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The Physics Teacher

Visualizing the Probability Density Function of a Classical Harmonic Oscillator

Mamraj Singh, Amanpal Singh, Sandeep Kumar

Published Sep 25, 2023• 0 citations
Publisher
DOI
10.1119/5.0094365

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The interpretation of the q-deformed 1-D quantum harmonic oscillator is investigated for two definitions of q-deformation.This investigation is achieved by using Zaslavskii's method to obtain the Heisenberg equations of motion (quantum Liouville equations) in the undeformed phase space.These quantum Liouville equations exhibit a non-commutative geometry produce from the existence of the dilatation operator which is inherent in the q-deformation process.The classical limits of these equations are obtained by applying a special classical limiting condition to produce the classical Liouville equations of the q-deformed oscillator.These classical Liouville equations are solved by using the method of characteristics in order to obtain the classical probability distribution functions for this system.The 2-D and 3-D behaviors of these functions were then investigated using a computer visualization method.The results of the mathematical derivations together with the computer visualization method show that the classical limit of the quantum Liouville equations for the q-deformed 1-D quantum harmonic oscillator are statistical in nature where the nonlinearity parameter for the q-deformed oscillator is connected with h .This result conforms to that obtained by Ghosh et al. for the undeformed 1-D quantum harmonic oscillator.The obtained classical probability distribution functions exhibit whorl shapes that evolve with time in phase space that are similar to the shapes obtained for the 1-D classical q-deformed oscillator.These whorl shapes in phase space are similar to those introduced by Milburn for the 1-D classical anharmonic oscillator.This similarity results from the fact that the anharmonicity itself represents a kind of deformation with a frequency that is a function of amplitude.
Abstract Using quantum theory and MATLAB software, the basic properties of two-dimensional linear harmonic oscillators in quantum mechanics are systematically studied in polar coordinate system, and obtain the visualized results. The results show that, in polar coordinate system, with the exception of special case nr =0, |m|=0, the degeneracy of two-dimensional linear harmonic oscillator is 2nr +|m|+1, and the corresponding energy eigenvalues is ħω (2nr +|m|+1). The number of intersection line between wave function and the plane with ψ=0 is 2nr +m. In the case of nr =0, the maximum number of probability density distributions is 2|m|. The results of this visualization are in complete agreement with the theoretical results. The visualization results in different coordinate systems can be verified with each other, which opens up a new research idea and also provides an idea for other quantum theoretical models to be studied.