Understanding Quantum Physics: An Advanced Guide for the Perplexed

mass m (Salzman 2005). This quantity represents the
strength of the influence of the self-interaction on the normal evolution of
the wave function; when ε 2 ≈ 1 the influence is significant.
Similarly, for a free charged system with charge Q, the measure of the
potential strength of the electrostatic self-interaction is ε 2 =
(4kQ2/hc) 2 . As a typical example, for a free electron the potential
strength of the electrostatic self-interaction will be ε 2 =
(4ke2/hc) 2 ≈ 1 × 10 −3 . This indicates that the
electrostatic self-interaction will have a remarkable influence on the
evolution of the wave function of a free electron [19] . If such an interaction indeed exists, it
should have been detected by precise interference experiments on electrons. On
the other hand, the superposition principle of quantum mechanics, which denies
the existence of the observable electrostatic self-interaction, has been
verified for microscopic particles with astonishing precision. As another
example, consider the electron in the hydrogen atom. Since the potential of the
electrostatic self-interaction is of the same order as the Coulomb potential
produced by the nucleus, the energy levels of hydrogen atoms will be remarkably
different from those predicted by quantum mechanics and confirmed by
experiments. Therefore, the electrostatic self-interaction cannot exist for a
charged quantum system.
    In conclusion,
although the gravitational self-interaction is too weak to be detected
presently, the existence of the 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