Reparametrization-invariant formulation of classical mechanics and the Schrödinger equation

A. Deriglazov, B. F. Rizzuti

Research output: Contribution to journalArticle

4 Citations (Scopus)

Abstract

Any classical-mechanics system can be formulated in reparametrization-invariant form. That is, we use the parametric representation for the trajectories, x = x(τ) and t = t(τ) instead of x = x(t). We discuss the quantization rules in this formulation and show that some of the rules become clearer. In particular, both the temporal and the spatial coordinates are subject to quantization, and the canonical Hamiltonian in the reparametrization-invariant formulation is proportional to H̃ = pt+H, where H is the usual Hamiltonian and pt is the momentum conjugate to the variable t. Due to reparametrization invariance, H̃ vanishes for any solution, and hence the corresponding quantum-mechanical operator has the property Ĥψ= 0, which is the time-dependent Schrödinger equation, ih∂tψ=Ĥψ. We discuss the quantum mechanics of a relativistic particle as an example.

Original languageEnglish
Pages (from-to)882-885
Number of pages4
JournalAmerican Journal of Physics
Volume79
Issue number8
DOIs
Publication statusPublished - 26 Jul 2011
Externally publishedYes

ASJC Scopus subject areas

  • Physics and Astronomy(all)

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