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Relativistic diffraction in time wave functions can be used as a basis for causal scattering waves. We derive such exact wave function for a beam of Dirac and Klein-Gordon particles. The transient Dirac spinors are expressed in terms of integral defined functions which are the relativistic equivalent of the Fresnel integrals. When plotted versus time the exact relativistic densities show transient oscillations which resemble a diffraction pattern. The Dirac and Klein-Gordon time oscillations look different, hence relativistic diffraction in time depends strongly on the particle spin.

Similarities between optics and quantum mechanics have long been recognized. One example of this symmetry was obtained by Moshinsky [

where

where the integral is the complex Fresnel function:

The right-hand side in Equation (3) is similar to the mathematical expression for the light intensity in the optical Fresnel diffraction by a straight edge [

These temporal oscillations are a pure quantum phenomenon, and similar oscillations arise at the moment of closing and opening gates in nanoscopic circuits [

One of the main problems in physics is to find, for the S matrix of an interaction, restrictions which proceed from general principles such as causality [

function for causal scattering. Indeed, for an arbitrary function,

condition given by:

then the free time evolution of the initial condition becomes

It is evident that if we want a relativistic solution for

As far as we know nobody has ever reported the exact relativistic solution to the shutter problem. Moshinsky worked this problem and gave an approximated answer. In a couple of articles [

of diffraction in time in the relativistic case. In the case of photons this is obviously true, the d’Alembert’s solution does not allow such time oscillations. However, for particles with mass different from zero, in full disagreement with Moshinsky’s conclusions, we report here that relativistic diffraction in time oscillations is indeed present.

The purpose of the present paper is to derive the exact solutions for the Dirac and Klein-Gordon shutter problems. The exact transient Dirac spinors are expressed in terms of integral-defined-functions which are the relativistic equivalent of the Fresnel integrals. In partial agreement with Moshinsky’s conclusions we find that indeed the relativistic densities do not resemble the mathematical expression for intensity of light that appears in the theory of diffraction in Optics. In spite of this, when our exact relativistic densities are plotted versus time, the plots show transient oscillations which resemble a diffraction pattern. For this reason in this article we claim that impressive diffractions in time oscillations do exist in the relativistic realm. Furthermore, the Dirac and Klein-Gordon densities look quite different, which implies that relativistic diffraction in time distinguishes between spin 0 and 1/2.

Consider, for relativistic particles of spin 1/2, the shutter problem. We want to find out the spinor wave function

where

we choose the initial state with a well defined direction of spin, for instance parallel to the direction of motion,

For free particles, the helicity

In terms of the remaining two components

with the initial condition:

where

We use the Compton length

where

Clearly the real and imaginary parts of

From Appendix A, for the right side of the shutter

Notice the function

Given the Dirac wave function

In

As expected, for a relativistic solution, the Dirac density vanishes for times

For relativistic particles with spin 0, the Klein-Gordon shutter problem is, by definition, the solution

where

where

ditions:

Similar to the Dirac problem, in terms of the dimensionless variables:

or in simplified notation

The presence of

Given the Klein-Gordon wave function

In

Notice that the asymptotic behavior of

Therefore, for

We derived the exact solutions for the Klein-Gordon and the Dirac shutter problems. In agreement with Moshinsky we find that the relativistic solutions do not resemble the analytic expression that appears in the theory of diffraction in Optics. In spite of this, we prove that when the exact Dirac and Klein-Gordon densities are plotted versus time, the following happens: 1) both densities show transient oscillations which in some way resemble the optical diffraction pattern; 2) the Dirac density looks quite different from the Klein-Gordon one, which implies that transient time oscillations depend strongly on the particle spin.

For these reasons and in total disagreement with Moshinsky’s conclusions [

Salvador Godoy,Karen Villa, (2016) A Basis for Causal Scattering Waves, Relativistic Diffraction in Time Functions. Journal of Modern Physics,07,1181-1191. doi: 10.4236/jmp.2016.710107

With the help of the dimensionless variables given by:

and the initial condition becomes:

Taking the Laplace transform (

which holds in the range

and for the right side,

Both functions

Because the matrix

has eigenvalues given by:

then, taking into account the boundary conditions at

where

Substituting Equation (32) into Equation (30) and Equation (33) into Equation (31), we get the solution for Dirac shutter problem in the (

Notice the singular points at

To simplify the final notation, we express the inverse Laplace transforms (

Using Laplace Transforms Tables [

Next, we use the relation

to obtain

Finally, using the identities

and

we obtain

Therefore, for

where each inverse Laplace transform has been previously calculated. We claim that Equation (43) is the exact Dirac wave function for the shutter problem, valid for

In a similar way to the Dirac solution, the Klein-Gordon shutter problem can be written in terms of dimen- sionless variables:

with initial conditions given by:

Taking the Laplace transform (

and

Here both functions

Taking into account the boundary conditions at

where

Substituting Equation (50) into Equation (48) and Equation (51) into Equation (49) we have the solutions:

Finally, we need to invert the Laplace transforms (

We have then the final solutions, for

and for

We claim that Equations (56) and (57) are the exact Klein-Gordon wave functions for the shutter problem. It’s

no surprising to find the Bessel functions

Green’s function and its derivative respectively for the Klein-Gordon equation [