Understanding Quantum Physics: An Advanced Guide for the Perplexed

system
during a finite period of time, but instantaneous properties of the system.
    2.4 How does the mass and charge of a
quantum system distribute?
    The fundamental assumption is that the
space density of electricity is given by the square of the wavefunction. —
Erwin Schrödinger, 1926 [11]
    According to
protective measurement, the expectation values of dynamical variables are
properties of a single quantum system. Typical examples of such properties are
the mass and charge density of a quantum system. In this section, we will
present a detailed analysis of this property, as it may have important
implications for the physical meaning of the wave function.
    2.4.1
A heuristic argument
    The mass and
charge of a classical system always localize in a definite position in space at
each moment. For a charged quantum system described by the wave function ψ(x,
t), how do its mass and charge distribute in space then? We can measure the
total mass and charge of the quantum system by the gravitational and
electromagnetic interactions and find them in some region of space. Thus it
seems that the mass and charge of a quantum system must also exist in space
with a certain distribution. Before we discuss the answer given by protective
measurement, we will first give a heuristic argument.
    The Schrödinger
equation of a charged quantum system under an external electromagnetic
potential may provide a clue to the answer. The equation is
    where m and Q are
the mass and charge of the system, respectively, ϕ and A are the electromagnetic potential,
and c is the speed of light. The electrostatic interaction term Q ϕ ψ(x, t) in the equation indicates that the
interaction exists in all regions where the wave function of the system, ψ(x,
t), is nonzero, and thus it seems to suggest that the charge of the system also
distributes throughout these regions. If the charge does not distribute in some
regions where the wave function is nonzero, then there will not exist an
electrostatic interaction there. Furthermore, since the integral ∫Q|ψ(x, t)| 2 d 3 x
is the total charge of the system, the charge density in space, if indeed
exists, will be Q|ψ(x, t)| 2 . Similarly, the mass density can be
obtained from the Schrödinger equation of a quantum system under an external
gravitational potential:

    The gravitational
interaction term mV G ψ(x, t) in the equation also suggests that the
(passive gravitational) mass of the quantum system distributes throughout the
whole region where its wave function ψ(x, t) is nonzero, and the mass density
in space is m|ψ(x, t)| 2 .
    2.4.2
The answer of protective measurement
    In the following,
we will show that protective measurement provides a more convincing argument
for the existence of mass and charge density. The mass and charge density of a
single quantum system, as well as its wave function, can be measured by
protective measurement as expectation values of certain observables (Aharonov
and Vaidman 1993). For example, a protective measurement of the flux of the
electric field of a charged quantum system out of a certain region will yield
the expectation value of its charge inside this region, namely the integral of
its charge density over this region. Similarly, we can also measure the mass
density of a quantum system by a protective measurement of the flux of its
gravitational field in principle (Anandan 1993).
    Consider a quantum
system in a discrete nondegenerate energy eigenstate ψ(x). We take the measured
observable A n to be (normalized) projection operators on small
spatial regions V n having volume v n :

    The protective
measurement of A n then yields

    where |ψ n | 2 is the average of the density ρ(x) = |ψ(x)| 2 over the small region V n .
Then when v n → 0 and after performing measurements in sufficiently
many regions V n we can measure ρ(x) everywhere in space.
    Since the physical
realization of the observable A n and the corresponding interaction
Hamiltonian must always resort to the electromagnetic or gravitational
interaction between the measured system and the measuring device, what the
above protective measurement measures is in fact the charge or mass density of
the quantum system [12] , and its result indicates that the mass and charge
density is proportional to the modulus square of the wave function of the
system, namely the density ρ(x). In the following, we will give a concrete
example to illustrate this important result (see also Aharonov, Anandan and
Vaidman