Understanding Quantum Physics: An Advanced Guide for the Perplexed

same energy, and their relative strength is
determined by the energy probability distribution of the particle. This is
satisfactory in logic, as there should exist two opposite tendencies in
general, and their relative strength is determined by certain condition. In
some sense, the two tendencies of a particle are related to the two parts of
its instantaneous state; the jumping tendency is related to the wave function,
and it is needed to manifest the superposition of different energy eigenstates,
while the staying tendency is related to the random stays. These two opposite
tendencies together constitute the complete "temperament" of a
particle.
    It can be argued
that the tendency to stay in the same energy for individual particles might be
the physical origin of the energy-conserved wavefunction collapse. For a
particle in a superposition of energy eigenstates, the particle stays in an
instantaneous state with definite energy at a discrete instant, and the staying
tendency of the particle will increase its probability of being in the
instantaneous states with the present energy at next instant. In other words,
the random stay of a particle in an instantaneous state with an energy
eigenvalue will increase the probability of the energy eigenvalue (and
correspondingly decrease the probabilities of other energy eigenvalues pro
rata). Moreover, the increase of probability may relate to the energy
probability distribution of the particle. By the continuity of change of
staying tendency, the particle will jump more readily among the instantaneous
states with small energy uncertainty and more hardly among the instantaneous
states with large energy uncertainty (which can also be regarded as a
restriction of energy change). Thus the larger the energy uncertainty of the
superposition is, the larger the increase of probability is during each random
stay. A detailed calculation, which will be given in the next subsection, shows
that such random changes of energy probability distribution can continuously
accumulate to lead to the collapse of the superposition of energy eigenstates
to one of them.
    It can be further
argued that the probability distribution of energy eigenvalues should remain
constant during the random evolution of an ensemble of identical systems, and
thus the resulting wavefunction collapse will satisfy Born’s rule. The reason
is as follows. When an initial superposition of energy eigenstates undergoes
the dynamical collapse process, the probability distribution of energy
eigenvalues should manifest itself through the collapse results for an ensemble
of identical systems. At a deeper level, it is very likely that the laws of
nature permit nature to manifest itself, or else we will be unable to find the
laws of nature and verify them by experiments, and our scientific
investigations will be also pointless. This may be regarded as a meta-law.
Since the collapse evolution of individual systems is completely random and
irreversible, the diagonal density matrix elements for an ensemble of identical
systems must be precisely the same as the initial probability distribution at
every step of the evolution. Otherwise the frequency distribution of the
collapse results in the ensemble cannot reflect the initial probability
distribution, or in other words, the probability information contained in the
initial state will be completely lost due to the random and irreversible
collapse [73] . As a consequence, the collapse evolution will
conserve energy at the ensemble level, and the collapse results will also
satisfy Born’s rule in quantum mechanics.
    Certainly, there
is still a question that needs to be answered. Why energy? Why not position or
momentum? If there is only one property that undergoes the random discontinuous
motion (e.g. position), then the above tendency argument for the unique
property may be satisfying. But if there are many properties that undergoes the
random discontinuous motion, then we need to answer why the tendency argument applies
only to energy. A possible answer is that energy is the property that
determines the linear evolution of the state of motion, and thus it seems
natural and uniform that energy also determines the nonlinear collapse
evolution. Moreover, energy eigenstates are the states of motion that no longer
evolve (except an absolute phase) for the linear evolution. Then by analogy, it
is likely that energy eigenstates are also the states that no longer evolve for
the nonlinear