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electric force between two
charges located in different spatial regions. It is incontrovertible that e is
the charge of the electron, and it exists in the position r. Then |a| 2 Q
should be the other charge that exists in the position r 1 . In other
words, there exists a charge |a| 2 Q in box 1.
    In conclusion,
protective measurement shows that a quantum system with mass m and charge Q,
which is described by the wave function ψ(x, t), has a mass density m|ψ(x, t)| 2 and a charge density Q|ψ(x, t)| 2 , respectively [16] .
    2.5 The origin of mass and charge density
    We have argued
that a charged quantum system has mass and charge density proportional to the modulus
square of its wave function. In this section, we will further investigate the
physical origin of the mass and charge density. Is it real or only effective?
As we will see, the answer may provide an important clue to the physical
meaning of the wave function.
    2.5.1 The mass and
charge density is not real
    If the mass and
charge density of a charged quantum system is real, that is, if the densities
at different locations exist at the same time, then there will exist
gravitational and electrostatic self-interactions of the density [17] .
    Interestingly, the
Schrödinger-Newton equation, which was proposed by Diosi (1984) and Penrose
(1998), just describes the gravitational self-interaction of the mass density.
The equation for a single quantum system can be written as

    where m is the
mass of the quantum system, V is an external potential, G is Newton’s
gravitational constant. Much work has been done to study the mathematical
properties of this equation (Moroz, Penrose and Tod 1998; Moroz and Tod 1999;
Harrison, Moroz and Tod 2003; Salzman 2005). Several experimental schemes have
been also proposed to test its physical validity (Salzman and Carlip 2006). As
we will see below, although such gravitational self-interactions cannot yet be
excluded by experiments [18] , the existence of the electrostatic self-interaction
for a charged quantum system already contradicts experimental observations.
    If there is also
an electrostatic self-interaction, then the equation for a free quantum system
with mass m and charge Q will be

    Note that the
gravitational self-interaction is attractive, while the electrostatic
self-interaction is repulsive. It has been shown that the measure of the potential
strength of the gravitational self-interaction is ε 2 = (4Gm2/hc) 2 for a free system with 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