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 $\tau$. 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 $D$ after a negative measurement as the solution of Schr\"odinger's equation at time $\tau$ with instantaneously collapsed initial condition and homogeneous Dirichlet condition on the boundary of $D$. 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., $[-\pi,\pi]$. 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

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

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

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