Revealing a quantum feature of dimensionless uncertainty in linear and quadratic potentials by changing potential intervals

TL;DR

This paper investigates how changing potential intervals affects the quantum uncertainty in linear and quadratic potentials, revealing that the dimensionless quantum uncertainty depends on quantum number n and aligns with the correspondence principle.

Contribution

It demonstrates that adjusting potential intervals exposes the n-dependence of quantum uncertainty, challenging previous assumptions of n-independence in certain potentials.

Findings

Quantum uncertainty depends on n when potential intervals are modified.

Dimensionless analysis can reveal quantum features hidden in standard approaches.

Higher power potentials also show n-dependence in quantum uncertainty.

Abstract

As an undergraduate exercise, in an article (2012 Am. J. Phys. $\bf{80}$ 780-14), quantum and classical uncertainties for dimensionless variables of position and momentum were evaluated in three potentials: infinite well, bouncing ball, and harmonic oscillator. While original quantum uncertainty products depend on ${\rm{\hbar}}$ and the number of states ($n$), a dimensionless approach makes the comparison between quantum uncertainty and classical dispersion possible by excluding ${\rm{\hbar}}$. But the question is whether the uncertainty still remains dependent on quantum number $n$. In the above-mentioned article, there lies this contrast, on the one hand, the dimensionless quantum uncertainty of the potential box approaches classical dispersion only in the limit of large quantum numbers ($n\to \infty $)-consistent with the correspondence principle. On the other hand, similar evaluations for bouncing ball and harmonic oscillator potentials are equal to their classical counterparts independent of $n$. This equality may hide the quantum feature of low energy levels. In the current study, we change the potential intervals in order to make them symmetric for the linear potential and non-symmetric for the quadratic potential. As a result, it is shown in this paper that the dimensionless quantum uncertainty of these potentials in the new potential intervals is expressed in terms of quantum number $n$. In other words, the uncertainty requires the correspondence principle in order to approach the classical limit. Therefore, it can be concluded that the dimensionless analysis, as a useful pedagogical method, does not take away the quantum feature of the n-dependence of quantum uncertainty in general. Moreover, our numerical calculations include the higher powers of the position for the potentials.