Dimensional analysis and the correspondence between classical and quantum uncertainty

Abstract

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.