Understanding Quantum Physics: An Advanced Guide for the Perplexed

the
motion. In logic the instantaneous condition can only be deterministic or
indeterministic. That the instantaneous condition is deterministic means that
it leads to a deterministic change of the position of a particle at a given
instant. That the instantaneous condition is indeterministic means that it only
determines the probability of the particle appearing in each position in space
at a given instant. If the instantaneous condition is deterministic, then the
simplest states of motion of the free particle will have two possible forms.
The first one is continuous motion with constant velocity, and the equation of
motion of the particle is x(t + dt) = x(t) + vdt, where the deterministic
instantaneous condition v is a constant [24] . The second one is discontinuous motion
with infinite average velocity; the particle performs a finite jump along a
fixed direction at every instant, where the jump distance is a constant,
determined by the constant instantaneous condition [25] . On the other hand, if the instantaneous
condition is indeterministic, then the simplest states of motion of the free
particle will be random discontinuous motion with even position probability
density. At each instant the probability density of the particle appearing in
every position is the same.
    In order to know
whether the instantaneous condition is deterministic or not, we need to
determine which sort of simplest states of motion are the solutions of the
equation of free motion in quantum mechanics (i.e. the free Schrödinger
equation) [26] . According to the analysis in the last subsection,
the momentum eigenstates of a free particle, which are the solutions of the
free Schrödinger equation, describe the ergodic motion of the particle with
even position probability density in space. Therefore, the simplest states of
motion with a constant probabilistic instantaneous condition are the solutions
of the equation of free motion, while the simplest states of motion with a
constant deterministic instantaneous condition are not.
    When assuming that
(1) the simplest states of motion of a free particle are the solutions of the
equation of free motion; and (2) the instantaneous condition determining the
position change of a particle is always deterministic or indeterministic for
any state of motion, the above result then implies that motion, no matter
whether it is free or forced, has no deterministic cause, and thus it is random
and discontinuous, only determined by a probabilistic cause. The argument may be
improved by further analyzing these two seemingly reasonable assumptions, but
we will leave this for future work.
    2.6 The wave function represents the state of
random discontinuous motion of particles
    The wavefunction gives not the density of
stuff, but gives rather (on squaring its modulus) the density of probability.
Probability of what exactly? Not of the electron being there, but of the
electron being found there, if its position is measured. Why this aversion to
being and insistence on finding? The founding fathers were unable to form a
clear picture of things on the remote atomic scale. — John Bell, 1990
    In classical
mechanics, we have a clear physical picture of motion. It is well understood
that the trajectory function x(t) in classical mechanics describes the
continuous motion of a particle. In quantum mechanics, the trajectory function
x(t) is replaced by a wave function ψ(x, t). If the particle ontology is still
viable in the quantum domain, then it seems natural that the wave function
should describe some sort of more fundamental motion of particles, of which
continuous motion is only an approximation in the classical domain, as quantum
mechanics is a more fundamental theory of the physical world, of which
classical mechanics is an approximation. The analysis in the last section
provides a strong support for this conjecture, and it suggests that what the
wave function describes is the more fundamental motion of particles, which is
essentially discontinuous and random. In this section, we will give a more
detailed analysis of this suggested interpretation of the wave function (Gao
1993, 1999, 2000, 2003, 2006b, 2008, 2011a, 2011b).
    2.6.1
An analysis of random discontinuous motion
    Let’s first make
clearer what we mean when we say a quantum system such as an electron is a
particle. The picture of particle appears from our analysis of the mass and
charge density of a quantum system. As we have shown in the last section,