Physical Review A

Quantization of motion in a velocity-dependent field: The v2 case

F. Negro, Angelo Tartaglia

Published Apr 1, 1981• 24 citations

Abstract

The case of the quantum unidimensional motion of a point particle subject to a viscous force proportional to the square of the speed of the particle is treated. Once a suitable Lagrangian has been chosen, the canonical quantization procedure is applied leading to a Schr\"odinger equation for quadratic friction and an arbitrary potential. The cases of an exponentially decreasing potential plus a linearly increasing term is explicitly solved.

DOI: 10.1103/physreva.23.1591

Related papers

Phase space Schrödinger equation and quadratic friction

H. Dessano, G. X. A. Petronilo, R. G. G. Amorim +1 more

A general Lagrangian for a point particle that moves across a viscous field quadratically dependent on the velocity is found. Afterward, the system is quantized via the Weyl quantization rule, and an explicit expression for the wave function in phase space is found.

Journal of Mathematical PhysicsFeb 1, 20221 citation

Quantization of a particle with a force quadratic in the velocity

Catherine Stuckens, Donald H. Kobe

A particle with a nonconservative force quadratic in the velocity is described by an appropriate Lagrangian. The particle can be canonically quantized, but ambiguities concerning the ordering of operators arise. These ambiguities are eliminated if the Feynman path-integral procedure is used to quantize the particle.

Physical Review ANov 1, 198621 citations

Quantization of Dissipative Systems with Friction Linear in Velocity

U. Das, Subrata K. Ghosh, Pranab Sarkar +1 more

It is shown that a dissipative system with friction linear in velocity can be analytically represented by two different Lagrangians which are harmless at the classical level. But they lead to different quantum mechanical systems. This is explicitly demonstrated by using both canonical quantization procedure as well as path integral method.

Physica ScriptaJan 1, 20056 citations

Embedding of velocity-power forces into a class of energy-dependent potentials and their quantization

J. Geicke

Physical Review AFeb 1, 19962 citations

Quantization of a particle with a force quadratic in the velocity

J. Sá Borges, L.N. Epele, Huner Fanchiotti A +2 more

The quantization of a system subject to a force quadratic in the velocity and position dependence is carried out in the phase-space path-integral framework. The resulting Hamiltonian coincides with that obtained by using the Weyl-ordering canonical prescription.

Physical Review ASep 1, 198821 citations

On the Space-Time Formulation of Non-Relativistic Quantum Mechanics

Izuru Fujiwara

Progress of Theoretical PhysicsJun 1, 195916 citations

Physical Review A

Quantization of motion in a velocity-dependent field: The v2 case

F. Negro, Angelo Tartaglia

Published Apr 1, 1981• 24 citations

Publisher

Abstract

The case of the quantum unidimensional motion of a point particle subject to a viscous force proportional to the square of the speed of the particle is treated. Once a suitable Lagrangian has been chosen, the canonical quantization procedure is applied leading to a Schr"odinger equation for quadratic friction and an arbitrary potential. The cases of an exponentially decreasing potential plus a linearly increasing term is explicitly solved.

DOI: 10.1103/physreva.23.1591

Related papers

Phase space Schrödinger equation and quadratic friction

H. Dessano, G. X. A. Petronilo, R. G. G. Amorim +1 more

Journal of Mathematical PhysicsFeb 1, 20221 citation

Quantization of a particle with a force quadratic in the velocity

Catherine Stuckens, Donald H. Kobe

Physical Review ANov 1, 198621 citations

Quantization of Dissipative Systems with Friction Linear in Velocity

U. Das, Subrata K. Ghosh, Pranab Sarkar +1 more

Physica ScriptaJan 1, 20056 citations

Embedding of velocity-power forces into a class of energy-dependent potentials and their quantization

J. Geicke

Using the classical trajectory we quantize the motion of a pointlike particle of mass m and energy 1/2${\mathit{mx}\mathit{\ifmmode \dot{}\else \.{}\fi{}}}_{0}^{2}$ under the influence of the force ${\mathit{F}}_{\mathit{n}}$ =-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}}$, \ensuremath{\alpha}\ensuremath{\gtrsim}0, n=1,2,.... Such forces are frictional only for x\ifmmode \dot{}\else \.{}\fi{}\ensuremath{\gtrsim}0 and antifrictional for x\ifmmode \dot{}\else \.{}\fi{}0 and can be simulated through the gradient of a one-parameter class of energy-dependent potentials ${\mathit{V}}_{\mathit{n}}$ (x;x${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}_{0}$), which increase monotonically in x. This permits one to construct a class of Lagrange and Hamilton functions that do not contain any integrating functions. We then proceed to quantization and study the quantum scattering problem assuming that ${\mathit{F}}_{\mathit{n}}$ acts on a finite interval of the x axis only. The stationary problem can be solved analytically for all n, and quantitative examples are shown for n=2. The time-dependent scattering problem is discussed with emphasis on Ehrenfest's theorems and on linear damping n=1. When the quantum-mechanical probability for reflection is small the method provides an approximation to the motion of a wave packet through a dissipative medium, with velocity-power forces ${\mathit{F}}_{\mathit{dn}}$ =-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}\mathrm{\ensuremath{-}}1}$x\ifmmode \dot{}\else \.{}\fi{}. \textcopyright{} 1996 The American Physical Society.Using the classical trajectory we quantize the motion of a pointlike particle of mass m and energy 1/2${\mathit{mx}\mathit{\ifmmode \dot{}\else \.{}\fi{}}}_{0}^{2}$ under the influence of the force ${\mathit{F}}_{\mathit{n}}$=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}}$, \ensuremath{\alpha}\ensuremath{\gtrsim}0, n=1,2,.... Such forces are frictional only for x\ifmmode \dot{}\else \.{}\fi{}\ensuremath{\gtrsim}0 and antifrictional for x\ifmmode \dot{}\else \.{}\fi{}0 and can be simulated through the gradient of a one-parameter class of energy-dependent potentials ${\mathit{V}}_{\mathit{n}}$(x;x${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}_{0}$), which increase monotonically in x. This permits one to construct a class of Lagrange and Hamilton functions that do not contain any integrating functions. We then proceed to quantization and study the quantum scattering problem assuming that ${\mathit{F}}_{\mathit{n}}$ acts on a finite interval of the x axis only. The stationary problem can be solved analytically for all n, and quantitative examples are shown for n=2. The time-dependent scattering problem is discussed with emphasis on Ehrenfest's theorems and on linear damping n=1. When the quantum-mechanical probability for reflection is small the method provides an approximation to the motion of a wave packet through a dissipative medium, with velocity-power forces ${\mathit{F}}_{\mathit{dn}}$=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}\mathrm{\ensuremath{-}}1}$x\ifmmode \dot{}\else \.{}\fi{}. \textcopyright{} 1996 The American Physical Society.

