Breakdown of Ergodic Theorem in Bohmian Quantum Theory

Aharonov, Scully et-al 2004 wrote in Physica Scripta

"In the literature it is often stated that in the Bohmian
picture, what is measured in a single measurement is the
particle position, whereas the wavefunction can be
measured only indirectly. However, the measurement
process is described by the same Von Neumann model in
which the interaction of the pointer of the measurement
device with the measured particle is the result of an
interaction term in the Hamiltonian governing the
dynamics of their common wavefunction. It is well known
that for wavefunctions which are eigenstates of certain
Hamiltonians, the particle’s Bohmian position is constant
in time. However, one might hope that the act of
measurement itself, by introducing a perturbing potential,
somehow causes the time distribution of the position to
resemble the ensemble distribution (which by hypothesis
coincides with the wavefunction distribution).

In classical statistical mechanics, the ergodic hypothesis
tells us that we have two ways of measuring the probability
of a particle to be in a region—to measure the appropriate
density for many particles (averaging over a Boltzmann
ensemble), or to track a single particle over a long time and
calculate the proportion of the time it spends there to the
total. In fact, for some quantum systems we can do
something similar: we can either measure the probability to
be in a region by measuring the projection operator onto
the region, for a large number of identically prepared
particles (the usual ensemble average), or we can gently
measure this operator for a single particle over a long time
(protective measurement, which will be described below).
Since the protective measurement [4] is gentle (weak and
adiabatic), it hardly changes the wavefunction, and so any
time averages are trivial. Could the Bohmian particle
position play the part of the ‘‘microscopic details’’ of
statistical mechanics? ...

In a protective measurement, we start with an eigenwavefunction,
and apply a weak adiabatic interaction with
a ‘‘pointer’’, which acts for a long time. It has been shown
[5] that (in the adiabatic limit), the wavefunction is
unchanged by the measurement, and it can therefore be
measured (one region at a time) for a single particle. It was
shown already in [3], for a particular Hamiltonian, that the
ergodic hypothesis fails for this kind of measurement ..."


Time and Ensemble Averages in Bohmian Mechanics
Yakir Aharonov 1,2, Noam Erez 1,3, and Marlan O. Scully 3,4
Physica Scripta. Vol. 69, 81–83, 2004