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Journal of Hunan University

Classical Limit of Quantum Motion and Fejér Mean Value

Qi Wei

Published Jan 1, 2001• 0 citations
OpenAlex

Abstract

The rigorous calculation to quantum motion in one infinite potential square well shows that the expectation value of a quantum mechanical quantity on equally weighted wave packet in the classical limit tends to the Fejer mean value of the Fourier series of corresponding classical quantity,rather than Fourier series itself.Furthermore,when the number of eigenstates N equals to the square root of quantum number n ,the equally weighted wave packet minimizes the uncertainty relation.

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The rigorous calculation to quantum motion in one infinite potential square well shows that the expectation value of a quantum mechanical quantity on equally weighted wave packet in the classical limit tends to the Fejer mean value of the Fourier series of corresponding classical quantity,rather than Fourier series itself.Furthermore,when the number of eigenstates N equals to the square root of quantum number n ,the equally weighted wave packet minimizes the uncertainty relation.
The first two period behaviors of a quantum wave packet in an infinite square well potential is studied. First, the short term behavior of expectation value of a quantity on an equally weighted wave packet (EWWP) is in classical limit proved to reproduce the Fej'{e}r average of the Fourier series decomposition of the corresponding classical quantity. Second, in order to best mimic the classical behavior, a nice relation between number NNN of stationary states in the EWWP with the average quantum number nnn as N≈nN\thickapprox \sqrt{n}N≈n​ is revealed. Third, since the Fejér average can only approximate the classical quantity, it carries an uncertainty which in large quantum number case is almost the same as the quantum uncertainty.
The mean value of a quantity in an equally weighted wave packet was recently found in the classical limit to be the Fejér average of partial sums of Fourier series expansion of the classical quantity, and the number of stationary states in it is equal to that of partial sums. The incompleteness of the Fejér average in representing a classical quantity enables us to define a classical uncertainty relation which turns out to be the counterpart of the quantum one. In this paper, two typical quantum systems, a harmonic oscillator and a particle in an infinite square well, are used to illustrate the above-mentioned points.