Understanding Quantum Physics: An Advanced Guide for the Perplexed

particle at instant t. From Eq. (2.34) we can see that ρ(x, t)
satisfies the normalization relation, namely ∫ρ(x, t)dx = 1.
    Since the position
density will change with time in general, we can further define the position
flux density j(x, t) through the relation j(x, t) = ρ(x, t)v(x, t), where v(x,
t) is the velocity of the local position density. It describes the change rate
of the position density. Due to the conservation of measure, ρ(x, t) and j(x,
t) satisfy the continuity equation:

    The position
density ρ(x, t) and position flux density j(x, t) provide a complete
description of the state of random discontinuous motion of a single particle.
    The description of
the motion of a single particle can be extended to the motion of many
particles. For the random discontinuous motion of N particles, we can define
joint position density ρ(x 1 , x 2 , ...x N , t) and
joint position flux density j(x 1 , x 2 , ...x N ,
t) = ρ(x 1 , x 2 , ...x N , t) v(x 1 , x 2 ,
...x N , t). They also satisfy the continuity equation:

    When these N
particles are independent, the joint position density can be reduced to the
direct product of the position density for each particle. Note that the joint
position density ρ(x 1 , x 2 , ...x N , t) and joint
position flux density j(x 1 , x 2 , ...x N , t) are
not defined in the real three-dimensional space, but defined in the
3N-dimensional configuration space.
    2.6.2
Interpreting the wave function
    Although the
motion of particles is essentially discontinuous and random, the discontinuity
and randomness of motion is absorbed into the state of motion, which is defined
during an infinitesimal time interval, by the descriptive quantities of
position density ρ(x, t) and position flux density j(x, t). Therefore, the
evolution of the state of random discontinuous motion of particles can be
described as a deterministic continuous equation. By assuming that the
nonrelativistic equation of random discontinuous motion is the Schrödinger
equation in quantum mechanics, both ρ(x, t) and j(x, t) can be expressed by the
wave function in a unique way [31] :

    Correspondingly,
the wave function ψ(x, t) can be uniquely expressed by ρ(x, t) and j(x, t)
(except for a constant phase factor):

    In this way, the
wave function ψ(x, t) also provides a complete description of the state of
random discontinuous motion of particles. For the motion of many particles, the
joint position density and joint position flux density are defined in the
3N-dimensional configuration space, and thus the many-particle wave function,
which is composed of these two quantities, is also defined in the
3N-dimensional configuration space.
    Interestingly, we
can reverse the above logic in some sense, namely by assuming the wave function
is a complete objective description for the motion of particles, we can also
reach the random discontinuous motion of particles, independent of our previous
analysis. If the wave function ψ(x, t) is a description of the state of motion
for a single particle, then the quantity |ψ(x, t)| 2 dx not only gives
the probability of the particle being found in an infinitesimal space interval
dx near position x at instant t (as in standard quantum mechanics), but also
gives the objective probability of the particle being there. This accords with
the common-sense assumption that the probability distribution of the
measurement results of a property is the same as the objective distribution of
the property in the measured state. Then at instant t the particle may appear
in any location where the probability density |ψ(x, t)| 2 is nonzero,
and during an infinitesimal time interval near instant t the particle will move
throughout the whole region where the wave function ψ(x, t) spreads. Moreover,
its position density is equal to the probability density |ψ(x, t)| 2 .
Obviously this kind of motion is essentially random and discontinuous.
    One important
point needs to be stressed here. Since the wave function in quantum mechanics
is defined at an instant, not during an infinitesimal time interval, it should
be regarded not simply as a description of the state of random discontinuous
motion of particles, but more suitably as a description of the probabilistic
instantaneous condition or dispositional property of the particles that
determines their random discontinuous motion at a deeper level      [32] . In particular, the modulus square of the
wave function determines the probability density of the