Journal of Modern Physics

On the Quantization of One-Dimensional Conservative Systems with Variable mass

Gustavo V. López

Published Jan 1, 2012• 3 citations

Abstract

The Hamiltonian associated to the mass variable system is constructed from first principles through finding a constant of motion of the system. A comparison is made of the classical motion of a body with its mass position depending in the (x,v) space and (x,p) space which are defined by the constant of motion and the Hamiltonian, for a particular model of mass variation. As one could expected, these motion looks different on these spaces. The quantization of the harmonic oscillator with this mass variation is done, and a comparison is made by using the usual Hamiltonian approach with the proposed quantization of the constant of motion approach. This comparison is done at first order in perturbation theory, and one sees a difference between both approaches which can, in principle, be measured.

DOI: 10.4236/jmp.2012.38102

Related papers

Quantization of the 1-D harmonic oscillator with variable mass using the operators $\hat v$ and $\hat p$

Gustavo V. López, Eric M. Reynaga

arXiv:1609.08383Sep 27, 20160 citations

Quantization of the 1-D Forced Harmonic Oscillator in the Space (x, v)

Gustavo V. López, Omar J. P. Bravo

Journal of Modern PhysicsJan 1, 20211 citation

Quantization of a one-dimensional time-dependent periodic system with Hamiltonian and constants of motion approaches

Gustavo V. López

For a particle moving in a one-dimensional space an under a periodic external force, its quantization is study using the Hamiltonian (generalized linear momentum quantization) and constant of motion (velocity quantization) approaches. it is shown a great difference on the quantization of both approaches and the ambiguities arisen by using the quantization on the constants of motion.

arXiv:quant-ph/0608149Aug 18, 20060 citations

Quantization of the 1-D forced harmonic oscillator in the space ($x, v$)

Gustavo V. López, Omar J. P. Bravo

arXiv:2005.00344Apr 29, 20200 citations

Ambiguities in Quantizing a Classical System

Ian H. Redmount, Wai-Mo Suen, K. Young

arXiv:gr-qc/9904042Apr 19, 19991 citation

Study about Forced Harmonic Oscillator in the Space (x, v)

Gustavo V. López, Omar J. P. Bravo

Book Publisher International (a part of SCIENCEDOMAIN International)Sep 1, 20231 citation

Journal of Modern Physics

On the Quantization of One-Dimensional Conservative Systems with Variable mass

Gustavo V. López

Published Jan 1, 2012• 3 citations

Publisher

Abstract

DOI: 10.4236/jmp.2012.38102

Related papers

Quantization of the 1-D harmonic oscillator with variable mass using the operators $\hat v$ and $\hat p$ Quantization of the 1-D harmonic oscillator with variable mass using the operators $\hat v$ and $\hat p$

Gustavo V. López, Eric M. Reynaga

For the 1-D harmonic oscillator with position depending variable mass, a Hamiltonian and constant of motion are given through a consistent approach. Then, the quantization of this system is carried out using the operator $\hat p$ , for the Hamiltonian, and the operator $\hat v$ for the constant of motion. We find that the result of both quantizations brings about different quantum dynamics.For the 1-D harmonic oscillator with position depending variable mass, a Hamiltonian and constant of motion are given through a consistent approach. Then, the quantization of this system is carried out using the operator $\hat p$, for the Hamiltonian, and the operator $\hat v$ for the constant of motion. We find that the result of both quantizations brings about different quantum dynamics.

arXiv:1609.08383Sep 27, 20160 citations

Quantization of the 1-D Forced Harmonic Oscillator in the Space (x, v)

Gustavo V. López, Omar J. P. Bravo

The quantization of the forced harmonic oscillator is studied with the quantum variable (x, v∧), with the commutation relation , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion K (x, v, t) of the classical system, The comparison with the quantization in the space (x, p) is done with the usual Schrödinger’s equation for the Hamiltonian H(x, p, t), and with the commutation relation . It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (x, v∧) quantization has fewer oscillations than the (x, p∧) quantization, the average energy of the system is higher in (x, p∧) quantization than on the (x, v∧) quantization, and the Boltzmann-Shannon entropy on the (x, p∧) quantization is higher than on the (x, v∧) quantization.The quantization of the forced harmonic oscillator is studied with the quantum variable (x, v∧), with the commutation relation , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion K (x, v, t) of the classical system, The comparison with the quantization in the space (x, p) is done with the usual Schrödinger’s equation for the Hamiltonian H(x, p, t), and with the commutation relation . It is found that for the non-resonant case, both forms of quantization bring about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (x, v∧) quantization has fewer oscillations than the (x, p∧) quantization, the average energy of the system is higher in (x, p∧) quantization than on the (x, v∧) quantization, and the Boltzmann-Shannon entropy on the (x, p∧) quantization is higher than on the (x, v∧) quantization.

