Probability and Mechanics

Ilya Prigogine

John Halpin

The nature of quantum mechanical probability has often seemed mysterious. To shed some light on this topic, the present paper analyzes the logical form of probability assignment in quantum mechanics. To begin the paper, I set out and criticize several attempts to analyze the form. I go on to propose a new form which utilizes a novel, probabilistic conditional and argue that this proposal is, overall, the best rendering of the quantum mechanical probability assignments. Finally, quantum mechanics aside, the discussion here has consequences for counterfactual logic, conditional probability, and epistemic probability.

L. E. Ballentine

The abstract theory of probability and its interpretation are briefly reviewed, and it is explicitly demonstrated that the formalism of quantum mechanics satisfies the axioms of probability theory. This refutes the suggestions which have occasionally been made that ‘‘classical’’ probability theory does not apply to quantum mechanics. Several erroneous applications of probability theory to quantum mechanics are examined, and the nature of the errors are exposed. It is urged that more attention be given to probability theory in the physics curriculum.

Lazar Mayants

Xijin Fu

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.