Classical mechanics in reparametrization-invariant formulation and the Schr\"odinger equation

TL;DR

This paper explores the reparametrization-invariant formulation of classical mechanics and its quantization, revealing clearer rules and a direct link to the Schrödinger equation, with implications for teaching quantum mechanics.

Contribution

It demonstrates how reparametrization-invariant classical mechanics naturally leads to the Schrödinger equation, offering a new perspective for quantum theory and education.

Findings

Reparametrization invariance simplifies quantization rules.

The quantum Hamiltonian operator enforces the Schrödinger equation.

Application to relativistic particles illustrates the approach.

Abstract

The dynamics of any classical-mechanics system can be formulated in the reparametrization-invariant (RI) form (that is we use the parametric representation for trajectories, ${\bf x}={\bf x}(\tau)$, $t=t(\tau)$ instead of ${\bf x}={\bf x}(t)$). In this pedagogical note we discuss what the quantization rules look like for the RI formulation of mechanics. We point out that in this case some of the rules acquire an intuitively clearer form. Hence the formulation could be an alternative starting point for teaching the basic principles of quantum mechanics. The advantages can be resumed as follows. a) In RI formulation both the temporal and the spatial coordinates are subject to quantization. b) The canonical Hamiltonian of RI formulation is proportional to the quantity $\tilde H=p_t+H$, where $H$ is the Hamiltonian of the initial formulation. Due to the reparametrization invariance, the quantity $\tilde H$ vanishes for any solution, $\tilde H=0$. So the corresponding quantum-mechanical operator annihilates the wave function, $\hat{\tilde H}\Psi=0$, which is precisely the Schr\"odinger equation $i\hbar\partial_t\Psi=\hat H\Psi$. As an illustration, we discuss quantum mechanics of the relativistic particle.