787 research outputs found
Continuous Observations and the Wave Function Collapse
We propose to modify the collapse axiom of quantum measurement theory by
replacing the instantaneous with a continuous collapse of the wave function in
finite time . We apply it to coordinate measurement of a free quantum
particle that is initially confined to a domain D\subset\rR^d and is observed
continuously by illuminating \rR^d-D. The continuous collapse axiom (CCA)
defines the post-measurement wave function (PMWF)in after a negative
measurement as the solution of Schr\"odinger's equation at time with
instantaneously collapsed initial condition and homogeneous Dirichlet condition
on the boundary of . The CCA applies to all cases that exhibit the Zeno
effect. It rids quantum mechanics of the unphysical artifacts caused by
instantaneous collapse and introduces no new artifacts.Comment: 12 pages and 2 figure
Feynman Integrals with Absorbing Boundaries
We propose a formulation of an absorbing boundary for a quantum particle. The
formulation is based on a Feynman-type integral over trajectories that are
confined to the non-absorbing region. Trajectories that reach the absorbing
wall are discounted from the population of the surviving trajectories with a
certain weighting factor. Under the assumption that absorbed trajectories do
not interfere with the surviving trajectories, we obtain a time dependent
absorption law. Two examples are worked out.Comment: 4 pages, revte
Stochastic resonance with applied and induced fields: the case of voltage-gated ion channels
We consider a charged Brownian particle in an asymmetric bistable
electrostatic potential biased by an externally applied or induced time
periodic electric field. While the amplitude of the applied field is
independent of frequency, that of the one induced by a magnetic field is.
Borrowing from protein channel terminology, we define the open probability as
the relative time the Brownian particle spends on a prescribed side of the
potential barrier. We show that while there is no peak in the open probability
as the frequency of the applied field and the bias (depolarization) of the
potential are varied, there is a narrow range of low frequencies of the induced
field and a narrow range of the low bias of the potential where the open
probability peaks. This manifestation of stochastic resonance is consistent
with experimental results on the voltage gated Iks and KCNQ1 potassium channels
of biological membranes and on cardiac myocytes.Comment: 17 figure
Brownian Motion in Dire Straits
The passage of Brownian motion through a bottleneck in a bounded domain is a
rare event and the mean time for such passage increases indefinitely as the
bottleneck's radius shrinks to zero. Its calculation reveals the effect of
geometry and smoothness on the flux through the bottleneck. We find new
behavior of the narrow escape time through bottlenecks in planar and spatial
domains and on a surface. Some applications are discussed.Comment: 32 pages, 14 figure
Stochastic model of a pension plan
Structuring a viable pension plan is a problem that arises in the study of
financial contracts pricing and bears special importance these days.
Deterministic pension models often rely on projections that are based on
several assumptions concerning the "average" long-time behavior of the stock
market. Our aim here is to examine some of the popular "average" assumptions in
a more realistic setting of a stochastic model. Thus, we examine the contention
that investment in the stock market is similar to gambling in a casino, while
purchasing companies, after due diligence, is safer under the premise that
acting as a holding company that wholly owns other companies avoids some of the
stock market risks. We show that the stock market index faithfully reflects its
companies' profits at the time of their publication. We compare the shifted
historical dynamics of the S\&P500's aggregated financial earnings to its
value, and find a high degree of correlation. We conclude that there is no
benefit to a pension fund in wholly owning a super trust. We verify, by
examining historical data, that stock earnings follow an exponential
(geometric) Brownian motion and estimate its parameters. The robustness of this
model is examined by an estimate of a pensioner's accumulated assets over a
saving period. We also estimate the survival probability and mean survival time
of the accumulated individual fund with pension consumption over the residual
life of the pensioner.Comment: 41 pages, 19 figure
Wave function collapse implies divergence of average displacement
We show that propagating a truncated discontinuous wave function by
Schr\"odinger's equation, as asserted by the collapse axiom, gives rise to
non-existence of the average displacement of the particle on the line. It also
implies that there is no Zeno effect. On the other hand, if the truncation is
done so that the reduced wave function is continuous, the average coordinate is
finite and there is a Zeno effect. Therefore the collapse axiom of measurement
needs to be revised
Nonlinear Filtering with Optimal MTLL
We consider the problem of nonlinear filtering of one-dimensional diffusions
from noisy measurements. The filter is said to lose lock if the estimation
error exits a prescribed region. In the case of phase estimation this region is
one period of the phase measurement function, e.g., . We show that
in the limit of small noise the causal filter that maximizes the mean time to
loose lock is Bellman's minimum noise energy filter
The case of escape probability as linear in short time
We derive rigorously the short-time escape probability of a quantum particle
from its compactly supported initial state, which has a discontinuous
derivative at the boundary of the support. We show that this probability is
liner in time, which seems to be a new result. The novelty of our calculation
is the inclusion of the boundary layer of the propagated wave function formed
outside the initial support. This result has applications to the decay law of
the particle, to the Zeno behavior, quantum absorption, time of arrival,
quantum measurements, and more, as will be discussed separately.Comment: 7 pages, 1 figures.Spelling and typo corrected. The appendix was
removed and for further mathematical detail A. Friedman's book was referred
to. Both sections (introduction and summary) have been extended (as the
referee suggested). Accepted for publication at Physics Letters
On Recovering the Shape of a Domain from the Trace of the Heat Kernel
The problem of recovering geometric properties of a domain from the trace of
the heat kernel for an initial-boundary value problem arises in NMR microscopy
and other applications. It is similar to the problem of ``hearing the shape of
a drum'', for which a Poisson type summation formula relates geometric
properties of the domain to the eigenvalues of the Dirichlet or Neumann problem
for the Laplace equation. It is well known that the area, circumference, and
the number of holes in a planar domain can be recovered from the short time
asymptotics of the solution of the initial-boundary value problem for the heat
equation. It is also known that the length spectrum of closed billiard ball
trajectories in the domain can be recovered from the eigenvalues or from the
solution of the wave equation. This spectrum can also be recovered from the
heat kernel for a compact manifold without boundary. We show that for a planar
domain with boundary, the length spectrum can be recovered directly from the
short time expansion of the trace of the heat kernel. The results can be
extended to higher dimensions in a straightforward manner
A subluminous Schroedinger equation
The standard derivation of Schroedinger's equation from a Lorentz-invariant
Feynman path integral consists in taking first the limit of infinite speed of
light and then the limit of short time slice. In this order of limits the light
cone of the path integral disappears, giving rise to an instantaneous spread of
the wave function to the entire space. We ascribe the failure of Schroedinger's
equation to retain the light cone of the path integral to the very nature of
the limiting process: it is a regular expansion of a singular approximation
problem, because the boundary conditions of the path integral on the light cone
are lost in this limit. We propose a distinguished limit, which produces an
intermediate model between non-relativistic and relativistic quantum mechanics:
it produces Schroedinger's equation and preserves the zero boundary conditions
on and outside the original light cone of the path integral. These boundary
conditions relieve the Schroedinger equation of several annoying, seemingly
unrelated unphysical artifacts, including non-analytic wave functions,
spontaneous appearance of discontinuities, non-existence of moments when the
initial wave function has a jump discontinuity (e.g., a collapsed wave function
after a measurement), the EPR paradox, and so on. The practical implications of
the present formulation are yet to be seen
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