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American Journal of Physics

Transition Probability of a Linearly Forced Harmonic Oscillator System

M. M. Ninan, Z. Stipčević

Published Jul 1, 1969• 4 citations
Publisher

Abstract

The transition probability of a linearly forced harmonic oscillator system is derived by expanding the evolution operator using an expansion theorem valid for operators obeying the commutation law, (A, B) = 1.
DOI
10.1119/1.1975791

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arXiv:2005.00344•Apr 29, 2020•0 citations
A unified treatment by the methods of the operator mechanics is given of the quantum theory of the harmonic oscillator subjected to an applied force which is a given, but arbitrary, function of time. Few of the results are new, and most have appeared before scattered among papers devoted to a variety of topics and using a variety of mathematical techniques. Emphasis is placed on numerical methods for the evaluation of the relevant transition probabilities, and on the equality, under certain conditions, of the classical and quantum values of the average energy transfer to an oscillator by an impressed force. Generating functions for the transition probabilities are derived. Some numerical results are presented for illustration.
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 transition probabilities of linearly driven parametric oscillators that model collinear collisions of an atom and a diatomic molecule are studied within the formulation for treating the time evolution of time-dependent quadratic Hamiltonian systems. It is shown that the transition probabilities of three oscillator models: (i.e. the oscillator with varying effective mass and frequency plus linear potential, the harmonic oscillator with linear potential and the harmonic oscillator with linear plus quadratic potentials) obey the Poisson distribution in common, and evaluations of these probabilities are reduced to performing a simple operation and numerical integrations on time in terms of the parameters of the considered systems. The reasons for the difference between the previous results and ours are analyzed.
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.