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Journal of Physics A Mathematical and General

Variational energy spectra of relativistic Hamiltonians

M. Moshińsky, Anju Sharma

Published Jan 9, 1998• 6 citations
Publisher

Abstract

In ordinary and relativistic quantum mechanics the energy spectra of most of the Hamiltonians cannot be obtained exactly. Approximate methods have to be used among which the variational one is particuarly popular. The purpose of this paper is to show that when the matrices appearing in a Dirac equation with interaction, are replaced by a direct product of matrices associated with ordinary and what we call sign spins, then a standard complete set of non-relativistic wavefunctions can be used to carry out the variational calculations. To illustrate the power of our method we analyse first the variational energies of ordinary and Dirac relativistic oscillator Hamiltonians, and then indicate the procedure for the general one-particle case. The extensions to higher spins, or to a larger number of particles, are briefly mentioned in the conclusion.

DOI
10.1088/0305-4470/31/1/033

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In ordinary and relativistic quantum mechanics the energy spectra of most of the Hamiltonians cannot be obtained exactly. Approximate methods have to be used among which the variational one is particuarly popular. The purpose of this paper is to show that when the matrices appearing in a Dirac equation with interaction, are replaced by a direct product of matrices associated with ordinary and what we call sign spins, then a standard complete set of non-relativistic wavefunctions can be used to carry out the variational calculations. To illustrate the power of our method we analyse first the variational energies of ordinary and Dirac relativistic oscillator Hamiltonians, and then indicate the procedure for the general one-particle case. The extensions to higher spins, or to a larger number of particles, are briefly mentioned in the conclusion.
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By a canonical transformation on the Dirac Hamiltonian for a free particle, a representation of the Dirac theory is obtained in which positive and negative energy states are separately represented by two-component wave functions. Playing an important role in the new representation are new operators for position and spin of the particle which are physically distinct from these operators in the conventional representation. The components of the time derivative of the new position operator all commute and have for eigenvalues all values between $\ensuremath{-}c$ and ccc. The new spin operator is a constant of the motion unlike the spin operator in the conventional representation. By a comparison of the new Hamiltonian with the non-relativistic Pauli-Hamiltonian for particles of spin \textonehalf{}, one finds that it is these new operators rather than the conventional ones which pass over into the position and spin operators in the Pauli theory in the non-relativistic limit. The transformation of the new representation is also made in the case of interaction of the particle with an external electromagnetic field. In this way the proper non-relativistic Hamiltonian (essentially the Pauli-Hamiltonian) is obtained in the non-relativistic limit. The same methods may be applied to a Dirac particle interacting with any type of external field (various meson fields, for example) and this allows one to find the proper non-relativistic Hamiltonian in each such case. Some light is cast on the question of why a Dirac electron shows some properties characteristic of a particle of finite extension by an examination of the relationship between the new and the conventional position operators.
Abstract A recently proposed approach to relativistic quantum mechanics (Grave de Peralta, Poveda, Poirier in Eur J Phys 42:055404, 2021) is applied to the problem of a particle in a quadratic potential. The methods, both exact and approximate, allow one to obtain eigenstate energy levels and wavefunctions, using conventional numerical eigensolvers applied to Schrödinger-like equations. Results are obtained over a nine-order-of-magnitude variation of system parameters, ranging from the non-relativistic to the ultrarelativistic limits. Various trends are analyzed and discussed—some of which might have been easily predicted, others which may be a bit more surprising.
Dirac's free particle equation originated in an attempt to express linearly the relativistic quadratic relation between energy and momentum. The authors introduce a Dirac equation which, besides the momentum, is also linear in the coordinates. They call it the Dirac oscillator because in the nonrelativistic limit it becomes a harmonic oscillator with a very strong spin-orbit coupling term. The eigenstates and eigenvalues of the Dirac oscillator can be obtained in an elementary fashion, with the degeneracy of the latter being quite different from that of the ordinary oscillator. They briefly mention the symmetry Lie algebra responsible for this degeneracy and the generalisation of the problem to many-particle systems.
This thesis is concerned with the development of a new spin-1/2 wave equation in relativistic quantum mechanics. This equation is the spin-1/2 analogue of the spin-0 Feshbach-Villars equation. The thesis begins in chapter 1 with a review of the subject of relativistic wave equations and its place in the development of modern physics. Chapter 2 sees the derivation of the new equation (hereafter referred to as the FV1/2 equation). This derivation is presented at three levels of understanding, and is combined with an analysis of the equation using elements of the underlying mathematics of the theories of special relativity and quantum mechanics. The analysis justifies the methods of derivation of the equation, and its suitability as a wave equation in relativistic quantum mechanics. It is shown th a t the FV1/2 equation is different to the Dirac equation and other equations in the literature. One of the striking features of the FV1/2 equation is th a t it contains a less restrictive dynamics than the conventional Dirac equation. Chapter 3 examines the solution space and proves some results using the discrete symmetries C, P and T, which enable a subset of the solution space to be used for physical applications. It is possible to use a subset of solutions which contain the usual Dirac negative energy states, or it is possible to use a subset which contains only positive energy states, but includes states of opposite signs of charge. The second subset corresponds to the physical properties of particles and antiparticles and suggests th a t the FV1/2 equation could be useful in atomic physics. Chapter 4 represents a start on the development of the FV1/2 equation for use in atomic physics. It contains a study of hydrogenic atoms. The energy spectra and wavefunctions for hydrogenic atoms are derived using the FV1/2 equation. The method of solution is of comparable difficulty to th a t using the Dirac equation. The spectra are found to be identical to th a t of the Dirac equation, but the wavefunctions differ. A calculation of the expectation value of the Coulomb energy is performed and the results obtained differ from the Dirac equation results by 6 x 10-3% for Z — 1, and 40% for Z = 70. This suggests th a t physically measureable quantities should be calculated and compared using the FV1/2 and Dirac equations. A calculation is presented to show the energy spectra and wavefunctions obtained are consistent with the literature. The results obtained in this chapter (again) suggest th a t the FV1/2 equation could be useful for calculations in atomic physics. Chapter 5 discusses important further work which should be undertaken if the FV1/2 equation is to become useful to physics. The development in the previous chapters concentrated on the FV1/2 equation in relativistic quantum mechanics. It should be also considered as a quantised field equation. The two electron problem and the calculation of physical quantities are crucial tests of the FV1/2 equation’s applicability in atomic physics. The mathematical analysis in chapter 2 can be extended. Suggestions for gaining a deeper mathematical understanding of the less restrictive dynamics of the FV1/2 equation are given. Finally some conclusions on what has been achieved in the thesis are presented in chapter 6.
We show that the first order form of the Schrodinger equation proposed in [1] can be obtained from the Dirac equation in the non-relativistic limit. We also show that the Pauli Hamiltonian is obtained from this equation by requiring local gauge invariance. In addition, we study the problem of a spin up particle incident on a finite potential barrier and show that the known quantum mechanical results are obtained. Finally, we consider the symmetric potential well and show that the quantum mechanical expression for the quantized energy levels of a particle is obtained with periodic boundary conditions. Based on these conclusions, we propose that the equation introduced in [1] is the non-relativistic limit of the Dirac equation and more appropriately describes spin 1/2 particles in the non-relativistic limit.