatially separated region, no matter whether there is an infinite
potential wall between them, and such discontinuous motion is not influenced by
the environment and setup between these regions either. Besides, discontinuous
motion can also solve the problems of infinite velocity and accelerating
radiation. The reason is that no classical velocity and acceleration can be
defined for discontinuous motion, and energy and momentum will require new
definitions and understandings as in quantum mechanics.
    In conclusion, we
have argued that the mass and charge density of a quantum system, which can be
measured by protective measurement, is not real but effective. Moreover, the
effective mass and charge density is formed by the discontinuous motion of a
localized particle, and the probability density of the particle appearing in
each position is proportional to the modulus square of its wave function there.
    2.5.3
An argument for random discontinuous motion
    Although the above
analysis demonstrates that the ergodic motion of a particle is discontinuous,
it doesn’t say that the discontinuous motion must be random. In particular, the
randomness of the result of a quantum measurement may be only apparent. In
order to know whether the motion of particles is random or not, we need to
analyze the cause of motion. For example, if motion has no deterministic cause,
then it will be random, only determined by a probabilistic cause. This may also
be the right way to find how particles move. Since motion involves change in
position, if we can find the cause or instantaneous condition determining the
change [23] , we will be able to find how particles move in
reality.
    Let’s consider the
simplest states of motion of a free particle, for which the instantaneous
condition determining the change of its position is a constant during