It is found that the dirac equation has solutions with both positive and negative energy. Substitution of this solution into the equation for the probability density yields. Formally this corresponds the upper solution corresponds to a particle with momentum p and energy e while the lower solution has. But im still a newbie on indices notation and contravariant and covariant objects. For a plane wave solution to klein gordon equation. From and, we see that each of the four general solutions to the dirac equation correspond to one of the columns of the above transformation matrix. Dirac proposed that the equation be considered as a matrix equation. Let us now look at some of the properties of the general solution to the dirac equation. Hypermonogenic plane wave solutions of the dirac equation. In quantum field theory, the dirac spinor is the bispinor in the plane wave solution. Volume 104, number 5 physics letters 10 september 1981 exact solutions of the dirac equation in pseudoscalar plane waves qiren mang department of technical physics, peking university, beijing, china received 6 april 1981 some simple exact solutions of the dirac equation useful in the study of ircondensation are reported. The dirac spacetime matrix equation when expanded is equivalent to the four vector equations,, and. Within the frameworks of classical and quantum electrodynamics, the behavior of relativistic charged particle in the field of a plane electromagnetic wave is considered.
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. Thus, the planewave ansatz can only solve the dirac equation if p is the four. Solutions of the dirac equation in spacetimedependent. Dirac equation for dummies or theory of elasticity for the. The dirac equation describes the behaviour of spin12 fermions in relativistic quantum. The kleingordon equation successfully describes spin 0 particles in relativistic. Exact solutions of the dirac equation in pseudoscalar.
Solution of the dirac equation in plane wave metric. We examine the relation between sections of the exterior and clifford bundles of a pseudoriemannian manifold. A new form of dirac equation of a second order partial differential equation is found. A spinor calculus is established and used to investigate a new solution of the dirac equation lying in a minimal left ideal characterized by a certain idempotent projector.
For particle physics need a relativistic formulation of quantum mechanics. Whatever the motivation, dirac sought a wave equation whose solutions would be solutions of 2, but that was linear in e. Shortly we will compare the properties of the electromagnetic plane wave solutions with those of the quantum mechanical plane wave solutions. We note that the solutions of the dirac equation can also be expressed in a closed form for a quantized electromagnetic plane wave, as has first. The general solution of the electromagnetic wave equation in homogeneous, linear, timeindependent media can be written as a linear superposition of plane waves of different frequencies and polarizations. However, pauli and weisskopf suggested 6 years after the development of dirac s equation to interpret the 4current as a 4current charge density. In particular, we look for freeparticle plane wave solutions of the form. In the absence of free charge and current densities the maxwell equations are. The dirac equation and the prediction of antimatter. Plane wave solutions to the dirac equation relativistic. A geometric discretisation scheme based on both forward and backward difference operators is used.
Because we have eliminated the matrices from the equation, this is an equation for each component of the dirac spinor. An exact solution of diracs equation in the field of. A quaternionic analogue of dirac equation is presented and compared with the ordinary dirac equation. Hornbuckle january 27, 2016 1 maxwells equations maxwells equations and the constitutive relations describe the. I think i can understand the solutions for rest particles, but the plane wave solutions appear to be a bit weird to me. The two equations become the same if we replace the particle rest mass, m0, in the latter by im0.
Pdf new exact solutions of the dirac equation of a. Indeed, we can take our plane wave solutions to the kg equation and substitute them back into the dirac equation, and. It must be admitted, however, that there seems to be no reason to believe that a quaternion or biquaternion formulation of the dirac equation contains any additional physics and the purpose of. This will give us a real solution of the kleingordon equation. Again, we refer to these equations as the dirac spacetime vector equations for free space.
This interpretation is not immediately available here since it may. Im trying to understand the plane wave solutions of the dirac equation. Exact solution to the dirac equation in the presence of an. Any solution of the free dirac equation is, componentwise, a solution of the free kleingordon equation.
Thus, the dirac equation no only describes spin but it also includes particle and the corresponding antiparticle solutions. Sinusoidal planewave solutions of the electromagnetic. New exact solutions of the dirac equation of a charged. In nonrelativistic quantum mechanics, wave functions are descibed by the timedependent schrodinger equation. With this wave equation the quivering motion zitterbewegung is satisfactorily explained. Chapter maxwells equations and electromagnetic waves.
Wave solution properties of the dirac particle majorana particle dirac equation elementary particle physics. Uniform plane wave solution to maxwells equations brian k. M, was somewhat of an accidental mystery for years. There is no known such theory for particles of any spin. In this video, i show you how to solve the dirac equation for plane waves. We present a new solution of the dirac equation in the background of a plane wave metric. New exact solutions of the dirac equation of a charged particle interacting with an electromagnetic plane wave in a medium article pdf available in laser physics letters 109. Dirac equation for dummies or theory of elasticity for the seriously advanced james burnett, olga chervova and dmitri vassiliev 30 january 2009 kcl colloquium. His approach was to hypothesize that 2 can be expressed as the product of conjugate linear factor. Solution of the dirac equation for a free particle. Solutions of the maxwell equations and photon wave functions. The conditions under which a discrete plane wave solution satisfies a discrete joyce equation are discussed. For a free fermion the wavefunction is the product of a plane wave and a dirac spinor, up.
Therefore, as long as energy and momentum follows the einsteins relation eq. These plane wave solutions are very useful, and non trivial enough to find, that i thought that it would be a good idea. Visualising plane wave solutions up to a rigid orthogonal transformation. In fact, his enigma so permeated all facets of his life that his own first name, shortened to p. Hi, im recently reading an introductory text about particle physics and there is a section about the dirac equation. Specifically, he postulated a set of basis variables. Plane electromagnetic waves and wave propagation 7. 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. A new matrix formulation of the maxwell and dirac equations. Plane wave solutions of the dirac equation stack exchange. The plane wave is solution of the equation for free particles namely with v0. The dirac equation is one of the two factors, and is conventionally taken to be p m 0 31. These plane wave solutions satisfy the ortogonality relations. Sommen, hypermonogenic plane wave solutions of the dirac equation in superspace, advances in applied clifford algebras, vol.
Plane wave solutions of the free dirac equation assume solutions of the form. For this case the right hand sides of the wave equations are zero. The angularmomentum operator and the corresponding eigenfunctions are given and shown to be complete in sec. It is shown that the imposition of natural initial conditions in relativistic motion equation of lorentz and the choice of initial wave function in dirac s equation describe the correspondence between the classical and quantum. It is the purpose of the present paper to find an exact solution of the dirac equation in the presence of the exact plane wave proposed by bondi, by passing, using the condition of chirality, we find a solution for a. Sinusoidal plane wave solutions are particular solutions to the electromagnetic wave equation. Solutions of dirac equation the wavefunctions can be written as.