  Heisenberg's uncertainty principle is often cited as an example of a "purely
quantum" relation with no analogue in the classical limit where $\hbar \to 0$.
However, this formulation of the classical limit is problematic for many
reasons, one of which is dimensional analysis. Since $\hbar$ is a dimensionful
constant, we may always work in natural units in which $\hbar = 1$. Dimensional
analysis teaches us that all physical laws can be expressed purely in terms of
dimensionless quantities. This indicates that the existence of a dimensionally
consistent constraint on $\Delta x \Delta p$ requires the existence of a
dimensionful parameter with units of action, and that any definition of the
classical limit must be formulated in terms of dimensionless quantities (such
as quantum numbers). Therefore, bounds on classical uncertainty (formulated in
terms of statistical ensembles) can only be written in terms of dimensionful
scales of the system under consideration, and can be readily compared to their
quantum counterparts after being non-dimensionalized. We compare the
uncertainty of certain coupled classical systems and their quantum counterparts
(such as harmonic oscillators and particles in a box), and show that they
converge in the classical limit. We find that since these systems feature
additional dimensionful scales, the uncertainty bounds are dependent on
multiple dimensionless parameters, in accordance with dimensional
considerations.
