Understanding Quantum Physics: An Advanced Guide for the Perplexed

of
Schrödinger’s idea to some extent. The twist is: that the charge density is not
classical does not imply its nonexistence; rather, its existence points to a
non-classical physical picture of motion hiding behind the mathematical wave
function.
    The charge
distribution of a charged quantum system such as an electron has two possible
existent forms: it is either real or effective. The charge distribution is real
means that it exists throughout space at the same time, and the charge
distribution is effective means that there is only a localized particle with
the total charge of the system at every instant, and its motion forms the
effective charge distribution. If the charge distribution is effective, then
there will exist no electrostatic self-interaction of the charge distribution,
as there is only a localized charged particle at every instant. By contrast, if
the charge distribution is real, then there will exist electrostatic
self-interaction of the charge distribution, as the distribution exists
throughout space at the same time. Since the superposition principle of quantum
mechanics prohibits the existence of electrostatic self-interaction, and
especially, the existence of the electrostatic self-interaction for the charge
distribution of an electron already contradicts experimental observations, the
charge distribution of a quantum system cannot be real but must be effective.
This means that for a quantum system, at every instant there is only a
localized particle with the total mass and charge of the system, and during an
infinitesimal time interval at a given instant the time average of the motion
of the particle forms the effective mass and charge density in every position,
which is proportional to the modulus square of the wave function of the system
there. Since the integral of the formed mass and charge density in any region
is equal to the expectation value of the total mass and charge in the region,
the motion of the particle is ergodic.
    The next question
is which sort of ergodic motion the particle undergoes. It can be argued that
the classical ergodic models, which assume continuous motion of particles, are
inconsistent with quantum mechanics, and the effective mass and charge density
of a quantum system is formed by discontinuous motion of a localized particle
with mass and charge. Moreover, the discontinuous motion is not deterministic
but random. Based on this result, we suggest that the wave function in quantum
mechanics describes the state of random discontinuous motion of particles, and
at a deeper level, it represents the property of the particles that determines
their random discontinuous motion. In particular, the modulus square of the
wave function (in position space) determines the probability density of the
particles appearing in certain positions in real space. In the following, we
will give a full exposition of this suggested interpretation of the wave
function.
    2.1 Standard quantum mechanics and conventional measurements
    The standard
formulation of quantum mechanics, which was first developed by Dirac and von
Neumann, is based on the following four basic principles.
    1. Physical states
    The state of a
physical system is represented by a normalized wave function or unit vector
|ψ(t)> in a Hilbert space [2] . The Hilbert space is complete in the sense that
every possible physical state can be represented by a state vector in the
space.
    2. Physical
properties
    Every measurable
property or observable of a physical system is represented by a Hermitian
operator on the Hilbert space associated with the system. A physical system has
a determinate value for an observable if and only if it is in an eigenstate of
the observable (this is often called the eigenvalue-eigenstate link).
    3. Composition
rule
    The Hilbert space
associated with a composite system is the tensor product of the Hilbert spaces
associated with the systems of which it is composed. Similarly, the Hilbert
space associated with independent properties is the tensor product of the
Hilbert spaces associated with each property.
    4. Evolution law
    (1). Linear
evolution
    The state of a
physical system |ψ(t)> obeys the linear Schrödinger equation
i∂|ψ(t)>/∂t=H|ψ(t)> (when it is not measured), where H is the Hamiltonian
operator that depends on the energy properties of the system.
    (2). Nonlinear
collapse evolution
    If a physical
system is in a quantum superposition of the eigenstates of an observable A,
i.e., |ψ>=Σ i c