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arXiv:1609.08383

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

Gustavo V. López, Eric M. Reynaga

Published Sep 27, 2016• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.1609.08383

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Quantization of the 1-D harmonic oscillator with variable mass using the operators v^\hat vv^ and p^\hat pp^​
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 p^\hat pp^​, for the Hamiltonian, and the operator v^\hat vv^ for the constant of motion. We find that the result of both quantizations brings about different quantum dynamics.
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.
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 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 (x\mathit{x}x , v^\hat{v}v^) , with the commutation relation [x\mathit{x}x , v^\hat{v}v^] = iℏ\mathit{i}\hbariℏ/m\mathit{m}m , and using a Schrödinger’s like equation on these variable, and associating a linear operator to a constant of motion K\mathit{K}K(x,v,t\mathit{x, v, t}x,v,t) of the classical system, The comparison with the quantization in the space (x,p\mathit{x, p}x,p) is done with the usual Schrödinger’s equation for the Hamiltonian H\mathit{H}H(x,p,t\mathit{x, p, t}x,p,t), and with the commutation relation [x\mathit{x}x , p^\hat{p}p^​] = iℏ\mathit{i}\hbariℏ. 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 (x\mathit{x}x , v^\hat{v}v^) quantization has fewer oscillations than the (x\mathit{x}x , p^\hat{p}p^​) quantization, the average energy of the system is higher in (x\mathit{x}x , p^\hat{p}p^​) quantization than on the (x\mathit{x}x , v^\hat{v}v^) quantization, and the Boltzmann- Shannon entropy on the (x\mathit{x}x , p^\hat{p}p^​) quantization is higher than on the (x\mathit{x}x , v^\hat{v}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.
Quantization of the 1-D forced harmonic oscillator in the space (x,vx, vx,v)
The quantization of the forced harmonic oscillator is studied with the quantum variable (x,v^x,\hat vx,v^), with the commutation relation [x,v^]=iℏ/m[x,\hat v]=i\hbar/m[x,v^]=iℏ/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)K(x,v,t)K(x,v,t) of the classical system, The comparison with the quantization in the space (x,px,px,p) is done with the usual Schrödinger's equation for the Hamiltonian H(x,p,t)H(x,p,t)H(x,p,t), and with the commutation relation [x,p^]=iℏ[x,\hat p]=i\hbar[x,p^​]=iℏ. 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,v^x,\hat vx,v^) quantization has less oscillations than the (x,p^x,\hat px,p^​) quantization, the average energy of the system is higher in (x,p^x,\hat px,p^​) quantization than on the (x,v^(x,\hat v(x,v^) quantization, and the Boltzmann-Shannon entropy on the (x,p^x,\hat px,p^​) quantization is higher than on the (x,v^x,\hat vx,v^) quantization.
We study classical and quantum dynamics of the symmetric harmonic oscillator and the symmetric bouncer defined in 2-D. For these systems we get each of them for two different constants of motion, two Lagrangians and two Hamiltonians which describe the same classical dynamics. However, the quantization of these systems (using Schrödinger equation), using their two equivalents Hamiltonian, describes different quantum dynamics for each of them. This represents an ambiguity on the Hamiltonian formulation of the Quantum Mechanics.