On the correspondence principle for the Klein-Gordon and Dirac Equations

TL;DR

This paper examines how solutions to the Klein-Gordon and Dirac equations behave asymptotically in high quantum number regimes, using spatial averaging to connect quantum results with classical distributions.

Contribution

It applies a local spatial averaging approach to analyze the correspondence principle for relativistic quantum equations in 1+1 dimensions, revealing quantum corrections in different potential scenarios.

Findings

Probability densities approach classical distributions with quantum corrections suppressed by powers of 7

Different structures of quantum corrections are observed in harmonic oscillator and particle in a box cases

Quantum corrections are context-dependent and vary with potential type

Abstract

We investigate the asymptotic behavior of the solutions to the Klein-Gordon and Dirac equations using the local spatial averaging approach to Bohr's correspondence principle in the large principal quantum number regime. The procedure is applied in two basic problems in $1+1$-dimensions, the relativistic quantum oscillator and the relativistic particle in a box. In the harmonic oscillator cases, we find that the corresponding probability densities reduce to their respective classical single-particle distributions plus a series of terms suppressed by powers of the $\hbar$ constant, while particle in a box cases show a different structure for the quantum corrections.