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Such a measurement cannot tell
us the charge distribution of the system in each box before the measurement.
    Fig.1 Scheme of a protective measurement of the charge density of a quantum
system
    Now let’s make a
protective measurement of A 1 . Since ψ(x, t) is degenerate with its
orthogonal state ψ (x, t) = b ∗ ψ 1 (x, t)−a ∗ ψ 2 (x, t), we need an artificial
protection procedure to remove the degeneracy, e.g. joining the two boxes with
a long tube whose diameter is small compared to the size of the box [13] . By this protection ψ(x, t) will be a
nondegenerate energy eigenstate. The adiabaticity condition and the weakly
interacting condition, which are required for a protective measurement, can be
further satisfied when assuming that (1) the measuring time of the electron is
long compared to /∆E, where ∆E is the smallest of the energy differences
between ψ(x, t) and the other energy eigenstates, and (2) at all times the
potential energy of interaction between the electron and the system is small
compared to ∆E. Then the measurement of A 1 by means of the electron
trajectory is a protective measurement, and the trajectory of the electron is
only influenced by the expectation value of the charge of the system in box 1.
In particular, when the size of box 1 can be ignored compared with the
separation between it and the electron wave packet, the wave function of the
electron will obey the following Schrödinger equation:

    where m e is the mass of electron, k is the Coulomb constant, r 1 is the
position of the center of box 1, and |a| 2 Q is the expectation value
of the charge Q in box 1. Correspondingly, the trajectory of the center of the
electron wave packet, r c (t), will satisfy the following equation by
Ehrenfest’s theorem:

    Then the electron
wave packet will reach the position “ |a| 2 ” between “0” and “1” on
the screen as denoted in Fig.1. This shows that the result of the protective
measurement is the 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