Physical Review AFeb 1, 19962 citations

Quantization of a particle with a force quadratic in the velocity

J. Sá Borges, L.N. Epele, Huner Fanchiotti A +2 more

Physical Review ASep 1, 198821 citations

On the Space-Time Formulation of Non-Relativistic Quantum Mechanics

Izuru Fujiwara

For the case of the quantization of the usual non-relativistic classical Lagrangian function quadratic in the velocity the validity is demonstrated of the non-canonical space-time formulation of quantum mechanics proposed recently by the author, which aims to evaluate, without appealing to the Schrödinger equation, the transformation function K(x,t′;y,t′) in the space representation on the basis of the composition rule K(x, t″; y, t′) = ∫K(x, t″; z, t)dzK(z, t; y, t′) (1) coupled with the supposition that it is approximated to zeroth order in the quantum of action h by the so-called semi-classical kernel Ko(x, t″; y, t′) = [(i/h)∂2S/∂x∂y]1/2 exp [(i/ℏ) S(x,t″; y, t′] (2) written in terms of the classical action S(x, t″; y, t′) alone. In the first place the action function corresponding to the above Lagrangian is expanded in power of the interval of time T = t″ − t′. Then the deviation of the semi-classical kernel (2) from the unitary transformation function is shown to be of the third order in T, and the corresponding correction term is evaluated by solving the integral equation (1). It is also shown that the semi-classical kernel is unitary for a free motion of a particle with its mass being a function in the space coordinate.For the case of the quantization of the usual non-relativistic classical Lagrangian function quadratic in the velocity the validity is demonstrated of the non-canonical space-time formulation of quantum mechanics proposed recently by the author, which aims to evaluate, without appealing to the Schrödinger equation, the transformation function K(x,t′;y,t′) in the space representation on the basis of the composition rule K(x, t″; y, t′) = ∫K(x, t″; z, t)dzK(z, t; y, t′) (1) coupled with the supposition that it is approximated to zeroth order in the quantum of action h by the so-called semi-classical kernel Ko(x, t″; y, t′) = [(i/h)∂2S/∂x∂y]1/2 exp [(i/ℏ) S(x,t″; y, t′] (2) written in terms of the classical action S(x, t″; y, t′) alone. In the first place the action function corresponding to the above Lagrangian is expanded in power of the interval of time T = t″ − t′. Then the deviation of the semi-classical kernel (2) from the unitary transformation function is shown to be of the third order in T, and the corresponding correction term is evaluated by solving the integral equation (1). It is also shown that the semi-classical kernel is unitary for a free motion of a particle with its mass being a function in the space coordinate.

Progress of Theoretical PhysicsJun 1, 195916 citations

Using the classical trajectory we quantize the motion of a pointlike particle of mass m and energy 1/2${\mathit{mx}\mathit{\ifmmode \dot{}\else \.{}\fi{}}}_{0}^{2}$ under the influence of the force

{\mathit{F}}_{\mathit{n}}

=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}}$, \ensuremath{\alpha}\ensuremath{\gtrsim}0, n=1,2,.... Such forces are frictional only for x\ifmmode \dot{}\else \.{}\fi{}\ensuremath{\gtrsim}0 and antifrictional for x\ifmmode \dot{}\else \.{}\fi{}0 and can be simulated through the gradient of a one-parameter class of energy-dependent potentials

{\mathit{V}}_{\mathit{n}}

(x;x${\mathrm{\ifmmode \dot{}\else \.{}\fi{}}}_{0}$), which increase monotonically in x. This permits one to construct a class of Lagrange and Hamilton functions that do not contain any integrating functions. We then proceed to quantization and study the quantum scattering problem assuming that

{\mathit{F}}_{\mathit{n}}

acts on a finite interval of the x axis only. The stationary problem can be solved analytically for all n, and quantitative examples are shown for n=2. The time-dependent scattering problem is discussed with emphasis on Ehrenfest's theorems and on linear damping n=1. When the quantum-mechanical probability for reflection is small the method provides an approximation to the motion of a wave packet through a dissipative medium, with velocity-power forces

{\mathit{F}}_{\mathit{dn}}

=-\ensuremath{\alpha}\ensuremath{\Vert}x\ifmmode \dot{}\else \.{}\fi{}${\mathrm{\ensuremath{\Vert}}}^{\mathit{n}\mathrm{\ensuremath{-}}1}$x\ifmmode \dot{}\else \.{}\fi{}. \textcopyright{} 1996 The American Physical Society.