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Physica Scripta

A modified Bohr’s quantization rule to estimate the energy levels of relativistic systems

Felan Amal, Tavarekere Shivalingaswamy, B. A. Kagali

Published Aug 11, 2023• 2 citations
Publisher

Abstract

Abstract Estimation of quantum energy levels is crucial to understand the physical systems in both non relativistic and relativistic domains. Here we put forward a new method to deduce quantized energy levels of some simple relativistic quantum systems without solving wave equations but using Bohr’s quantization rule along with a few suitable modifications. The results obtained are in good agreement with formal calculations available in the literature.

DOI
10.1088/1402-4896/acef76

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Abstract Estimation of quantum energy levels is crucial to understand the physical systems in both non relativistic and relativistic domains. Here we put forward a new method to deduce quantized energy levels of some simple relativistic quantum systems without solving wave equations but using Bohr’s quantization rule along with a few suitable modifications. The results obtained are in good agreement with formal calculations available in the literature.
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Based on the exact quantization rule for the nonrelativistic Schrodinger equation, the exact quantization rule for the relativistic one-dimensional Klein-Gordon equation is suggested. Using the new quantization rule, the exact relativistic energies for exactly solvable potentials, e.g. harmonic oscillator, Morse, and Rosen-Morse II type potentials, are obtained. Consequently the new quantization rule is found to be exact for one-dimensional spinless relativistic quantum systems. Though the physical meanings of the new quantization rule have not been fully understood yet, the present formal derivation scheme may shed light on understanding relativistic quantum systems more deeply.
In this article we review some results obtained from a generalization of quantum mechanics obtained from modification of the canonical commutation relation [q,p]=iℏ[q,p]={\rm i}\hbar[q,p]=iℏ. We present some new results concerning relativistic generalizations of previous works, and we calculate the energy spectrum of some simple quantum systems, using the position and momentum operators of this new formalism.
Abstract. – An exact quantization rule for the Schrödinger equation is presented. In the exact quantization rule, in addition to Nπ, there is an integral term, called the quantum correction. For the exactly solvable systems we find that the quantum correction is an invariant, independent of the number of nodes in the wave function. In those systems, the energy levels of all the bound states can be easily calculated from the exact quantization rule and the solution for the ground state, which can be obtained by solving the Riccati equation. With this new method, we re-calculate the energy levels for the one-dimensional systems with a finite square well, with the Morse potential, with the symmetric and asymmetric Rosen-Morse potentials, and with the first and the second Pöschl-Teller potentials, for the harmonic oscillators both in one dimension and in three dimensions, and for the hydrogen atom. In the development of quantum mechanics, the Bohr-Sommerfeld quantization rules of the old quantum theory [1] occupy a position intermediate between classical and quantum mechanics. The WKB approximation [2–4] is a method for the approximate treatment of the Schrödinger wave function with another quantization rule [1]. Both quantization rules are approximate. We are going to derive an exact quantization rule for the one-dimensional Schrödinger equation: d2 ψ(x) = −2M dx2 2 [E − V (x)] ψ(x), (1) ¯h where M is the mass of the particle, and the potential V (x) is a piecewise continuous real function of x satisfying for definiteness V (x)> E, − ∞ < x < xA or xB < x < ∞, V (x) = E, x = xA or x = xB, V (x) < E, xA < x < xB, (2) where xA and xB are two turning points. Between two turning points, the momentum is k(x) = √ 2M [E − V (x)]/¯h.