Exact solution of a time-dependent quantum harmonic oscillator with two frequency jumps via the Lewis-Riesenfeld dynamical invariant method

• URL: http://arxiv.org/abs/2211.06756v1
• Date: Sat, 12 Nov 2022 22:20:11 GMT
• Title: Exact solution of a time-dependent quantum harmonic oscillator with two frequency jumps via the Lewis-Riesenfeld dynamical invariant method
• Authors: Stanley S. Coelho, Lucas Queiroz, Danilo T. Alves
• Abstract summary: We reobtain exact analytical formulas of Tibaduiza et al. for the squeeze parameters, quantum fluctuations of the position and momentum operators, and the probability amplitude of a transition from the fundamental state to an arbitrary energy eigenstate. We also present original expressions for the mean energy value, for the mean number of excitations, and for the transition probabilities, considering the initial state different from the fundamental.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: In a recent paper, Tibaduiza et al. [Braz. J. Phys. 50, (2020)] studied, by means of an exact algebraic method, the dynamics of a quantum harmonic oscillator that, initially with frequency $\omega_0$, undergoes an abrupt jump to a frequency $\omega_1$ and, after a certain time interval, another jump returning to its initial frequency $\omega_0$. In the present paper, using another method, namely the Lewis-Riesenfeld method of dynamical invariants, we investigate the same physical system and reobtain the exact analytical formulas of Tibaduiza et al. for the squeeze parameters, the quantum fluctuations of the position and momentum operators, and the probability amplitude of a transition from the fundamental state to an arbitrary energy eigenstate. This not only confirms our results, obtained via the LR method, but also those found by Tibaduiza et al.. In addition, we also present original expressions for the mean energy value, for the mean number of excitations, and for the transition probabilities, considering the initial state different from the fundamental. We show that the mean energy of the oscillator, after the jumps, is equal or greater than that before these jumps, even when $\omega_1<\omega_0$. We also show that, for special values of the time interval between the jumps, the oscillator returns to the same initial state.

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