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

Abstract

The quantization of the forced harmonic oscillator is studied with the quantum variable ($x,\hat{v}$), with the commutation relation $[x,\hat{v}]=i\hslash /m$, 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 $[x,\hat{p}]=i\hslash$. 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.

3 figures, 11 pages

References in corpus (3)

• Statistical Physics on the space (x,v) for dissipative systems and study of an ensemble of harmonic oscillators in a weak linear dissipative medium
• Ambiguities on the Hamiltonian formulation of the free falling particle with quadratic dissipation
• Ambiguity appearing on the Hamiltonian formulation of Quantum Mechanics