Understanding Quantum Physics: An Advanced Guide for the Perplexed

of identical systems for wavefunction collapse
(See Pearle 2000 for a more detailed analysis). Note that for the linear
Schrödinger evolution under an external potential, energy is conserved but
momentum is not conserved even at the ensemble level, and thus it is not
momentum conservation but energy conservation that is a more universal
restriction for wavefunction collapse.
    The conservation
of energy can not only help to solve the preferred basis problem, but also
further determine the law of dynamical collapse to a large extent. For each
system in the same quantum state in an ensemble, in order that the probability
distribution of energy eigenvalues of the state can keep constant for the whole
ensemble (i.e. energy is conserved at the ensemble level), the random stay of
the system at each discrete instant can only change its (objective) energy
probability distribution [72] , and moreover, the change must also satisfy a certain
restriction. Concretely speaking, the random stay in a definite energy E i will increase the probability of the energy branch |E i > and
decrease the probabilities of all other energy branches pro rata. Moreover, the
increasing amplitude must be proportional to the total probability of all other
energy branches, and the coefficient is related to the energy uncertainty of
the state. We will demonstrate this result in the next subsection.
    A more important
problem is whether this energy-conserved collapse model can explain the
emergence of definite measurement results and our macroscopic experience. At
first sight the answer appears negative. For example, the energy eigenstates
being collapse states seems apparently inconsistent with the localization of
macroscopic objects. However, a detailed analysis given in the following
subsections will demonstrate that the model can be consistent with existing
experiments and our macroscopic experience. The key is to realize that the
energy uncertainty driving the collapse of the entangled state of a many-body
system is not the uncertainty of the total energy of all subsystems, but the
sum of the absolute energy uncertainty of every sub-system. As a result, the
collapse states are the product states of the energy eigenstates of the
Hamiltonian of each sub-system for a non-interacting or weakly-interacting
many-body system. This provides a further collapse rule for the superpositions
of degenerate energy eigenstates of a many-body system.
    4.2.3
In search of a deeper basis
    In this
subsection, we will investigate the possible physical basis of the energy
conservation restriction for wavefunction collapse.
    It is well known
that the conservation of energy and momentum refers to an ensemble of identical
systems in standard quantum mechanics. However, this standard view seems
unnatural when assuming an objective interpretation of the wave function of a
single system, e.g. our suggested interpretation in terms of random
discontinuous motion of particles. An ensemble is not an actual system after
all, and the conservation of something for an ensemble seems physically
meaningless. Moreover, since a single system in the ensemble does not ‘know’
the other systems and the whole ensemble, there must exist some underlying
mechanism that can ensure the conservation of energy for an ensemble. Then the
conservation of energy for an ensemble of identical systems is probably a
result of the laws of motion for individual systems in the ensemble. Here is a
possible scheme. First of all, energy is conserved for the evolution of
individual energy eigenstates. Next, a superposition of energy eigenstates will
dynamically collapse to one of these energy eigenstates, and the probability of
the collapse result satisfies Born’s rule. Then the wavefunction collapse will
satisfy the conservation of energy for an ensemble of identical systems.
    In the following,
we will further suggest a possible physical basis for this scheme of
energy-conserved wavefunction collapse. According to the picture of random
discontinuous motion, for a particle in a superposition of energy eigenstates,
the particle stays in an instantaneous state with a definite energy eigenvalue
at a discrete instant, and at another instant it may jump to another
instantaneous state with another energy eigenvalue. It seems to be a reasonable
assumption that the particle has both the tendency to jump among the
instantaneous states with different energies and the tendency to stay in the
instantaneous states with the