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Physica E Low-dimensional Systems and Nanostructures

From probabilities to quantum and classical mechanics

Lubomír Skála, V. Kapsa

Published Jul 2, 2005• 10 citations
Publisher
DOI
10.1016/j.physe.2005.05.009

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•Sep 26, 2020•0 citations
Abstract Based on the theory of quantum mechanics, this paper systematically analyzes the basic characteristics of n-dimensional linear harmonic oscillator in quantum mechanics, focuses on the eigenfunction and probability density of one-dimensional harmonic oscillator, and simulates the eigenfunction and probability density of some energy levels with MATLAB software. Finally, MATLAB software was used to compare the probability distribution of linear harmonic oscillator in classical mechanics and quantum mechanics. The results indicate that the number of points of intersection between a wave function and a φ=0 line is n; The probability distribution satisfies the normalization condition; Taking different values of n, the probability distribution function of harmonic oscillator in quantum mechanics has n different nodes, and the amplitude of harmonic oscillator in classical mechanics also changes accordingly. φ0 by the ground state probability distribution of quantum mechanics and classical mechanics distribution probability of the simulation image can be seen that the shape of the two distribution curve is the opposite, but when n is large, the probability density of quantum mechanics |ϕn (ξ)|2 local average and classical probability distribution P(ξ), that is, classical mechanics and quantum mechanics ZhongZhen gradually increase the probability distribution of similarity. These results are reflected in the image, and the characteristics shown in the image are consistent with the theoretical results.
The classical and quantum mechanical solutions to three simple physical systems are directly compared by using the classical equations of motion to determine a probability density function. This function is compared to the probability density function for the quantum mechanical stationary state with the same energy. The nonclassical behavior of the quantum mechanical systems is thus made more distinct. This comparison allows students to gain experience using probability functions to describe physical systems and to see how the quantum mechanical solutions reduce to classical behavior in the high-energy/short-wavelength limit.
When quantum mechanics is understood as a new generalized theory of probability - to be called the quantum probability theory - mysteries and controversies regarding quantum mechanics are dissolved. In the classical probability theory, that a measurement of some system requires an additional measurement apparatus is of insignificant importance - in the quantum probability theory, this comes to change. For one central single reason around a particular classical probability equation, the generalized probability view has not gained much traction, despite the fact that this essentially echoes (and provides logical underpinnings of) the conventional wisdom that `quantum mechanics just works as it is.' A classical probability axiom is just an initial intuition - there is no reason why we have to dogmatically cling onto axioms that can clearly be generalized. Issues with the principle of indifference in the classical probability theory are emphasized, along with the quantum reconstruction project of deriving quantum mechanics from epistemic requirements and potential quantum gravity consequences from the principle of maximum entropy.