Journal of Modern PhysicsJan 1, 20211 citation

Quantization of a one-dimensional time-dependent periodic system with Hamiltonian and constants of motion approaches

Gustavo V. López

arXiv:quant-ph/0608149Aug 18, 20060 citations

Quantization of the 1-D forced harmonic oscillator in the space ( $x, v$ )Quantization of the 1-D forced harmonic oscillator in the space ($x, v$)

Gustavo V. López, Omar J. P. Bravo

The quantization of the forced harmonic oscillator is studied with the quantum variable ( $x,\hat v$ ), with the commutation relation $[x,\hat v]=i\hbar/m$ , and using a Shrödinger's like equation on these variable, and associating a linear operator to a constant of motion $K(x,v,t)$ of the classical system, The comparison with the quantization in the space ( $x,p$ ) is done with the usual Schrödinger's equation for the Hamiltonian $H(x,p,t)$ , and with the commutation relation $[x,\hat p]=i\hbar$ . It is found that for the non resonant case, both forms of quantization brings about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the ( $x,\hat v$ ) quantization has less oscillations than the ( $x,\hat p$ ) quantization, the average energy of the system is higher in ( $x,\hat p$ ) quantization than on the $(x,\hat v$ ) quantization, and the Boltzmann-Shannon entropy on the ( $x,\hat p$ ) quantization is higher than on the ( $x,\hat v$ ) quantization.The quantization of the forced harmonic oscillator is studied with the quantum variable ($x,\hat v$), with the commutation relation $[x,\hat v]=i\hbar/m$, and using a Shrödinger's like equation on these variable, and associating a linear operator to a constant of motion $K(x,v,t)$ of the classical system, The comparison with the quantization in the space ($x,p$) is done with the usual Schrödinger's equation for the Hamiltonian $H(x,p,t)$, and with the commutation relation $[x,\hat p]=i\hbar$. It is found that for the non resonant case, both forms of quantization brings about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the ($x,\hat v$) quantization has less oscillations than the ($x,\hat p$) quantization, the average energy of the system is higher in ($x,\hat p$) quantization than on the $(x,\hat v$) quantization, and the Boltzmann-Shannon entropy on the ($x,\hat p$) quantization is higher than on the ($x,\hat v$) quantization.

arXiv:2005.00344Apr 29, 20200 citations

Ambiguities in Quantizing a Classical System

Ian H. Redmount, Wai-Mo Suen, K. Young

One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.One classical theory, as determined by an equation of motion or set of classical trajectories, can correspond to many unitarily {\em in}equivalent quantum theories upon canonical quantization. This arises from a remarkable ambiguity, not previously investigated, in the construction of the classical (and hence the quantized) Hamiltonian or Lagrangian. This ambiguity is illustrated for systems with one degree of freedom: An arbitrary function of the constants of motion can be introduced into this construction. For example, the nonrelativistic and relativistic free particles follow identical classical trajectories, but the Hamiltonians or Lagrangians, and the canonically quantized versions of these descriptions, are inequivalent. Inequivalent descriptions of other systems, such as the harmonic oscillator, are also readily obtained.

arXiv:gr-qc/9904042Apr 19, 19991 citation

Study about Forced Harmonic Oscillator in the Space (x, v)

