Wavefunctios of log-periodic oscillators

TL;DR

This paper derives exact quantum wave functions for time-dependent harmonic oscillators with log-periodic behavior using invariant and unitary transformation methods, analyzing quantum fluctuations and correlations.

Contribution

It introduces a method to obtain exact solutions for log-periodic oscillators and analyzes their quantum properties, which is novel in the study of such systems.

Findings

Oscillator with m=m0t/t0 and ω=ω0t0/t behaves as a constant-parameter harmonic oscillator.

Quantum fluctuations in coordinate and momentum are calculated explicitly.

Quantum correlations between coordinate and momentum are analyzed.

Abstract

We use the Lewis and Riesenfeld invariant method [\textit{J. Math. Phys.} \textbf{10}, 1458 (1969)] and a unitary transformation to obtain the exact Schr\"{o}dinger wave functions for time-dependent harmonic oscillators exhibiting log-periodic-type behavior. For each oscillator we calculate the quantum fluctuations in the coordinate and momentum as well as the quantum correlations between the coordinate and momentum. We observe that the oscillator with $m=m_0t/t_0$ and $\omega= \omega_0t_0/t$, which exhibits an exact log-periodic oscillation, behaves as the harmonic oscillator with $m$ and $\omega$ constant.