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 ∆t is O ∆t 3/2 and the initial uni-directional current is O ∆t 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 ∆t is O ∆t 1/2 and the initial uni-directional current is O ∆t −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 (∆t). This gives a decay law.

A general solution to the "shutter" problem is presented. The propagation of an arbitrary initially bounded wavefunction is investigated, and the general solution for any such function is formulated. It is shown that the exact solution can be written as an expression that depends only on the values of the function (and its derivatives) at the boundaries. In particular, it is shown that at short times ( h / 2 2 mx t << , where x is the distance to the boundaries) the wavefunction propagation depends only on the wavefunction's values (or its derivatives) at the boundaries of the region. Finally, we generalize these findings to a non-singular wavefunction (i.e., for wavepackets with finite-width boundaries) and suggest an experimental verification.

We propose a formulation of an absorbing boundary for a quantum particle. The formulation is based on a Feynmantype 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.PACS numbers: 02.50.-r, 05.40.+j, 05.60.+w.The purpose of this letter is to propose a Feynman-type integral to describe absorption of particles in a surface bounding a domain. The need for such description arises for example in scattering theory, in the description of a photographic plate, in the double slit experiment, in neutron optics, and so on. The optical model [1,2] is often used to describe absorption in quantum systems. This model is based on analogy with electro-magnetic wave theory. It is not obvious that the methods of describing absorption in Maxwell's equations carry over to quantum mechanics because the wave function of a particle does not interact with the medium the way an electromagnetic wave does. In particular, in classical quantum theory, unlike in electromagnetic theory, the wave function does not transfer energy to the medium.The main tenet of our Feynman-type integral description of absorption is that trajectories that propagate into the absorbing surface for the first time are considered to be instantaneously absorbed and are therefore terminated at that surface. The population of the surviving trajectories is therefore discounted by the probability of the absorbed trajectories at each time step.The process of discounting can be explained as follows. In general, if the trajectories are partitioned into two subsets, the part of the wave function obtained from the Feynman integral over one cannot be used to calculate the probability of this subset, due to interference between the wave functions of the two subsets. However, in the physical situation under consideration, such a calculation may be justified. Our procedure, in effect, assumes a partition of all the possible trajectories at any given time interval [t, t+ ∆t] into two classes. One is a class of bounded trajectories that have not reached the surface by time t + ∆t and remain in the domain, and the other is a class of trajectories that hit the surface for the first time in the interval [t, t+ ∆t]. We assume that the part of the wave function obtained from the Feynman integral over trajectories that hit the surface in this time interval no longer interferes with the part of the wave function obtained from the Feynman integral over the class of bounded trajectories in a significant way. That is, the interference is terminated at this point so that the general population of trajectories can be discounted by the probability of the terminated trajectories. This assumption makes it possible to calculate separately the probability of the absorbed ...

The problem of diffusion with absorption and trapping sites arises in the theory of molecular signaling inside and on the membranes of biological cells. In particular, this problem arises in 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 role of killing sites, while the end of the dendritic shaft is an absorbing boundary. We develop a general mathematical framework for diffusion in the presence of absorption and killing sites and apply it to the computation of the time-dependent survival probability of ions. We also compute the ratio of the number of absorbed particles at a specific location to the number of killed particles. We show that the ratio depends on the distribution of killing sites. The biological consequence is that the position of the pumps regulates the fraction of calcium ions that reach the dendrite.

Although a simple spring/mass system damped by a friction force of constant magnitude shares many of the characteristics of the simple and damped harmonic oscillators, its solution is not presented in most texts. Closed form solutions for the turning and stopping points can be found using an energy-based approach. A dynamical approach leads to a closed form solution for the position of the mass as a function of time. The main result is that the amplitude of the oscillator damped by a constant magnitude friction force decreases by a constant amount each swing and the motion dies out after a finite time. We present these two solutions and suggest ways that the system can be used in the classroom.