Understanding Quantum Physics: An Advanced Guide for the Perplexed

experiments.
    Next, if the
collapse states are position eigenstates [78] , then the collapse time formula Eq. (4.13)
will be replaced by something like τ c ≈l 2 t P /(∆x) 2 , where l is certain length scale relating to the collapsing state. No matter
what length scale l is, the collapse time of a momentum eigenstate will be zero
as its position uncertainty is infinite. This means that the momentum
eigenstates of any quantum system will collapse instantaneously to one of its
position eigenstates and thus cannot exist. Moreover, the superposition states
with very small momentum uncertainty will also collapse very quickly even for
microscopic particles. These results are apparently inconsistent with quantum
mechanics. Although it may be possible to adjust the length scale l to make the
model consistent with experience, the collapse time formula will be much more
complex than that in the above energy-conserved collapse model. Let’s give a
little more detailed analysis here. There are two universal length scales for a
quantum system: its Compton wavelength λ c and the Planck length l P .
It is obvious that both of them cannot be directly used as the length scale in
the collapse time formula τ c ≈l 2 t P /(∆x) 2 .
Then the formula can only be written in a more complex form: τ c ≈ (λ c /l P ) α ·λ c 2 t P /(∆x) 2 .
Moreover, experiments such as the SQUID experiments and our everyday
macroscopic experience require α ≈ 8. It seems very difficult to explain this
unusually large exponent in theory. To sum up, the collapse states can hardly
be position eigenstates when considering the consistency with experiments and
the simplicity of theory.
    Based on the above
analysis, the state of the multi-level system at instant t = nt P will be:

    Besides the linear
Schrödinger evolution, the collapse dynamics adds a discrete stochastic
evolution for P i (t) ≡ |c i (t)| 2 :
    where ∆E is the
energy uncertainty of the state at instant t defined by Eq. (4.10), E s is a random variable representing the random stay of the system, and its
probability of assuming E i at instant t is P i (t). When E s = E i , δ EsEi = 1, and when E s = E i ,
δ EsEi = 0.
    This equation of
dynamical collapse can be directly extended to the entangled states of a
many-body system. The difference only lies in the definition of the energy
uncertainty ∆E. According to our analysis in the last subsection, for a
non-interacting or weakly-interacting many-body system in an entangled state,
for which the energy uncertainty of each sub-system can be properly defined, ∆E
is the sum of the absolute energy uncertainty of all sub-systems, namely

    where n is the
total number of the entangled sub-systems, m is the total number of energy
branches in the entangled state, and E li is the energy of sub-system
l in the i-th energy branch of the state. Correspondingly, the collapse states
are the product states of the energy eigenstates of the Hamiltonian of each
sub-system. It should be stressed here that ∆E is not defined as the
uncertainty of the total energy of all sub-systems as in the energy-driven
collapse models (see, e.g. Percival 1995, 1998a; Hughston 1996). For each
sub-system has its own energy uncertainty that drives its collapse, and the
total driving “force” for the whole entangled state should be the sum of the driving
“forces” of all sub-systems, at least in the first order approximation.
Although these two kinds of energy uncertainty are equal in numerical values in
some cases (e.g. for a strongly-interacting many-body system), there are also
some cases where they are not equal. For example, for a superposition of
degenerate energy eigenstates of a non-interacting many-body system, which may
arise during a common measurement process, the uncertainty of the total energy
of all sub-systems is exactly zero, but the absolute energy uncertainty of each
sub-system and their sum may be not zero. As a result, the superpositions of
degenerate energy eigenstates of a many-particle system may also collapse. As
we will see later, this is an important feature of our model, which can avoid
Pearle’s (2004) serious objections to the energy-driven collapse models.
    It can be seen
that the equation of dynamical collapse, Eq.(4.15), has an interesting
property, scale invariance. After one discrete instant t P , the
probability increase of the staying branch |E i > is ∆P i =(1 − P i )∆E/E P , and the probability decrease of