Understanding Quantum Physics: An Advanced Guide for the Perplexed

according to our model. The reason can be summarized as
follows. The wave function of the measured particle is usually a spherical wave
(e.g. a spherically symmetric wave function) in three-dimensional space. Its
momentum is along the radial direction, but the local and random measurement
result distributes along the sphere, perpendicular to the radial direction.
During the detection, the measured particle interacts with a single atom of the
detector by an ionizing process in each local branch of the entangled state of
the whole system including the particle and the atoms in the detector. The
particle is usually absorbed by the atom or bound in the atom, and its energy
is wholly transferred to the newly-formed atom and the ejecting electrons
during the ionizing process in each branch. Then the amplification process such
as an avalanche process of atoms introduces very large energy difference
between the detected branch and the empty branch, and as a result, the whole
superposition will soon collapse into one of its local branches in a random way
according to the energy-conserved collapse model [85] . After the collapse, the state of the
measured particle is localized in the spatial region of one atom. Moreover,
since each local branch of the entangled state of the particle and the detector
has the same energy spectrum, the collapse process also conserves energy at the
individual level.
    4.4.3 Emergence of
the classical world
    Now let’s see
whether the discrete model of energy-conserved wavefunction collapse is
consistent with our macroscopic experience. It seems that there is an apparent
inconsistency here. According to the model, when there is a superposition of a
macroscopic object in an identical physical state (an approximate energy
eigenstate) at two different, widely separated locations, the superposition
does not collapse. The reason is that there is no energy difference between the
two branches of the superposition. However, the existence of such
superpositions is obviously inconsistent with our macroscopic experience; the
macroscopic objects are localized. This common objection has been basically
answered by Adler (2002). The crux of the matter lies in the influences of
environment. The collisions and especially the accretions of environmental
particles will quickly increase the energy uncertainty of the entangled state
of the whole system including the object and environmental particles, and thus
the initial superposition will soon collapse to one of the localized branches
according to our model. Accordingly, the macroscopic objects can always be
localized due to the environmental influences. Note that the energy uncertainty
here denotes the sum of the absolute energy uncertainty of each sub-system in
the entangled state as defined in our model [86] .
    As a typical
example, we consider a dust particle of radius a ≈ 10 −5 cm and mass m
≈ 10 −7 g. It is well known that localized states of macroscopic
objects spread very slowly under the free Schrödinger evolution. For instance,
for a Gaussian wave packet with initial (mean square) width ∆, the wave packet
will spread so that the width doubles in a time t = 2m∆ 2 /h. This
means that the double time is almost infinite for a macroscopic object. If the
dust particle had no interactions with environment and its initial state is a
Gaussian wave packet with width ∆ ≈ 10 −5 cm, the doubling time would
be about the age of the universe. However, if the dust particle is in
interaction with environment, the situation turns out to be very different.
Although the different components that couple to the environment will be
individually incredibly localised, collectively they can have a spread that is
many orders of magnitude larger. In other words, the state of the dust particle
and the environment could be a superposition of zillions of very well localised
terms, each with slightly different positions, and which are collectively
spread over a macroscopic distance (Bacciagaluppi 2008). According to Joos and
Zeh (1985), the spread in an environment full of thermal radiation only is
proportional to mass times the cube of time for large times, namely (∆x) 2 ≈
Λmτ 3 , where Λ is the localization rate depending on the environment,
defined by the evolution equation of density matrix ρ t (x, x ) = ρ 0 (x,
x )e −Λt(x−x )2 . For example, if the above dust particle interacts
with thermal radiation at T = 300K, the localization rate is Λ = 10 12