arXiv subjects

A. Marchewka

Publications and source records attributed to A. Marchewka.

8 recordsLinked to original sources

arXiv2011

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 $τ$. 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 Schrödinger's equation at time $τ$ 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.

quant-ph↗

arXiv2005

Wave function collapse implies divergence of average displacement

We show that propagating a truncated discontinuous wave function by Schrödinger'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.

quant-ph↗

arXiv2005

The survival probability of diffusion with killing

We present a general framework to study the effect of killing sources on moving particles, trafficking inside biological cells. We are merely concerned with the case of spine-dendrite communication, where the number of calcium ions, modeled as random particles is regulated across the spine microstructure by pumps, which play the killing role. In particular, we study here the survival probability of ions in such environment and we present a general theory to compute the ratio of the number of absorbed particles at specific location to the number of killed particles during their sojourn inside a domain. In the case of a dendritic spine, the ratio is computed in terms of the survival probability of a stochastic trajectory in a one dimensional approximation. We show that the ratio depends on the distribution of killing sources. The biological conclusion follows: changing the position of the pumps is enough to regulate the calcium ions and thus the spine-dendrite communication.

math-ph↗

arXiv2001

Total Absorption in Finite Time in an $iδ$ Potential

We consider the evolution of Green's function of the one-dimensional Schrödinger equation in the presence of the complex potential $-ikδ(x)$. Our result is the construction of an explicit time-dependent solution which we use to calculate the time-dependent survival probability of a quantum particle. The survival probability decays to zero in finite time, which means that the complex delta potential well is a total absorber for quantum particles. This potential can be interpreted as a killing measure with infinite killing rate concentrated at the origin.

quant-ph↗

arXiv1999

Measurement as Absorption of Feynman Trajectories: Collapse of the Wave Function Can be Avoided

We define a measuring device (detector) of the coordinate of quantum particle as an absorbing wall that cuts off the particle's wave function. The wave function in the presence of such detector vanishes on the detector. The trace the absorbed particles leave on the detector is identifies as the absorption current density on the detector. This density is calculated from the solution of Schrödinger's equation with a reflecting boundary at the detector. This current density is not the usual Schrödinger current density. We define the probability distribution of the time of arrival to a detector in terms of the absorption current density. We define coordinate measurement by an absorbing wall in terms of 4 postulates. We postulate, among others, that a quantum particle has a trajectory. In the resulting theory the quantum mechanical collapse of the wave function is replaced with the usual collapse of the probability distribution after observation. Two examples are presented, that of the slit experiment and the slit experiment with absorbing boundaries to measure time of arrival. A calculation is given of the two dimensional probability density function of a free particle from the measurement of the absorption current on two planes.

quant-ph↗

arXiv1999

Feynman Integral Approach to Absorption in Quantum Mechanics

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 by the absorbing boundary. Trajectories that reach the absorbing wall are instantaneously terminated and their probability is discounted from the population of the surviving trajectories. This gives rise to a unidirectional absorption current at the boundary. We calculate the survival probability as a function of time. Several modes of absorption are derived from our formalism: total absorption, absorption that depends on energy levels, and absorption of non-interacting particles. Several applications are given: the slit experiment with an absorbing screen and with absorbing lateral walls, and one dimensional particle between two absorbing walls. The survival probability of a particle between absorbing walls exhibits decay with beats.

quant-ph↗

arXiv1999

A Path Intergal Approach to Current

Discontinuous initial wave functions or wave functions with discontintuous derivative and with bounded support arise in a natural way in various situations in physics, in particular in measurement theory. The propagation of such initial wave functions is not well described by the Schrödinger current which vanishes on the boundary of the support of the wave function. This propagation gives rise to a uni-directional current at the boundary of the support. We use path integrals to define current and uni-directional current and give a direct derivation of the expression for current from the path integral formulation for both diffusion and quantum mechanics. Furthermore, we give an explicit asymptotic expression for the short time propagation of initial wave function with compact support for both the cases of discontinuous derivative and discontinuous wave function. We show that in the former case the probability propagated across the boundary of the support in time $Δt$ is $O(Δt^{3/2})$ and the initial uni-directional current is $O(Δt^{1/2})$. This recovers the Zeno effect for continuous detection of a particle in a given domain. For the latter case the probability propagated across the boundary of the support in time $Δt$ is $O(Δt^{1/2})$ and the initial uni-directional current is $O(Δt^{-1/2})$. This is an anti-Zeno effect. However, the probability propagated across a point located at a finite distance from the boundary of the support is $O(Δt)$. This gives a decay law.

quant-ph↗

arXiv1997

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.

quant-ph↗