Understanding Quantum Physics: An Advanced Guide for the Perplexed

level.
    [73] Note that the reversible Schrödinger evolution conserves the information
even for individual isolated systems.
    [74] Strictly speaking, the description “branch” should be replaced by
“instantaneous state”, e.g. the branch |E i > should be replaced by
the instantaneous state with energy E i . Yet the branch description
may be more succinct and visual, and we will use it in the following
discussions.
    [75] The density matrix describes the ensemble of states which arise from all
possible random stays.
    [76] Note that the common RMS (mean square root) uncertainty also satisfies
the swap symmetry. Thus it still needs to be studied what the exact form of k
is.
    [77] This collapse time formula indicates that there is no wavefunction
collapse in continuous time because t P → 0 leads to τ c → ∞.
One premise of this conclusion is that the influence of each random stay is
proportional to the duration of stay.
    [78] In continuous space and time, a position eigenstate has infinite average
energy and cannot be physically real. But in discrete space and time, position
eigenstates will be the states whose spatial dimension is about the Planck
length, and they may exist.
    [79] Note that most collapse states in an ensemble of identical systems keep
the shape of the wavepacket almost precisely.
    [80] There might exist a subtle connection here. It seems that the
energy-conserved wavefunction collapse in discrete time requires a finite event
horizon to ensure the energy eigenvalues of any system are discrete. On the
other hand, it seems that discrete spacetime permits the existence of dark
energy as quantum fluctuations of  spacetime to lead to acceleration and
finite event horizon (Gao 2005). In any case, the existence of a cosmological
constant also leads to the existence of a finite event horizon.
    [81] A potentially more promising case is provided by certain long-lived
nuclear isomers, which have large energy gaps from their ground states (see
Adler 2002 and references therein). For example, the metastable isomer of 180 Ta, the only nuclear isomer to exist naturally on earth, has a
half-life of more than 10 15 years and an energy gap of 75keV from
the ground state. According to Eq. (4.13), a coherent superposition of the
ground state and metastable isomer of 180 Ta will spontaneously
collapse to either the isomeric state or the ground state, with a collapse time
of order 20 minutes. It will be a promising way to test our collapse model by
examining the maintenance of coherence of such a superposition.
    [82] Since the uncertainty of the total energy of the whole entangled system
is still zero, the energy-driven collapse models will predict that no
wavefunction collapse happens and no definite measurement result appears for
the above measurement process, which contradicts experimental observations
(Pearle 2004).
    [83] In more general measurement situations, the measured particle (e.g.
electron) is not annihilated by the detector. However, in each local branch of
the entangled state of the whole system, the particle also interacts with a
single atom of the detector by an ionizing process, and its total energy is
also wholly transferred to the atom and the ejecting electrons.
    [84] We take the widely-used Geiger counter as another illustration of the
amplification process during measurement. A Geiger counter is an instrument
used to detect particles such as α particles, β particles and γ rays etc. It
consists of a glass envelope containing a low-pressure gas (usually a mixture
of methane with argon and neon) and two electrodes, with a cylindrical mesh
being the cathode and a fine-wire anode running through the centre of the tube.
A potential difference of about 10 3 V relative to the tube is
maintained between the electrodes, therefore creating a strong electric field
near the wire. The counter works on the mechanism of gas multiplication.
Ionization in the gas is caused by the entry of a particle. The ions are
attracted to their appropriate electrode, and they gain sufficient energy to
eject electrons from the gas atoms as they pass through the gas. This further
causes the atoms to ionize. Therefore, electrons are produced continuously by
this process and rapid gas multiplication takes place (especially in the
central electrode because of its strong electric field strength). Its effect is
that more than 10 6 electrons are collected by the central electrode
for every ion produced in the primary absorption