Understanding Quantum Physics: An Advanced Guide for the Perplexed

the
neighboring branch |E i+1 > is ∆P i+1 =P i+1 ∆E/E P .
Then the probability increase of these two branches is

    Similarly, the
equation ∆P =(1 − P )∆E/E P holds true for the total probability of
arbitrarily many branches (one of which is the staying branch). This property
of scale invariance may simplify the analysis in many cases. For example, for a
superposition of two wavepackets with energy difference, ∆E 12 , much
larger than the energy uncertainty of each wavepacket, ∆E 1 = ∆E 2 , we can calculate the collapse dynamics in two steps. First, we use Eq.(4.15)
and Eq.(4.11) with |E 1 − E 2 | = ∆E 12 to
calculate the time of the superposition collapsing into one of the two
wavepackets [79] . Here we need not to consider the almost infinitely
many energy eigenstates constituting each wavepacket and their probability
distribution. Next, we use Eq.(4.15) with ∆E = ∆E 1 to calculate the
time of the wavepacket collapsing into one of its energy eigenstates. In
general, this collapse process is so slow that its effect can be ignored.
    Lastly, we want to
stress another important point. As we have argued before, the discontinuity of
motion requires that the collapse dynamics must be discrete in nature, and
moreover, the collapse states must be energy eigenstates in order that the
collapse dynamics satisfies the conservation of energy at the ensemble level.
As a result, the energy eigenstates and their corresponding eigenvalues must be
also discrete for any quantum system. This result seems to contradict quantum
mechanics, but when considering that our universe has a finite size (i.e. a
finite event horizon), the momentum and energy eigenvalues of any quantum
system in the universe may be indeed discrete [80] . The reason is that all quantum systems in
the universe are limited by the finite horizon, and thus no free quantum
systems exist in the strict sense. For example, the energy of a massless
particle (e.g. photon) can only assume

    This indicates
that the probability change during each random stay is still very tiny. Only
when the energy uncertainty is larger than 10 23 eV or 10 −5 E P ,
will the probability change during each random stay be sharp. Therefore, the
collapse evolution is still very smooth for the quantum states with energy
uncertainty much smaller than the Planck energy.
    4.4 On the consistency of the model and
experiments
    In this section,
we will analyze whether the discrete model of energy-conserved wavefunction
collapse is consistent with existing experiments and our macroscopic
experience. Note that Adler (2002) has already presented a detailed consistency
analysis in the context of energy-driven collapse models, and as we will see
below, most of his analysis also applies to our model.
    4.4.1
Maintenance of coherence
    First of all, the
model satisfies the constraint of predicting the maintenance of coherence when
this is observed. Since the energy uncertainty of the state of a microscopic
particle is very small in general, its collapse will be too slow to have any
detectable effect in present experiments on these particles. For example, the
energy uncertainty of a photon emitted from an atom is in the order of 10 −6 eV
, and the corresponding collapse time is 10 25 s according to Eq.
(4.13) of our collapse model, which is much longer than the age of the
universe, 10 17 s. This means that the collapse states (i.e. energy
eigenstates) are never reached for a quantum system with small energy
uncertainty even during a time interval as long as the age of the universe. As
another example, consider the SQUID experiment of Friedman et al (2000), where
the coherent superpositions of macroscopic states consisting of oppositely circulating
supercurrents are observed. In the experiment, each circulating current
corresponds to the collective motion of about 10 9 Cooper pairs, and
the energy uncertainty is about 8.6 × 10 −6 eV . Eq. (4.13) predicts a
collapse time of 10 23 s, and thus maintenance of coherence is
expected despite the macroscopic structure of the state [81] . For more examples see Adler (2002).
    4.4.2 Localization
in measurement situations
    In the following,
we will investigate whether the discrete model of energy-conserved wavefunction
collapse can account for the emergence of definite measurement results. Let’s
first see a simple position measurement experiment. Consider an initial state
describing a particle in a superposition of two locations (e.g. a