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electrostatic self-interaction for a charged
quantum system such as an electron already contradicts experimental
observations. Accordingly, the mass and charge density of a quantum system
cannot be real but be effective [20] . This means that at every instant there is
only a localized particle with the total mass and charge of the system, and
during a time interval the time average of the ergodic motion of the particle
forms the effective mass and charge density [21] . There exist no gravitational and
electrostatic self-interactions of the density in this case.
    2.5.2
The ergodic motion of a particle is discontinuous
    Which sort of
ergodic motion then? If the ergodic motion of the particle is continuous, then
it can only form the effective mass and charge density during a finite time
interval. However, the mass and charge density of a particle, which is
proportional to the modulus square of its wave function, is an instantaneous
property of the particle. In other words, the ergodic motion of the particle
must form the effective mass and charge density during an infinitesimal time
interval (not during a finite time interval) at a given instant. Thus it seems
that the ergodic motion of the particle cannot be continuous. This is at least
what the existing quantum mechanics says. However, there may exist a possible
loophole here. Although the classical ergodic models that assume continuous
motion are inconsistent with quantum mechanics due to the existence of a finite
ergodic time, they may be not completely precluded by experiments if only the
ergodic time is extremely short. After all quantum mechanics is only an
approximation of a more fundamental theory of quantum gravity, in which there
may exist a minimum time scale such as the Planck time. Therefore, we need to
investigate the classical ergodic models more thoroughly.
    Consider an
electron in a one-dimensional box in the first excited state ψ(x) (Aharonov and
Vaidman 1993). Its wave function has a node at the center of the box, where its
charge density is zero. Assume the electron performs a very fast continuous
motion in the box, and during a very short time interval its motion generates
an effective charge density distribution. Let’s see whether this density can assume
the same form as e|ψ(x)| 2 , which is required by protective
measurement [22] . Since the effective charge density is proportional
to the amount of time the electron spends in a given position, the electron
must be in the left half of the box half of the time and in the right half of
the box half of the time. But it can spend no time at the center of the box
where the effective charge density is zero; in other words, it must move at
infinite velocity at the center. Certainly, the appearance of velocity faster
than light or even infinite velocity may be not a fatal problem, as our
discussion is entirely in the context of non-relativistic quantum mechanics,
and especially the infinite potential in the 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