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The Dirac Equation for a Particle in a Spherical Box Potential with Application in Bag Modeling

Emil Blomquist, Trotte Boman

Published Jan 1, 2015• 2 citations
OpenAlex

Abstract

The Dirac equation is a relativistic wave equation and was the first equation to capture spin in relativistic quantum mechanics. Here, the Dirac equation will be derived and solved for a particle in a spherical box potential. Comparisons to the non-relativistic Schrödinger equation as well as to a relativistically corrected Schrödinger equation will be made. Applications of the Dirac equation, such as the Bogoliubov model, are examined and discussed and it is validated that the Dirac equation can provide knowledge about elementary particles.

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•Nov 24, 2015•0 citations
The Dirac equation is a relativistic wave equation and was the first equation to capture spin in relativistic quantum mechanics. Here, the Dirac equation will be derived and solved for a particle in a spherical box potential. Comparisons to the non-relativistic Schrödinger equation as well as to a relativistically corrected Schrödinger equation will be made. Applications of the Dirac equation, such as the Bogoliubov model, are examined and discussed and it is validated that the Dirac equation can provide knowledge about elementary particles.
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.
Abstract This paper offers educational insight into the Dirac equation, examining its historical context and contrasting it with the earlier Schrödinger and Klein–Gordon equations. The comparison highlights their Lorentz transformation symmetry and potential probabilistic interpretations. We explicitly solve the free particle dynamics in Dirac’s model, revealing the emergence of solutions with negative energy. In this discussion, we examine the Dirac Sea Hypothesis and explore the inherent helicity of the solutions. Additionally, we demonstrate how the Dirac equation accounts for spin and derive the Pauli equation in the non-relativistic limit. Through the Foldy–Wouthuysen transformation, we reveal how the equation incorporates spin–orbit interaction and other relativistic effects, ultimately leading to the hydrogen fine structure. A section on relativistic covariant notation is included to emphasize the invariance of the Dirac equation, along with more refined formulations of both the Klein–Gordon and Dirac equations. Designed for undergraduate students with an interest in the Dirac equation, this resource provides a historical perspective without being purely theoretical. Our approach underscores the significance of a pedagogical method that combines historical and comparative elements to profoundly understand the role of the Dirac equation in modern physics.
Abstract This chapter is seeking relativistic wave equations, which are invariant under Lorentz transformations, in an attempt to obtain a quantum mechanical description of relativistic particles. First, the chapter starts with the Klein–Gordon equation and then it discusses the Dirac equation. It also takes the non-relativistic limit of the Dirac equation to derive the Schrödinger equation with two additional terms, the Zeeman term and the spin–orbit coupling term. These two terms emerge naturally from the Dirac equation, and thus the spin, as an internal quantum number which behaves like angular momentum, is clearly identified. Finally, the existence of antimatter is clearly supported by the nature of the solutions to the Dirac equation.
Abstract The problem of a particle in a box is probably the simplest problem in quantum mechanics which allows for significant insight into the nature of quantum systems and thus is a cornerstone in the teaching of quantum mechanics. In relativistic quantum mechanics this problem allows also to highlight the implications of special relativity for quantum physics, namely the effect that spin has on the quantised energy spectra. To illustrate this point, we solve the problem of a spin zero relativistic particle in a one- and three-dimensional box using the Klein–Gordon equation in the Feshbach–Villars formalism. We compare the solutions and the energy spectra obtained with the corresponding ones from the Dirac equation for a spin one-half relativistic particle. We note the similarities and differences, in particular the spin effects in the relativistic energy spectrum. As expected, the non-relativistic limit is the same for both kinds of particles, since, for a particle in a box, the spin contribution to the energy is a relativistic effect.
I believe that physicists gain much of their physics intuition from solving simple model problems explicitly. Apart from the hydrogen atom, this is something that is not often done in relativistic quantum theory books (except the book by Greiner (1990)). In this chapter we set up and solve five simple models that have exact analytical solutions. Furthermore, they are all related to well-known non-relativistic counterparts, and most find application in many areas of physics, particularly solid state physics. These relationships will be pointed out and where appropriate we will also mention the applications and consequences of the models. There are very few models in quantum mechanics that yield exact solutions, hence any that do are of fundamental interest, the hydrogen atom being, perhaps, the most famous example. The hydrogen atom with the potential V(r) = — Ze 2 /4πε 0 r is unique in that it can be solved analytically classically and in both non-relativistic and relativistic quantum theory, with and without spin. This has been discussed in detail in chapters 3 and 8. The five examples we choose to consider here are the following. Firstly we will solve the Dirac equation for an electron in a one-dimensional well, a relativistic generalization of the non-relativistic particle in a box problem.
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.