Understanding Quantum Physics: An Advanced Guide for the Perplexed

particles appearing in
every position in space at a given instant. By contrast, the position density
and position flux density, which are defined during an infinitesimal time
interval at a given instant, are only a description of the state of the
resulting random discontinuous motion of particles, and they are determined by
the wave function. In this sense, we may say that the motion of particles is
"guided" by their wave function in a probabilistic way.
    We have been
discussed random discontinuous motion of particles in real space. The picture
of random discontinuous motion may exist not only for position but also for
other dynamical variables such as momentum and energy, and thus the suggested
interpretation of the wave function in position space may also apply to the
wave function in momentum space etc. Due to the randomness of motion for each
variable, the probability distributions of all variables for an arbitrary wave
function can be consistent with quantum mechanics [33] . However, it is worth stressing that spin
is a distinct property. Since the spin of a particle is always definite along
one direction (though the spin state can always be decomposed into two
eigenstates of spin along another direction), the spin of the particle, unlike
its position, does not undergo random discontinuous motion for any spin state [34] .

 
    Chapter 3
    How Come the Schrödinger Equation?
    The motion of particles follows probability
law but the probability itself propagates according to the law of causality.
    — Max Born
    After
investigating the physical meaning of the wave function, we will further
analyze the linear evolution law for the wave function in this chapter. It is demonstrated
that the linear non-relativistic evolution of the wave function of an isolated
system obeys the free Schrödinger equation due to the requirements of spacetime
translation invariance and relativistic invariance. In addition, we also
investigate the meaning and implications of the conservation laws in quantum
mechanics.
    Many quantum
mechanics textbooks provide a heuristic "derivation" of the
Schrödinger equation. It begins with the assumption that the state of a free
quantum system has the form of a plane wave e i(kx−ωt) . When
combining with the de Broglie relations for momentum and energy p = hk and E =
hω, this state becomes e i(px−Et)/h . Then it uses the
nonrelativistic energy-momentum relation E = p 2 /2m to obtain the
free particle Schrödinger equation. Lastly, this equation is generalized to
include an external potential, and the end result is the Schrödinger equation.
In the following sections, we will show that the heuristic
"derivation" of the free Schrödinger equation can be turned into a
real derivation by resorting to spacetime translation invariance and
relativistic invariance. Spacetime translation gives the definitions of
momentum and energy, and spacetime translation invariance entails that the
state of a free quantum system with definite momentum and energy assumes the
plane wave form e i(px−Et)/h . Moreover, the relativistic invariance
of the free states further determines the relativistic energy-momentum
relation, whose nonrelativistic approximation is E = p 2 /2m. Though
the requirements of these invariances are already well known, an explicit and
complete derivation of the free Schrödinger equation using them seems still
missing in the literature and textbooks. The new integrated analysis may be
helpful in understanding the physical origin of the Schrödinger equation, and
moreover, it is also helpful for understanding momentum and energy and their
conservation for random discontinuous motion of particles.
    3.1 Spacetime translation and its
invariance
    In this section,
we will show that the free states of motion for a quantum system can be
basically determined by spacetime translation invariance. The spacetime
translation invariance of natural laws reflects the homogeneity of space and
time. The homogeneity of space ensures that the same experiment performed at
two different places gives the same result, and the homogeneity in time ensures
that the same experiment repeated at two different times gives the same result.
There are in general two different pictures of translation: active
transformation and passive transformation. The active transformation
corresponds to displacing the studied system, and the passive transformation
corresponds to moving the coordinate system. Physically, the equivalence of