Understanding Quantum Physics: An Advanced Guide for the Perplexed

collapse evolution, i.e., that the energy eigenstates are the
collapse states. However, we may never be able to reach (and know we reach) the
end point of explanation. Another important task is to develop a concrete model
and compare it with experiments. We do this in the subsequent sections.
    4.3 A discrete model of energy-conserved wavefunction
collapse
    After giving a
speculative analysis of the origin of wavefunction collapse in terms of the
random discontinuous motion of particles, we will propose a discrete model of
energy-conserved wavefunction collapse based on results obtained from the
analysis.
    Consider a
multi-level system with a constant Hamiltonian. Its initial state is:
    where |E i >
is the energy eigenstate of the Hamiltonian of the system, E i is the
corresponding energy eigenvalue, and c i (0) satisfies the
normalization relation Σ i |c i (0)| 2 = 1.
    According to our
conjecture on the origin of wavefunction collapse, this superposition of energy
eigenstates will collapse to one of the eigenstates after a discrete dynamical
process, and the collapse evolution satisfies the conservation of energy at the
ensemble level. The physical picture of the dynamical collapse process is as
follows. At the initial discrete instant t 0 = t P (where t P is the Planck time), the system randomly stays in a branch |E i with
probability P i (0) ≡ |c i (0)| 2 . [74] This finite stay slightly increases the
probability of the staying branch and decreases the probabilities of all other
branches pro rata. Similarly, at any discrete instant t = nt P the
system randomly stays in a branch |E i with probability P i (t)
≡ |c i (t)| 2 , and the random stay also changes the
probabilities of the branches slightly. Then during a finite time interval much
larger than t P , the probability of each branch will undergo a
discrete and stochastic evolution. In the end, the probability of one branch
will be close to one, and the probabilities of other branches will be close to
zero. In other words, the initial superposition will randomly collapse to one
of the energy branches in the superposition.

    P i (t).
Then we can work out the diagonal density matrix elements of the evolution [75] :
    Here we shall
introduce the first rule of dynamical collapse, which says that the probability
distribution of energy eigenvalues for an ensemble of identical systems is
constant during the dynamical collapse process. As we have argued in the last
subsection, this rule is required by the principle of energy conservation at
the ensemble level, and it may also have a physical basis relating to the
manifestability of nature. By this rule, we have ρ ii (t + t P )
= ρ ii (t) for any i. This leads to the following set of equations:

    By solving this
equations set (e.g.by subtracting each other), we find the following relation
for any i:

    where k is an
undetermined dimensionless quantity that relates to the state |ψ(t)>.
    By using Eq.
(4.6), we can further work out the non-diagonal density matrix elements of the
evolution. But it is more convenient to calculate the following variant of
non-diagonal density matrix elements:

    Since the usual
collapse time, τ c , is defined by the relation ρ ij (τ c )
=1/2ρ ij (0), we may use a proper approximation, where k is assumed
to be the same as its initial value during the time interval [0, τ c ],
to simplify the calculation of the collapse time. Then we have:

    The corresponding
collapse time is in the order of:

    In the following,
we shall analyze the formula of k defined by Eq.(4.6). To begin with, the
probability restricting condition 0 ≤ P i (t)≤1 for any i requires
that 0 ≤ k ≤ 1. When k = 0, no collapse happens, and when k = 1, collapse
happens instantaneously. Note that k cannot be smaller than zero, as this will
lead to the negative value of P i (t) in some cases. For instance,
when k is negative and P i (t)< |k|/(1+|k|), P i (t + t P )
= P i (t) + k[1 − P i (t)] will be negative and violate the
probability restricting condition. That k is positive indicates that each
random stay increases the probability of the staying branch and decreases the
probabilities of other branches, which is consistent with the analysis given in
the last subsection.
    Next, k is
proportional to the duration of stay. The influence of each stay on the
probability of the staying branch is an accumulating process. When the duration
of stay is zero as in continuous space and time, no influence exists and no
collapse