  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.
