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arXiv:1711.06313 · European Journal of Physics

Relativistic particle in a box: Klein–Gordon versus Dirac equations

Pedro Alberto, Saurya Das, Elias C Vagenas

Published Nov 17, 2017• 15 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1088/1361-6404/aa9b43

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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.
Present work is devoted to studying the spatio-temporal of the Dirac and the Klein Gordon (KG) particles confined in a one-dimensional box. We discuss the quantum carpets and revivals time for each particle. Moreover, we explain that the relativistic phenomena not only decrease the density of the corresponding carpets but also affect the revivals time of each particle within a box. Further, we impose different limiting conditions on the obtained Dirac energy solution and explain the wellknown slight-relativistic and non-relativistic quantum carpets, respectively. In addition, we extend our study to explain the quantum carpets of the KG particle at different energy states and compare its revivals time with the Dirac particle.
As in non-relativistic quantum theory, the simplest problem to solve in relativistic quantum theory is that of describing a free particle. Much can be learned from this case which will be of use in interpreting the topics covered in later chapters. Furthermore, some of the most profound features of relativistic quantum theory are well illustrated by the free particle, so it is a very instructive problem to consider in detail. Another advantage of the free-particle problem is that the mathematics involved in solving it is not nearly as involved as that necessary for solving problems involving particles under the influence of potentials. Firstly we shall look briefly at the formulae for the current and probability density, then we shall go on to examine the solutions of the Dirac equation and investigate their behaviour. This leads us to a discussion of spin, the Pauli limit, and the relativistic spin operator. Next we consider the negative energy solutions and show how relativistic quantum theory predicts the existence of antiparticles. Some of the dilemmas this concept introduces and their resolution are discussed. At the end of the chapter we go back to the Klein paradox, and examine it for an incident spin-1/2 particle. We find that the Klein paradox exists for Dirac particles in exactly the same way as it existed for Klein—Gordon particles and has the same resolution and interpretation.
This chapter introduces the concepts of the relativistic theory to basic problems of quantum physics, in particular, the extension of the Schrodinger equation to a relativistic form. Starting from the fundamental laws of relativity, the chapter evaluates the Klein–Gordon equation and investigates if the equation can be considered a generalization of the non-relativistic Schrodinger equation to the case of the relativistic energy. The chapter shows that the wave function, which is a solution to the Klein–Gordon equation, cannot be connected with the probability wave function. For this reason, the Dirac equation is used to solve the problems faced by the Klein–Gordon equation. The Dirac equation also includes the spin. The advantage of the Dirac equation over the Schrodinger and Klein–Gordon equations is that it naturally includes the spin. In the Schrodinger and Klein–Gordon equations, the spin is not present and has to be added to the wave function manually whenever the wave function determines a system with the spin.
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 “particle-in-a-box” problem is investigated for a relativistic particle obeying the Klein–Gordon equation. To find the bound states, the standard methods known from elementary non-relativistic quantum mechanics can only be employed for “shallow” wells. For deeper wells, when the confining potentials become supercritical, we show that a method based on a scattering expansion accounts for Klein tunneling (undamped propagation outside the well) and the Klein paradox (charge density increase inside the well). We will see that in the infinite well limit, the wave function outside the well vanishes, and Klein tunneling is suppressed: Quantization is, thus, recovered, similar to the non-relativistic particle in a box. In addition, we show how wave packets can be constructed semi-analytically from the scattering expansion, accounting for the dynamics of Klein tunneling in a physically intuitive way.