Understanding Quantum Physics: An Advanced Guide for the Perplexed

superposition
of two Gaussian wavepackets separated by a certain distance). After the
measurement interaction, the position measuring device evolves to a
superposition of two macroscopically distinguishable states:
    (c 1 ψ 1 + c 2 ψ 2 )ϕ 0 → c 1 ψ 1 ϕ 1 + c 2 ψ 2 ϕ 2 , (4.18)
    where ψ 1 ,
ψ 2 are the states of the particle in different locations, ϕ 0 is the initial state of the position measuring device, and ϕ 1 , ϕ 2 are the different outcome states of the device. For an ideal measurement, the
two particle/device states ψ 1 ϕ 1 and ψ 2 ϕ 2 have precisely the same energy spectrum. Then it appears that this
superposition will not collapse according to the energy- conserved collapse
model.
    However, this is
not the case. The key is to see that the two states of the particle in the
superposition are detected in different parts of the measuring device, and they
interact with the different atoms or molecules in these parts. Thus we should
rewrite the device states explicitly as ϕ 0 = χ A (0)χ B (0),
ϕ 1 = χ A (1)χ B (0), and ϕ 2 = χ A (0)χ B (1),
where χ A (0) and χ B (0) denote the initial states of the
device in the parts A and B, respectively, and χ A (1) and χ B (1)
denote the outcome states of the device in the parts A and B, respectively.
Then we have
    ( c 1 ψ 1 +c 2 ψ 2 )χ A (0)χ B (0)
→ c 1 ψ 1 χ A (1)χ B (0)+c 2 ψ 2 χ A (0)χ B (1)
(4.19)
    This reformulation
clearly shows that there exists energy difference between the sub-systems in
the different outcome states of the device. Since there is always some kind of
measurement amplification from the microscopic state to the macroscopic outcome
in the measurement process, there will be a large energy difference between the
states χ A (0), χ B (0) and χ A (1), χ B (1).
As a result, the total energy difference ∆E = |∆E A | + |∆E B |
is also very large, and it will result in the rapid collapse of the above
superposition into one of its branches according to the energy-conserved
collapse model [82] .
    Let’s see a more
realistic example, a photon being detected via photoelectric effect (e.g. by a
single- photon avalanche diode). In the beginning of the detection, the
spreading spatial wave function of the photon is entangled with the states of a
large number of surface atoms of the detector. In each local branch of the
entangled state, the total energy of the photon is wholly absorbed by the
electron in the local atom interacting with the photon. This is clearly
indicated by the term δ(E f − E i − ω) in the transition rate
of photoelectric effect. The state of the ejecting electron is a (spherical)
wavepacket moving outward from the local atom, whose average direction and
momentum distribution are determined by the momentum and polarization of the
photon. The small energy uncertainty of the photon will also be transferred to
the ejecting electron [83] .
    This microscopic
effect of ejecting electron is then amplified (e.g. by an avalanche process of
atoms) to form a macroscopic signal such as the shift of the pointer of a
measuring device. During the amplification process, the energy difference is
constantly increasing between the branch in which the photon is absorbed and
the branch in which the photon is not absorbed near each atom interacting with
the photon. This large energy difference will soon lead to the collapse of the
whole superposition into one of the local branches, and thus the photon is only
detected locally. Take the single photon detector avalanche photodiode as a
typical example [84] . Its energy consumption is sharply peaked in a very
short measuring interval. One type of avalanche photodiode operates at 10 5 cps and has a mean power dissipation of 4mW (Gao 2006a). This corresponds to an
energy consumption of about 2.5 × 10 11 eV per measuring interval 10 −5 s.
By using the collapse time formula Eq. (4.13), where the energy uncertainty is
∆E ≈ 2.5 × 10 11 eV , we find the collapse time is τ c ≈ 1.25
× 10 −10 s. This collapse time is much smaller than the measuring
interval.
    One important
point needs to be stressed here. Although a measured particle is detected
locally in a detector (e.g. the spatial size of its collapse state is in the
order of the size of an atom), its wave function does not necessarily undergo
the position collapse assumed in an ideal position measurement by standard
quantum mechanics, and especially, energy can be conserved during the
localization process