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Revista Mexicana de Física E

A variational approach to ground state energy estimation in relativistic quantum systems

Felan Amal, O Arundathi, Tavarekere Shivalingaswamy

Published Jul 1, 2024• 0 citations
Publisher

Abstract

The variational method in quantum mechanics plays a crucial role in estimating upper bound values of eigenenergies, particularly in the efficient determination of ground state energies. This paper introduces a formalism for the calculation of ground state energies in relativistic quantum systems employing the variational method. The proposed approach is applied to specific physical systems, allowing us to evaluate its effectiveness. Furthermore, we compare the results obtained through this method with existing literature, shedding light on its accuracy and applicability in the context of relativistic quantum systems.

DOI
10.31349/revmexfise.21.020212

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The variational method in quantum mechanics plays a crucial role in estimating upper bound values of eigenenergies, particularly in the efficient determination of ground state energies. This paper introduces a formalism for the calculation of ground state energies in relativistic quantum systems employing the variational method. The proposed approach is applied to specific physical systems, allowing us to evaluate its effectiveness. Furthermore, we compare the results obtained through this method with existing literature, shedding light on its accuracy and applicability in the context of relativistic quantum systems.
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Variational methods in quantum mechanics are customarily presented as invaluable techniques to find approximate estimates of ground state energies. In the present paper a short catalogue of different celebrated potential distributions (both 1D and 3D), for which an exact and complete (energy and wavefunction) ground state determination can be achieved in an elementary way, is illustrated. No previous knowledge of calculus of variations is required. Rather, in all presented cases the exact energy functional minimization is achieved by using only a couple of simple mathematical tricks: 'completion of square' and integration by parts. This makes our approach particularly suitable for undergraduates. Moreover, the key role played by particle localization is emphasized through the entire analysis. This gentle introduction to the variational method could also be potentially attractive for more expert students as a possible elementary route toward a rather advanced topic on quantum mechanics: the factorization method. Such an unexpected connection is outlined in the final part of the paper.
Variational calculations play an important role in quantum mechanics, particularly in determining the ground-state energy of physical systems. Didactic examples in which one can see how the exact energy and wave function are approached by using a family of trial functions are not very numerous. An example was given in a recent book [M. Moshinsky, The Harmonic Oscillator in Modern Physics: From Atoms to Quarks (Gordon and Breach, New York, 1969)] which discusses the ground state of the hydrogen atom using as trial wave function a linear combination of harmonic-oscillator states. In the present paper we carry out a similar analysis for the ground state of a particle of mass m in a three-dimensional square-well potential. We discuss not only the approach to the exact energy when we vary the frequency and the number of oscillator states but also analyze the overlap of the exact and variational wave functions and compare the exact and approximate form factors.
We study the ground state energy and the critical screening parameter of the Yukawa potential in nonrelativistic quantum mechanics. After a short review of the existing literature on these quantities, we apply fifth-order perturbation theory to the calculation of the ground state energy, using the exact solutions of the Coulomb potential together with a cutoff on the principal number summations. We also perform a variational calculation of the ground state energy using a Coulomb-like radial wave function and the exact solution of the corresponding minimization condition. For not too large values of the screening parameter, close agreement is found between the perturbative and variational results. For the critical screening parameter, we devise a novel method that permits us to determine it to 10 digits. This is the most precise calculation of this quantity to date, and allows us to resolve some discrepancies between previous results.