We consider multiple collisions of quantum wave packets in one dimension. The system under investigation consists of an impenetrable wall and two hard-core particles with very different masses. The lighter particle bounces between the heavier one and the wall. Both particles are initially represented by narrow Gaussian wave packets. A complete analytical solution of this problem is presented. The idea of the method used is to decompose the two-particle wave function into a continuous superposition of terms (channels), such that the multiple collisions within each channel do not lead to subsequent entanglement between the two particles. For each channel, the time evolution of the two-particle wave function is completely determined by the motion of the corresponding classical pointlike particles; therefore the whole quantum problem is reduced to a classical calculation. The calculation based on the above method reveals the following unexpected result: The entanglement between the two particles first increases with time due to the collisions, but then it begins to decrease, disappearing completely when the light particle becomes too slow to catch up with the heavy one.
We consider multiple collisions of quantum wave packets in one dimension. The system under investigation consists of an impenetrable wall and two hard-core particles with very different masses. The lighter particle bounces between the heavier one and the wall. Both particles are initially represented by narrow Gaussian wave packets. A complete analytical solution of this problem is presented. The idea of the method used is to decompose the two-particle wave function into a continuous superposition of terms (channels), such that the multiple collisions within each channel do not lead to subsequent entanglement between the two particles. For each channel, the time evolution of the two-particle wave function is completely determined by the motion of the corresponding classical pointlike particles; therefore the whole quantum problem is reduced to a classical calculation. The calculation based on the above method reveals the following unexpected result: The entanglement between the two particles first increases with time due to the collisions, but then it begins to decrease, disappearing completely when the light particle becomes too slow to catch up with the heavy one.
We consider time-dependent Gaussian wave packet solutions of the Schrödinger equation, with arbitrary initial central position, x 0 , and momentum, p 0 , for an otherwise free particle, but with an infinite wall at x = 0, so-called bouncing wave packets . We show how difference or mirror solutions of the form ψ( x , t ) − ψ(− x , t ) can, in this case, be normalized exactly, allowing for the evaluation of a number of time-dependent expectation values and other quantities in closed form. For example, we calculate ⟨ p 2 ⟩ t explicitly which illustrates how the free-particle kinetic (and hence total energy) is affected by the presence of the distant boundary. We also discuss the time dependence of the expectation values of position, ⟨ x ⟩ t , and momentum, ⟨ p ⟩ t , and their relation to the impulsive force during the `collision' with the wall. Finally, the x 0 , p 0 → 0 limit is shown to reduce a special case of a non-standard free-particle Gaussian solution. The addition of this example to the literature then expands of the relatively small number of Gaussian solutions to quantum mechanical problems with familiar classical analogs (free particle, uniform acceleration, harmonic oscillator, unstable oscillator, and uniform magnetic field) available in closed form.
We consider a simple one-dimensional quantum system consisting of a heavy and a light particle interacting via a point interaction. The initial state is chosen to be a product state, with the heavy particle described by a coherent superposition of two spatially separated wave packets with opposite momentum and the light particle localized in the region between the two wave packets. We characterize the asymptotic dynamics of the system in the limit of small mass ratio, with an explicit control of the error. We derive the corresponding reduced density matrix for the heavy particle and explicitly compute the (partial) decoherence effect for the heavy particle induced by the presence of the light one for a particular set up of the parameters.
A known limitation of time-dependent mean-field approaches is a lack of quantum tunneling for collective motions such as in sub-barrier fusion reactions. As a first step toward a solution, a time-dependent model is considered using a superposition of Gaussian wave packets, to describe the relative motion between two colliding nuclei, which may be simplified to a problem for one particle in one dimension. In this article, how the model describes the potential-barrier transmission is investigated by paying attention to the time evolution of the phase space distribution, which in particular reveals that the behavior of the free propagation of the incoming state is not trivial, depending on the number of superposed wave packets. Passage over the barrier can occur due to the high-momentum components in the incoming state corresponding to energies above the barrier height, which is, however, of classical nature and needs to be distinguished from the true quantum tunneling. Although a transmitted wave packet in some case may end up with an energy lower than the barrier, a difficulty is noticed in guaranteeing the energy conservation when the energies of different exit channels, e.g. of transmission and reflection, are individually measured. To overcome these issues for a description of quantum tunneling is still a challenging problem. This article mainly treats the same system with the same model as in the paper Hasegawa, Hagino and Tanimura (2020) [1]. However, we reach different conclusions.
On the basis of the full analytical solution of the overall unitary dynamics, the time evolution of entanglement is studied in a simple bipartite model system evolving unitarily from a pure initial state. The system consists of two particles in one spacial dimension bound by harmonic forces and having its free center of mass initially localized in space in a minimum uncertainty wave packet. The existence of such initial states in which the bound particles are not entangled is pointed out. The entanglement of the two particles is shown to be independent of the wavepacket mean momentum, and to increase monotonically in a time scale distinct from that of the spreading of the center of mass wavepacket.
The quantum measurement problems are revisited from a new perspective. One of the main ideas of this work is that the basic entities of our world are various types of particles, elementary or composite. It follows that each elementary process, hence each measurement process at its core, is a space–time, pointlike, event. Another key idea is that, when a microsystem [Formula: see text] gets into contact with the experimental device, factorization of [Formula: see text] rapidly fails and entangled mixed states appear. The wave functions for the microsystem-apparatus coupled systems for different measurement outcomes then lack overlapping space–time support. It means that the aftermath of each measurement is a single term in the sum: a “wave function collapse.” Our discussion leading to a diagonal density matrix, [Formula: see text] shows how the information encoded in the wave function [Formula: see text] gets transcribed, via entanglement with the experimental device and environment, into the relative frequencies [Formula: see text] for various experimental results [Formula: see text]. These results represent the first, significant steps towards filling in the logical gaps in the standard interpretation based on Born’s rule, and replacing it with a more natural one. Accepting objective reality of quantum fluctuations, independent of any experiments, and independently of human presence, one renounces the idea that in a fundamental, complete theory of Nature the result of each single experiment must necessarily be predictable. A few well-known puzzles such as the Schrödinger cat conundrum and the Einstein–Podolsky–Rosen (EPR) paradox are briefly reviewed: they can all be naturally explained away.