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Optics and Spectroscopy

Short time propagation of a singular wave function: Some surprising results

Avi Marchewka, Er’el Granot, Z. Schuss

Published Aug 1, 2007• 10 citations
Publisher

Abstract

The Schrödinger evolution of an initially singular wave function was investigated. First it was shown that a wide range of physical problems can be described by initially singular wave function. Then it was demonstrated that outside the support of the initial wave function the time evolution is governed to leading order by the values of the wave function and its derivatives at the singular points. Short-time universality appears where it depends only on a single parameter—the value at the singular point (not even on its derivatives). It was also demonstrated that the short-time evolution in the presence of an absorptive potential is different than in the presence of a nonabsorptive one. Therefore, this dynamics can be harnessed to the determination whether a potential is absorptive or not simply by measuring only the transmitted particles density.

DOI
10.1134/s0030400x07080255

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The Schrödinger evolution of an initially singular wave function was investigated. First it was shown that a wide range of physical problems can be described by initially singular wave function. Then it was demonstrated that outside the support of the initial wave function the time evolution is governed to leading order by the values of the wave function and its derivatives at the singular points. Short-time universality appears where it depends only on a single parameter—the value at the singular point (not even on its derivatives). It was also demonstrated that the short-time evolution in the presence of an absorptive potential is different than in the presence of a nonabsorptive one. Therefore, this dynamics can be harnessed to the determination whether a potential is absorptive or not simply by measuring only the transmitted particles density.
We investigate the short-time dynamics of a delta-function potential barrier on an initially confined wave packet. There are mainly two conclusions: (A) At short times the probability density of the first particles that passed through the barrier is unaffected by it. (B) When the barrier is absorptive (i.e., its potential is imaginary) it affects the transmitted wave function at shorter times than a real potential barrier. Therefore, it is possible to distinguish between an imaginary and a real potential barrier by measuring its effect at short times only on the transmitting wave function.
The fall of a particle to the center of a singular potential U(r) is one of a few fundamental problems of quantum mechanics. Nonetheless, its solution is not complete yet. The known results just indicate that if U(r) decays fast enough at r tends to zero, the spectrum of the Schrodinger equation is not bounded from below. However, the wave functions of the problem are singular at r = 0 and do not admit the limiting transition to the wave function of the ground state. Therefore, the unboundedness of the spectrum is only the necessary condition. To prove that a quantum particle indeed can fall to the center, a wave function describing the fall should be obtained explicitly. This is done in the present paper. Specifically, an exact solution of the time-dependent Schrodinger equation describing the fall to the center is obtained and analyzed. The law describing the collapse to a single point of the region of the wave function localization is obtained explicitly, as well as the temporal dependences of the average kinetic and potential energy of the falling particle and its momentum. It is shown that the known necessary conditions for the particle to fall simultaneously are sufficient.
Discontinuous initial wave functions, or wave functions with a discontinuous derivative and a 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 Schr\"odinger current which vanishes on the boundary of the support of the wave function. This propagation gives rise to a unidirectional current at the boundary of the support. We use path integrals to define current and unidirectional current, and to provide 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 an initial wave function with compact support for cases of both a discontinuous derivative and a discontinuous wave function. We show that in the former case the probability propagated across the boundary of the support in time $\ensuremath{\Delta}t$ is $O(\ensuremath{\Delta}{t}^{3/2}),$ and the initial unidirectional current is $O(\ensuremath{\Delta}{t}^{1/2}).$ 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 $\ensuremath{\Delta}t$ is $O(\ensuremath{\Delta}{t}^{1/2}),$ and the initial unidirectional current is $O(\ensuremath{\Delta}{t}^{\ensuremath{-}1/2}).$ 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 $O(\ensuremath{\Delta}t).$ This gives a decay law.
The Schrödinger equation governs the probabilistic behavior of quantum particles through the wave function. Microlocal singularities denote regions with significantly high probability density or abrupt changes therein. By visualizing the probability distribution in time and space, we discern regions with higher probability density, indicative of potential microlocal singularities. These regions probably correspond to areas with a greater probability of particle presence. Such analysis aligns with Theorem 1, predicting characteristics of microlocal singularities of wave functions. Furthermore, Theorem 2 postulates that semiclassical path integrals along these singularities contribute significantly to solving the Schrödinger equation. Interpreting the temporal evolution of the probability density in the probability distribution visualization reveals the propagation of the particle over time. Regions of high density mean likely presence of particles at specific times, aligning with the predictions of Theorem 2. Consequently, the analysis of the contribution of high-density regions to the temporal evolution of the wave function resembles semi-classical path integral calculations. Thus, our findings demonstrate that visualization of probability distributions obtained from the numerical resolution of the Schrödinger equation allows a comprehensive interpretation of the behavior of quantum particles, consistent with the theorems.
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time-dependent Schrödinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper, we give a unified approach to determine the supershift property for the solution of the time-dependent one-dimensional Schrödinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Green’s function, but not its explicit form. With this efficient general technique, we are able to treat various potentials.
Superoscillating functions and supershifts appear naturally in weak measurements in physics. Their evolution as initial conditions in the time-dependent Schrödinger equation is an important and challenging problem in quantum mechanics and mathematical analysis. The concept that encodes the persistence of superoscillations during the evolution is the (more general) supershift property of the solution. In this paper, we give a unified approach to determine the supershift property for the solution of the time-dependent one-dimensional Schrödinger equation. The main advantage and novelty of our results is that they only require suitable estimates and regularity assumptions on the Green's function, but not its explicit form. With this efficient general technique, we are able to treat various potentials.