Understanding Quantum Physics: An Advanced Guide for the Perplexed

happens. When the duration of stay, t P , is longer, the
probability of the staying branch will increase more. Thus we have k ∝ t P .
    Thirdly, k is also
proportional to the energy uncertainty of the superposition of energy
eigenstates. First, from a dimensional analysis k should be proportional to an
energy term in order to cancel out the dimension of time. Next, the energy term
should be the energy uncertainty of the superposition defined in an appropriate
way according to the analysis of the last subsection. When the energy
uncertainty is zero, i.e., when the state is an energy eigenstate, no collapse
happens. When the energy uncertainty is not zero, collapse happens. Moreover,
the larger the energy uncertainty is, the larger the increase of the
probability of the staying branch for each random stay is, namely the larger k
is. Therefore, k will be proportional to the energy uncertainty of the
superposition. How to define the energy uncertainty then? Since k is invariant
under the swap of any two branches (P i , E i ) and (P j ,
E j ) according to Eq. (4.6), the most natural definition of the
energy uncertainty of a superposition of energy eigenstates is [76] :

    For the simplest
two-level system, we have

    It seems a little
counterintuitive that k contains the energy uncertainty term that relates to
the whole energy distribution. The puzzle is two-fold. First, this means that
the increase of the probability of the staying branch relates not to the energy
difference between the staying branch and all other branches, but to the energy
uncertainty of the whole state. This is reflected in the formula of ∆E in the
existence of the energy difference between any two branches, |E i − E j |
for any i and j. Next, the increase of the probability of the staying branch
relates also to the energy probability distribution that determines the energy
uncertainty. This is reflected in the formula of ∆E in the existence of P i P j .
In fact, these seemingly puzzling aspects are still understandable. The first
feature is required by the first rule of dynamical collapse that ensures energy
conservation at the ensemble level. This can be clearly seen from Eq. (4.6). If
the increase of the probability of the staying branch relates to the difference
between the energy of the staying branch and the average energy of all other
branches, then Eq. (4.6) will not hold true because the swap symmetry of k will
be violated, and as a result, the first rule of dynamical collapse will be
broken. The second feature can be understood as follows. In the picture of
random discontinuous motion, the probability distribution contains the
information of staying time distribution. An energy branch with small
probability means that the system jumps through it less frequently. Thus this
energy branch only makes a small contribution to the restriction of energy
change or the increase of the staying tendency. As a result, k or the increase
of the probability of the staying branch will relate not merely to energy
difference, but also to the energy probability distribution.
    Then after
omitting a coefficient in the order of unity, we can get the formula of k in the
first order:
    This is the second
rule of dynamical collapse. By inputting Eq. (4.12) into Eq. (4.9), we can
further get the collapse time formula:

    where E P = h/t P is the Planck energy, and ∆E is the energy uncertainty of the
initial state [77] .
    Here it is worth
pointing out that k must contain the first order term of ∆E. For the second
order or higher order term of ∆E will lead to much longer collapse time for
some common measurement situations, which contradicts experiments (Gao 2006a,
2006b). Besides, a similar analysis of the consistency with experiments may
also provide a further support for the energy-conserved collapse model in which
the collapse states are energy eigenstates. First of all, if the collapse
states are not energy eigenstates but momentum eigenstates, then the energy
uncertainty will be replaced by momentum uncertainty in the collapse time
formula Eq. (4.13), namely τ c ≈hE P /(∆pc) 2 . As a
result, the collapse time will be too short to be consistent with experiments
for some situations. For example, for the ground state of hydrogen atom the
collapse time will be about several days. Note that the second order or higher
order term of ∆p will also lead to much longer collapse time for some common
measurement situations, which contradicts