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arXiv:1110.5146 · Physical Review A

Salpeter equation and probability current in the relativistic Hamiltonian quantum mechanics

K. Kowalski, Jakub Rembieliński

Published Jul 11, 2011• 50 citations
arXivHTMLPDF

Abstract

The probability current for a quantum spinless relativistic particle is introduced based on the Hamiltonian dynamics approach utilizing the Salpeter equation as an alternative for the Klein-Gordon equation. The correctness of the presented formalism is illustrated by examples of exact solutions to the Salpeter equation including the new ones introduced in this work.

DOI
10.1103/physreva.84.012108

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The probability current for a quantum spinless relativistic particle is introduced based on the Hamiltonian dynamics approach utilizing the Salpeter equation as an alternative for the Klein-Gordon equation. The correctness of the presented formalism is illustrated by examples of exact solutions to the Salpeter equation including the new ones introduced in this work.
The chapter covers some of the essential aspects of relativistic quantum mechanics. The Klein-Gordon equation and its solution in coulombic fields, how to interpret probability density and probability current derived from the Klein-Gordon equation, derivation of the Dirac equation and its covariant form, probability density and probability current for Dirac equation, positive and negative energy eigenvalues in the case of a free particle, constants of motion for relativistic treatment, presence of spin magnetic moment and spin-orbit interaction energy in the Dirac equation, and the solution of the Dirac equation for central potentials and for the hydrogen atom are presented with elegant mathematical treatment and physical interpretations. A modern and reader friendly approach is adopted for derivation of the Dirac equation, constants of motion, and for the solution of the Dirac equation for central potentials. Solved examples and unsolved exercises relating to fundamental concepts of relativistic quantum mechanics are added towards the end of chapter.
The relativistic wave equations currently used in physics are the Dirac equation and the Klein-Gordon equation. Only recently, the Salpeter equation has stimulated the interest of the scientific community. In this paper a new relativistic wave equation for particles with non-zero rest mass, the Pearcey equation, is presented. There is great formal similarity between the Pearcey equation, the Schrödinger equation and the Salpeter equation. In particular, the Pearcey equation allows us to understand the onset of the relativistic behaviour in a similar way as the Salpeter equation. It is possibile also to keep the parallelism between optics and quantum mechanics at which we are used in non relativistic quantum mechanics.
In this paper, we shall derive a new conserved current for Klein-Gordon equation. The first component of this current is non-negative and reduces to ∣ϕ∣2|\phi|^2∣ϕ∣2 in the non-relativistic limit. Therefore, it can be interpreted as a suitable probability density for spinless particles. In addition, this current is time-like and so prevents faster than light particle propagation. We will see the probability density has a considerable deviation from ∣ϕ∣2|\phi|^2∣ϕ∣2 providing the uncertainty in momentum is much greater than m0cm_0cm0​c.
In this paper, a conserved current for Klein--Gordon equation is derived. It is shown, for (1+11+11+1) dimensions, the first component of this current is non-negative and reduces to ${|\ensuremath{\phi}|}^{2}$ in nonrelativistic limit. Therefore, it can be interpreted as the probability density of spinless particles. In addition, main issues pertaining to localization in relativistic quantum theory are discussed, with a demonstration on how this definition of probability density can overcome such obstacles. Our numerical study indicates that the probability density deviates significantly from ${|\ensuremath{\phi}|}^{2}$ only when the uncertainty in momentum is greater than m0c{m}_{0}cm0​c.