Understanding Quantum Physics: An Advanced Guide for the Perplexed

,
and the overall spread of its state is of the order of 10m after a second (Joos
and Zeh 1985). If the dust particle interacts with air molecules, e.g. floating
in the air, the spread of its state will be much faster.
    Let’s see whether
the energy-conserved collapse in our model can prevent the above spreading of
the wave packet. Suppose the dust particle is in a superposition of two
identical localized states that are separated by 10 −5 cm in space.
The particle floats in the air, and its average velocity is about zero. At
standard temperature and pressure, one nitrogen molecule accretes in the dust
particle, which area is 10 −10 cm 2 , during a time interval
of 10 −14 s in average (Adler 2002). Since the mass of the dust
particle is much larger than the mass of a nitrogen molecule, the velocity
change of the particle is negligible when compared with the velocity change of
the nitrogen molecules during the process of accretion. Then the kinetic energy
difference between an accreted molecule and a freely moving molecule is about
∆E = 3 kT ≈ 10 −2 eV . When one nitrogen molecule accretes
in one localized branch of the dust particle (the molecule is freely moving in
the other localized branch), it will increase the energy uncertainty of the
total entangled state by ∆E ≈ 10 −2 eV . Then after a time interval of
10 −4 s, the number of accreted nitrogen molecules is about 10 10 ,
and the total energy uncertainty is about 10 8 eV . According to Eq.
(4.13) in our collapse model, the corresponding collapse time is about 10 −4 s.
Since the two localized states in the superposition have the same energy
spectra, the collapse also conserves energy.
    In the
energy-conserved collapse model, the collapse states are energy eigenstates,
and in particular, they are nonlocal momentum eigenstates for free quantum
systems. Thus it is indeed counterintuitive that the energy-conserved collapse
can make the states of macroscopic objects local. As shown above, this is due
to the constant influences of environmental particles. When the spreading of
the state of a macroscopic object becomes larger, its interaction with
environmental particles will introduce larger energy difference between its
different local branches, and this will then collapse the spreading state again
into a more localized state [87] . As a result, the states of macroscopic objects in an
environment will never reach the collapse states, namely momentum eigenstates,
though they do continuously undergo the energy-conserved collapse. To sum up,
there are two opposite processes for a macroscopic object constantly
interacting with environmental particles. One is the spreading process due to
the linear Schrödinger evolution, and the other is the localization process due
to the energy-conserved collapse evolution. The interactions with environmental
particles not only make the spreading more rapidly but also make the
localization more frequently. In the end these two processes will reach an
approximate equilibrium. The state of a macroscopic object will be a wave
packet narrow in both position and momentum, and this narrow wave packet will
follow approximately Newtonian trajectories (if the external potential is
uniform enough along the width of the packet) by Ehrenfest’s theorem (See
Bacciagaluppi 2008 for a similar analysis in the context of decoherence) [88] . In some sense, the emergence of the
classical world around us is "conspired" by environmental particles
according to the energy-conserved collapse model.
    Ultimately, the
energy-conserved collapse model should be able to account for our definite
conscious experience. According to recent neuroscience literature, the
appearance of a (definite) conscious perception in human brains involves a
large number of neurons changing their states from resting state (resting
potential) to firing state (action potential). In each neuron, the main
difference of these two states lies in the motion of 10 6 Na + s
passing through the neuron membrane. Since the membrane potential is in the
order of 10 −2 V, the energy difference between firing state and
resting state is ∆E ≈ 10 4 eV. According to the energy-conserved
collapse model, the collapse time of a quantum superposition of these two
states of a neuron is
    τ c ≈ hE P /(∆E) 2 ≈ ( 2.8MeV/0.01MeV ) 2 ≈ 10 5 s, (4.20)
    where the Planck
energy E P ≈ 10 19 GeV . When considering the number of
neurons that can form a definite conscious perception