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Journal of Physics A Mathematical and Theoretical

Special function solutions of the free particle Dirac equation

P. Strange

Published Jun 26, 2012• 0 citations
Publisher

Abstract

The Dirac equation is one of the fundamental equations in physics. Here we present and discuss two novel solutions of the free particle Dirac equation. These solutions have an exact analytical form in terms of Airy or Mathieu functions and exhibit unexpected properties including an enhanced Doppler effect, accelerating wavefronts and solutions with a degree of localization.

DOI
10.1088/1751-8113/45/28/285303

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The Dirac equation is one of the fundamental equations in physics. Here we present and discuss two novel solutions of the free particle Dirac equation. These solutions have an exact analytical form in terms of Airy or Mathieu functions and exhibit unexpected properties including an enhanced Doppler effect, accelerating wavefronts and solutions with a degree of localization.
The Dirac equation has been studied in which the Dirac matrices $\\hat{\\boldmath$\\alpha$}, \\hat\\beta$ have space factors, respectively fff and f1f_1f1​, dependent on the particle's space coordinates. The fff function deforms Heisenberg algebra for the coordinates and momenta operators, the function f1f_1f1​ being treated as a dependence of the particle mass on its position. The properties of these functions in the transition to the Schr\\"odinger equation are discussed. The exact solution of the Dirac equation for the particle motion in the Coulomnb field with a linear dependence of the fff function on the distance rrr to the force centre and the inverse dependence on rrr for the f1f_1f1​ function has been found.
The Dirac equation has resided among the greatest successes of modern physics since its emergence as the first quantum mechanical theory fully compatible with special relativity. This compatibility ensures that the expectation value of the velocity is less than the vacuum speed of light. Here, we show that the Dirac equation admits free-particle solutions where the peak amplitude of the wave function can travel at any velocity, including those exceeding the vacuum speed of light, despite having a subluminal velocity expectation value. The solutions are constructed by superposing basis functions with correlations in momentum space. These arbitrary velocity wave functions feature a near-constant profile and may impact quantum mechanical processes that are sensitive to the local value of the probability density as opposed to expectation values. Published by the American Physical Society 2024
The usual choice of an orthogonal set of four plane-wave solutions of the free-particle Dirac equation does not lend itself readily to direct and complete physical interpretation except in low-energy approximation. A different choice of solutions can be made which yields a direct physical interpretation at all energies. Besides the separation of positive and negative energy states there is a further separation of states for which the spin is respectively parallel or antiparallel to the direction of the momentum vector.
In 1928, the brilliance of Paul Dirac granted him the insight to develop a particle equation that was fully consistent with both the principles of quantum mechanics and the theory of special relativity. In this bachelor thesis we have studied the fundamentals of relativistic quantum mechanics including the derivation of the Dirac equation in a similar fashion to the way originally done by Paul Dirac. Additionally we have studied the analytical solution for hydrogen-like atoms. Furthermore we have numerically analysed the energy spectrum, radial characteristics and correctional terms using programs written in Python and MATLAB. Finally we have investigated the implications and applications ofthe Dirac Hamiltonian and its use to effectively describe the attributes of gold atoms.We find that relativistic contribution scales approximately by Z2 for large Z. In addition, we conclude that the Dirac Hamiltonian is of vital importance for an accurate description of chemical and physical properties of large atoms. Furthermore, we have completed calculations of the radial solution for the Coulomb potential and also for a free particle.
Chapter 19 introduces relativistic quantum mechanics, with the aim of providing the student with a basic knowledge of the subject and a link to more advanced courses in relativistic quantum theory. It begins by showing that when the Schrödinger equation is modified to make it consistent with special relativity, the result is the Klein–Gordon equation, which gives better qualitative agreement with experiment. Introducing the electron spin into the Schrödinger equation leads to the Pauli equation, which gives rise to important spin effects. By considering both special relativity and spin, one obtains the Dirac and the van der Waerden equations. The main properties and implications of Dirac’s equation are discussed, including the electron’s zitterbewegung. The solution of Dirac’s equation for the free particle is presented in detail, the hydrogen-like atom is shown to be exactly solvable, and the particle in an external electromagnetic field is included as a complementary material.