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Journal of Experimental and Theoretical Physics Letters

Quasiclassical Quantization of the Motion of a Particle in the Presence of a Drag Force Proportional to the Square of the Velocity

С. В. Сазонов

Published Aug 1, 2023• 6 citations
Publisher

Abstract

The quasiclassical one-dimensional motion of a particle in a medium, where the drag force is proportional to the square of the particle velocity, is considered using the Caldirola–Kanai approach. The coherent state of the particle in the presence of a constant conservative force in addition to the drag force is studied. It has been shown that the wave packet undergoes quantum extension to a certain limit, forming a steady propagating profile. Thus, the drag force suppresses the quantum properties of the particle, and the classical features become more pronounced in its motion with time. This property allows one to consider such a medium as a classical instrument continuously measuring the state of the particle. For this reason, the restriction of the spatial extension of the wavefunction can be interpreted as one of the manifestations of the quantum Zeno effect.

DOI
10.1134/s0021364023602257

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The quasiclassical one-dimensional motion of a particle in a medium, where the drag force is proportional to the square of the particle velocity, is considered using the Caldirola–Kanai approach. The coherent state of the particle in the presence of a constant conservative force in addition to the drag force is studied. It has been shown that the wave packet undergoes quantum extension to a certain limit, forming a steady propagating profile. Thus, the drag force suppresses the quantum properties of the particle, and the classical features become more pronounced in its motion with time. This property allows one to consider such a medium as a classical instrument continuously measuring the state of the particle. For this reason, the restriction of the spatial extension of the wavefunction can be interpreted as one of the manifestations of the quantum Zeno effect.
Abstract Based on the formalism of quasi-classical canonical quantization, the translational motion of a particle in a viscous medium is studied. The case is considered when, in addition to the resistance forces, an external conservative force acts on the particle. It is shown that due to the forces of viscous resistance of the medium, the initial coherent state of the particle does not decay. This behavior of the wave function is accompanied by a partial suppression of the quantum properties of the particle. In this case, a viscous medium should be considered as a classical device that continuously measures the state of a particle. Therefore, the stability of the coherent state of a particle can be interpreted as a manifestation of the quantum Zeno effect.
Abstract A version of the quasi-classical approach is proposed, which makes it possible to describe the straight-line motion of a micro-particle in a bulk dissipative medium. From the side of the medium, the particle is acted upon by the force of viscous friction and the drag force, which are proportional to the velocity and the square of the velocity, respectively. In addition, an external conservative force is applied to the particle. The Green’s function of the particle and the quasi-classical coherent state in the form of a three-dimensional localized wave packet are found. It is shown that the translational motion of the wave packet is accompanied by a monotonic increase in the uncertainties of the particle coordinates up to certain maximum asymptotic values. Due to the drag force, these asymptotic uncertainties contain information about the corresponding initial uncertainties in the coordinates of the micro-particle, about its initial velocity, as well as about the external conservative force.
Abstract On the basis of the canonical quantization procedure, the motion of a micro-particle in a viscous medium is considered, where the drag force consists of two components proportional to the first and second degrees of velocity, respectively. The wave function of a particle corresponding to its quantum coherent state is found using the quasi-classical approach. It is shown that due to the viscous of the medium, a static localized domain of the probability density is formed at relatively long times. It is shown that the parameters of this domain contain the information about the parameters of the particle on the input to medium.
Abstract A self-consistent quasi-classical generalization of the axiomatic quantization of the translational motion of a particle in a dissipative medium is proposed, which is modeled by a drag force F d , which depends in a power-law manner on the velocity v : F d ∼ v n , where n is an arbitrary real number. A detailed analysis of the quasi-classical coherent state is carried out. It is shown that under condition n < 3 the dissipative medium suppresses the quantum properties of particle motion, performing the function of a classical measuring device. Otherwise, there is no suppression of quantum properties.
Abstract Based on the quasi-classical approximation, a general approach is proposed for constructing non-stationary quantum states of a charged particle in a magnetic field, when the dissipative forces of viscous friction and drag, proportional to the velocity and the square of the velocity, respectively, are also significant. The corresponding quasi-classical Green’s function is found, with the help of which the squeezed and coherent states of the particle are studied. It is shown that the dissipation and a magnetic field suppress the quantum properties of the particle. This is especially true for the transverse motion with respect to the magnetic field. Over time, the coherent and squeezed states transform into the same static state, which is characterized by a zero uncertainty of the transverse coordinates and an uncertainty of the longitudinal coordinate, which contains information about the initial velocity of the particle.
In quantum mechanics an incoming particle wave packet with sufficient energy will undergo both transmission and reflection when encountering a barrier of lower energy, but in classical mechanics there is no reflection, only transmission. In this paper we seek to explain the disappearance of quantum-mechanical reflection in the quasiclassical limit, using the standard machinery of decoherence through environmental interaction. We consider two models. In the first, the incoming particle is classicalized by coupling to an environment and modeled using a standard master equation of Lindblad form with Lindblad operator proportional to position (the simplest version of quantum Brownian motion). We find, however, that suppression of reflection is achieved only for environmental interaction so strong that large fluctuations in momentum are generated, which blurs the distinction between incoming and reflected wave packets. This negative conclusion also holds for a complex potential which has similar implications for attempts to understand the suppression of the Zeno effect using the same mechanism (discussed in more detail elsewhere). A different master equation with Lindblad operator proportional to momentum is shown to be successful in suppressing reflection without large fluctuations but such a master equation is unphysical. We consider a second model in which the barrier is modeled quantum mechanically by a massive target particle coupled to an environment to maintain it in a quasiclassical state. This avoids the fluctuations problem since the incoming particle is not coupled to the environment directly. We find that reflection is significantly suppressed as long as the decoherence time scale of the target particle is much smaller than certain characteristic scattering time scales of the incoming particle, or equivalently, as long as the velocity fluctuations in the target are larger than the velocity of the incoming particle.