Understanding Quantum Physics: An Advanced Guide for the Perplexed

expectation value of the projection operator A 1 ,
namely the integral of the density |ψ(x)| 2 in the region of box 1.
When multiplied by Q, it is the expectation value of the charge Q in the state
ψ 1 (x, t) in box 1, namely the integral of the charge density Q|ψ(x)| 2 in the region of box 1. In fact, as Eq. (2.29) and Eq. (2.30) clearly show,
this is what the protective measurement really measures.
    As we have argued
in the last section, the result of a protective measurement reflects an
objective property of the measured system. Thus the result of the above
protective measurement, namely the expectation value of the charge Q in the
state ψ 1 (x, t), |a| 2 Q, will reflect the actual charge
distribution of the system in box 1. In other words, the result indicates that
there exists a charge |a| 2 Q in box 1. [14] In the following, we will give another two
arguments for this conclusion.
    First of all,
let’s analyze the result of the protective measurement. Suppose we can
continuously change the measured state from |a| 2 = 0 to |a| 2 = 1. When |a| 2 = 0, the single electron will reach the position “0”
of the screen one by one, and it is incontrovertible that no charge is in box
1. When |a| 2 = 1, the single electron will reach the position “1” of
the screen one by one, and it is also incontrovertible that there is a charge Q
in box 1. Then when |a| 2 assumes a numerical value between 0 and 1
and the single electron reaches the position “|a| 2 ” between “0” and
“1” on the screen one by one, the results should similarly indicate that there
is a charge |a| 2 Q in the box by continuity. The point is that the
definite deviation of the trajectory of the electron will reflect that there
exists a definite amount of charge in box 1. [15] Next, let’s analyze the equation that
determines the result of the protective measurement, namely Eq. (2.30). It gives
a more direct support for the existence of a charge |a| 2 Q in box 1.
The r.h.s of Eq. (2.30) is the formula of the 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