Pdf advantageous numerical methods for solving the dirac equations are derived. The kleingordon equation successfully describes spin 0 particles in relativistic. Methods and applications sigma 8 2012, 082, 10 pages solutions of the. A great success of the dirac equation is that these components naturally give rise to the property of intrinsic spin. Fermions and the dirac equation in 1928 dirac proposed the following form for the electron wave equation. In fact, his enigma so permeated all facets of his life that his own first name, shortened to p. To nd solutions to this equation, it is convenient to rewrite it in the form similar to that of schr. Using a recently developed approach for solving the three dimensional dirac equation with spherical symmetry, we obtain the twopoint greens function of the relativistic diracmorse problem. The dirac spinor for the positivefrequency solution can be written as. Thus, we can use the superposition principle to solve the problem. Thus, the dirac equation no only describes spin but it also includes particle and the corresponding antiparticle solutions.
It is as if solutions 3 and 4 are moving backward in time. Dirac equation for dummies or theory of elasticity for the. Introduction the theory of in a previous paper 1 we have derived a general form of. The plane wave solutions of the dirac equation for positive and negative energy have the form. Relativistic quantum mechanics kleingordon equation dirac. We have shown recently that, in the noninteracting case, solutions of the and dkp equations are generalized solutions of the dirac equation. Answers and replies related high energy, nuclear, particle physics news on. The dirac equation for a particle in a spherical box.
Solution of the dirac equation for a free particle. Klein gordon equation f eynman stuec k elburg in terpretation dirac equation an tiparticles f ermion spin co v arian t notation massless fermions learning outcomes be able to deriv e k g equation and explain ph ysical meaning of v e e solutions. It is found that the dirac equation has solutions with both positive and negative energy. Nonrelativistic approximation of the dirac equation in an electromagnetic field. Dirac equation for dummies or theory of elasticity for the seriously advanced james burnett, olga chervova and dmitri vassiliev 30 january 2009 kcl colloquium. Solution of the dirac equation for potential interaction.
In addition, the solutions to the dirac equation are the four component dirac spinors. The dirac equation is of fundamental importance for relativistic quantum mechanics and quantum electrodynamics. Note that you here have an outer matrix product with. Because of translation invariance, we can use plane wave solutions of the form. The dirac equation is one of the two factors, and is conventionally taken to be p m 0 31 making the standard substitution, p. The solutions for nonzero momentum are not as simple. To do this the dirac spinor is transformed according to. Generalized solutions of the dirac equation, bosons, and. The dirac equation and the prediction of antimatter david vidmar throughout the history of physics, there may be no scientist quite so genuinely strange as paul allen maurice dirac. Quantization of the free dirac field university of illinois. See dirac spinor for details of solutions to the dirac equation.
Moreover, it is a linear partial di erential equation. Solution of the dirac equation in presence of an uniform magnetic. It was later suggested that the backward propagating solutions should correspond to antiparticle solutions. Thus, the interpretation is that the negative energy solutions correspond to antiparticles, the the components, and of correspond to the particle and antiparticle components, respectively. I get the general idea that this is the transformation of the solution of the equation, corresponding to a lorentz transformation of the coordinates. Physics 221b spring 2020 notes 49 solutionsof the dirac. Solution of the dirac equation for hydrogen the standard hydrogen atom problem can be solved exactly using relativistic quantum mechanics. It derives a series of solutions of the bound states near the boundary, and describes the current status of these solutions. Note that since the dirac operator acts on 4tuples of squareintegrable functions, its solutions should be members of the same hilbert space. To obtain solutions, we x our convention paulidirac representation for cli ord algebra to the following one.
Generalized solutions of the dirac equation in the interacting case. In order to see how to solve the inhomogenuous dirac equation 30 for electrons in an electromagnetic field lets first consider the example from electrostaticssolution of poisson equation. Unlike the kg equation, the dirac equation has probability densities which are always positive. This equation is manifestly lorentz invariant, and, therefore, forms a more convenient starting point for relativistic quantum mechanics. Solutions of the dirac equation in a magnetic field and intertwining operators. The full solution is a bit long but short compared to the complete effort we made in nonrelativistic qm. In fact, only those solutions of equation belonging to positive values of are. Helicity, chirality, and the dirac equation in the non.
Pdf iterative solutions to the dirac equation nasser. In the past people have studied the dirac equation in particular contigurations without any strategy for finding exact solutions. Lets start with one that gives a spin up electron in the first two components and plug it into the dirac equation to see. Note, however, that equation is not entirely equivalent to equation, because, although each solution of equation is also a solution of equation, the converse is not true. In this section, we consider the free dirac equation and the dirac equation for a charged particle in an external magnetic.
The dirac equation a great deal more was hidden in the dirac equation than the author had expected when he wrote it down in 1928. Green functions the dirac equation for electrons in an electromagnetic field. The dirac equation in solid state physics and nonlinear optics. For small speeds the solutions degenerate into the two spinors, something that we would expect.
In relativistic quantum mechanics, the dirac equation is referred to as oneparticle wave equation of motion for electron in an external electromagnetic field. The negative energy solutions, which were caused by the second derivative of time in the klein gordon equation, motivated dirac to develop an equation with a single derivative of time. Diracs equation is a model for a electron and positron massive case. Also, the equation should account for electron spin, which was demonstrated by the sterngerlach experiment 7 years earlier, in 1922.
If we change the charge on the electron from to and change the sign of the exponent, the dirac equation remains the invariant. Topological insulators dirac equation in condensed. To obtain solutions, we fix our convention paulidirac representation for clifford algebra to the following one. This new edition presents a unified description of these insulators from one to three dimensions based on the modified dirac equation.
In our derivation we have split the dkp equations for into two hagenhurley equations 1618. Pdf solutions of the dirac equation in a magnetic field. It should be added, however, that it was dirac who found most of the additional insights. Dirac equation, exact solutions, relativistic potentials, energy spectrum. The dirac hamiltonian takes the form where using, in the coordinate basis, the dirac equation for a free particle reads since the operator on the left side is a 4 4 matrix, the wave function is actually a fourcomponent vector of functions of and. This equation has nontrivial solutions only if the massshell condition is obeyed p2. The fact that the energies of the solutions do not have a lower bound is unexpected see the hole theory section below for more details. Relativistic quantum mechanics applications using the time. Exactly complete solutions of the dirac equation with. The equation predicts that antimatter must exist, which was unsuspected and unprovenatthetimeofformulation,buthassincebeenproventoexist3. Thus, we can turn the negative exponent solution going backward in time into the conventional positive exponent solution if we change the charge to.
It was formulated by the british physicist paul dirac in 1928. Dirac equation in 1928 dirac tried to understand negative energy solutions by taking the squareroot of the kleingordon equation. Supersymmetry and the dirac equation 3 hamiltonians are related by susy, which allows a simple calculation of the anomaly. Now lets look at some solutions to the dirac equation. In quantum field theory, the dirac spinor is the bispinor in the planewave solution. Dirac himself remarked in one of his talks that his equation was more intelligent than its author. The rst one we will look at is for a particle at rest.
February 1, 2008 abstract in this work we discuss the properties of the solutions of the dirac equation. M, was somewhat of an accidental mystery for years. The dirac equation is invariant under charge conjugation, defined as changing electron states into the opposite charged positron states with the same momentum and spin and changing the sign of external fields. The dirac equation and the prediction of antimatter. Show these relations by using the dirac representation. Lorentz covariance of the dirac equation means that the matrices are the same in both frames. The dirac equation is a relativistic wave equation that describes the quantum dynamics of spinors.