Gustavo V. López, Omar J. P. Bravo

The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. The quantization of the forced harmonic oscillator is studied with the quantum variable ( $\mathit{x}$ , $\hat{v}$ ) , with the commutation relation [ $\mathit{x}$ , $\hat{v}$ ] = $\mathit{i}\hbar$ / $\mathit{m}$ , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion $\mathit{K}$ ( $\mathit{x, v, t}$ ) of the classical system, The comparison with the quantization in the space ( $\mathit{x, p}$ ) is done with the usual Schrödinger’s equation for the Hamiltonian $\mathit{H}$ ( $\mathit{x, p, t}$ ), and with the commutation relation [ $\mathit{x}$ , $\hat{p}$ ] = $\mathit{i}\hbar$ . It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the ( $\mathit{x}$ , $\hat{v}$ ) quantization has fewer oscillations than the ( $\mathit{x}$ , $\hat{p}$ ) quantization, the average energy of the system is higher in ( $\mathit{x}$ , $\hat{p}$ ) quantization than on the ( $\mathit{x}$ , $\hat{v}$ ) quantization, and the Boltzmann- Shannon entropy on the ( $\mathit{x}$ , $\hat{p}$ ) quantization is higher than on the ( $\mathit{x}$ , $\hat{v}$ ) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.The time-dependent Schrödinger equation is a cornerstone of quantum physics and governs all phenomena of the microscopic world. However, despite its importance, its origin is still not widely appreciated and properly understood. The quantization of the forced harmonic oscillator is studied with the quantum variable ($\mathit{x}$ , $\hat{v}$) , with the commutation relation [$\mathit{x}$ , $\hat{v}$] = $\mathit{i}\hbar$/$\mathit{m}$ , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion $\mathit{K}$($\mathit{x, v, t}$) of the classical system, The comparison with the quantization in the space ($\mathit{x, p}$) is done with the usual Schrödinger’s equation for the Hamiltonian $\mathit{H}$($\mathit{x, p, t}$), and with the commutation relation [$\mathit{x}$ , $\hat{p}$] = $\mathit{i}\hbar$. It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the ($\mathit{x}$ , $\hat{v}$) quantization has fewer oscillations than the ($\mathit{x}$ , $\hat{p}$) quantization, the average energy of the system is higher in ($\mathit{x}$ , $\hat{p}$) quantization than on the ($\mathit{x}$ , $\hat{v}$) quantization, and the Boltzmann- Shannon entropy on the ($\mathit{x}$ , $\hat{p}$) quantization is higher than on the ($\mathit{x}$ , $\hat{v}$) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.

Book Publisher International (a part of SCIENCEDOMAIN International)Sep 1, 20231 citation

Quantization of the 1-D harmonic oscillator with variable mass using the operators

\hat v

and

\hat p

\hat p

, for the Hamiltonian, and the operator

\hat v

for the constant of motion. We find that the result of both quantizations brings about different quantum dynamics.

Quantization of the 1-D forced harmonic oscillator in the space (

x, v

)

The quantization of the forced harmonic oscillator is studied with the quantum variable (

x,\hat v

), with the commutation relation

[x,\hat v]=i\hbar/m

, and using a Shrödinger's like equation on these variable, and associating a linear operator to a constant of motion

K(x,v,t)

of the classical system, The comparison with the quantization in the space (

x,p

) is done with the usual Schrödinger's equation for the Hamiltonian

H(x,p,t)

, and with the commutation relation

[x,\hat p]=i\hbar

. It is found that for the non resonant case, both forms of quantization brings about the same result. However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (

x,\hat v

) quantization has less oscillations than the (

x,\hat p

) quantization, the average energy of the system is higher in (

x,\hat p

) quantization than on the

(x,\hat v

) quantization, and the Boltzmann-Shannon entropy on the (

x,\hat p

) quantization is higher than on the (

x,\hat v

) quantization.

\mathit{x}

\hat{v}

) , with the commutation relation [

\mathit{x}

\hat{v}

] =

\mathit{i}\hbar

\mathit{m}

, and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion

\mathit{K}

(

\mathit{x, v, t}

) of the classical system, The comparison with the quantization in the space (

\mathit{x, p}

) is done with the usual Schrödinger’s equation for the Hamiltonian

\mathit{H}

(

\mathit{x, p, t}

), and with the commutation relation [

\mathit{x}

\hat{p}

] =

\mathit{i}\hbar

. It is found that for the non-resonant case, both forms of quantization bring about the same result. The difference means that the H-approach case brings about more complex behavior in the quantum dynamics than the K-approach case, and that the H-approach case losses more information than the K-approach case.However, for the resonant case, both forms of quantization are different, and the probability for the system to be in the exited state for the (

\mathit{x}

\hat{v}

) quantization has fewer oscillations than the (

\mathit{x}

\hat{p}

) quantization, the average energy of the system is higher in (

\mathit{x}

\hat{p}

) quantization than on the (

\mathit{x}

\hat{v}

) quantization, and the Boltzmann- Shannon entropy on the (

\mathit{x}

\hat{p}

) quantization is higher than on the (

\mathit{x}

\hat{v}

) quantization. The parameter does not appear to be suitable for experimental measurement because the variation in the average value of the energy is quite modest. Entropy is a nice parameter to investigate experimentally because it differs significantly.