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**De Leo S ^{1*}, Ducati G^{2} and Giardino S^{3}**

^{1}Department of Applied Mathematics, State University of Campinas, Brazil

^{2}CMCC, Universidade Federal do ABC, S˜ao Paulo, Brazil

^{3}Department of Applied Mathematics, State University of Campinas, Brazil

- *Corresponding Author:
- Stefano De Leo

Department of Applied Mathematics

State University of Campinas, Brazil

**Tel:**5519 352159

**E-mail:**[email protected]

**Received Date:** November 28, 2014; **Accepted Date:** February 17, 2015; **Published Date:** March 10, 2015

**Citation:** De Leo S, Ducati G, Giardino S (2015) Quaternionic Dirac Scattering. J Phys Math 6:130. doi:10.4172/2090-0902.1000130

**Copyright:** © 2015 De Leo S, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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The scattering of a Dirac particle has been studied for a quaternionic potential step. In the potential region an additional diffusion solution is obtained. The quaternionic solution which generalizes the complex one presents an amplification of the reflection and transmission rates. A detailed analysis of the quaternionic spinorial velocities shed new light on the additional solution. For pure quaternionic potentials, the interesting and surprising result of total transmission is found. This suggests that the presence of pure quaternionic potentials cannot be seen by analyzing the reflection or transmission rates. It has been observed by measuring the mean value of some operator

Quaternions; Dirac equation; Scattering

There are several proposals to formulate quantum mechanics [1,2] by generalizing the complex number field. The most obvious generalization is done by introducing quaternions [3–20], but there are other possibilities, like Clifford algebras [21-25] and p−adic numbers [26]. Even inside each proposal, there may be internal divisions. In this article, we study a quaternionic quantum mechanics endowed with a quaternionic scalar product [3]. Our interest in the quaternionic formulation of Dirac equation is motivated by the recent interesting results obtained in the standard Dirac theory for the diffusion [27- 30], tunneling [31-33] and Klein zones [34-38]. The generalization to quaternionic potentials [20] could be useful to understand many hidden aspects of the Dirac solutions. Even simple solutions, as those involving the potential step, contain interesting results. In the case of quaternionic quantum mechanics, there are fewer studies which investigate the quaternionic formulation of Dirac equation. In Adler’s book [3], the theoretical foundations have been laid down, nevertheless explicit physical examples are still quite rare. More recently, the Dirac solutions in the presence of quaternionic potentials have been detailed discussed in ref. [20]. In the potential region, the quaternionic solutions allow two values for the wave-function momentum. This represents the first important difference with respect to the complex case in which only one value is found. A better understanding of the additional quaternionic solution can be achieved by studying an explicit example of quaternionic potential and calculating the reflection and transmission coefficients. In order to do this, we recall the general solution presented in a previous article [20] and study the scattering by a quaternionic potential step. In the complex scattering, for one-dimensional motion, we have not spin flip in the reflected and transmitted beams [2,29,30]. Thus, the spin flip is not considered too for the quaternionic case. The additional quaternionic solutions and the new reflection and transmission coefficients satisfy conservation of probability and in the complex limit reproduce the standard Dirac scattering. A novel and intriguing behavior is observed for the scattering by a purely quaternionic step. In this case, there is no reflection and the wave-function shows an oscillating spatial pattern similar to the one observed for circularly polarized light. The article is organized as follows. In section II, for the convenience of the reader we fix our notation and recall the Dirac solution in the presence of a quaternionic potential. In section III, we obtain the reflection and transmission coefficients and calculate the probability current. In section IV, we discuss the effect of the quaternionic potential on the particle velocities. The last sections round off the article with a brief discussion on the total transmission for pure quaternionic potentials (section V), final considerations and future possible investigations (section VI).

**The Quaternionic Dirac Equation**

As usual in quaternionic quantum mechanics [3], the Dirac equation is written in terms of an anti- hermitian Hamiltonian operator. In this spirit, we introduce the anti-hermitian quaternionic potential step

{0 for z < 0 (REGION I), or z > 0 (REGION II)}

Where V _{0,1,2} are real constants. In the potential region (region II),
the Dirac equation then reads

(1)

For V_{1} =V_{2} = 0 , the complex step is recovered. The Dirac
Hamiltonian conatins the matrices α andβ which satisfy the algebra

,,, and (2)

In the Dirac representation [2], we have

, (3)

Where σ are the well-known complex Pauli matrices. For convenience of notation, the quaternionic Potential can be rewritten as follows

exp [i arctan ]=

Considering a motion along the z-axis, p = ( 0 , 0 , Q ), introducing the quaternionic spinor

exp

Where u and v are complex spinors, we obtain two coupled complex matrix equations,

(5)

The first equation is obtained from the complex part of equation (1), and the second one is obtained taking the pure quaternionic part. Equations (5) can be rewritten as eigenvalue equations,

(6)

The determinants of the matrices on the left hand side of equations (6) are equal, and consequently the eigenvalues obtained from any of them are also equal. Non-trivial solutions have to satisfy

(7)

From equation (7), we obtain two squared momenta for particles moving in the region II, namely

, (8)

Where

,and

From equation (8), andmay be either positive or negative. Hence, we find three energy zones determined by the value,

Diffusion zone and ,

Tunneling zone and

Klein zone and

These regions generalize the diffusion, tunneling, and Klein zones obtained for the complex case [27-38],

1) Diffusion zone and

2) Tunneling zone and

3) Klein zone and

For diffusion [27-30] and Klein [34-38] zones, we find oscillatory solutions interpreted as particles or anti-particles. In the tunneling zone [31-33] the wave function becomes evanescent and thus we have not particle or anti-particle propagation. In order to explicitly determine the quaternionic wave functions, we calculate the eigenvectors of equations (6). The four component eigenvectors can be conveniently written by defining the two- dimensional vectors

After simple algebraic manipulation, for, we find

and (9)

And for Q = Q_{+} ,

and (10)

In order to study the scattering of a Dirac particle by the quaternionic
step, for simplicity of calculation we choose a spin orientation for the
incoming particle, i.e. χ = (10)^{t} and explicit in the complex spinors and their dependence on Q,V_{0} and W_{0} . For example, in
region I, we have an incident particle with momentum thus represented by There is no spin flip in the reflection
or transmission, thus we can eliminate the zero component in our
wave functions and work with two-component quaternionic spinors.
In region I (potential free region), the quaternionic wave function is
then given by

expexp , (11)

where R and are complex coefficients. Note that the presence of
complex exponential in (4) requires multiplication from the right by
complex coefficients. In region II (potential region), the quaternionic
wave function is characterized by two possible momenta, Q_{+} and Q_{−} ,
and its explicit expression is

expexp (12)

As happens for the reflection coefficients, the transmission coefficients T and are also complex. The matching condition, , implies

=

(13)

From these coupled equations, we can obtain the complex coefficients , in terms of the wave function parameters and consequently calculate the transmission and reflection rates.

**The reflection and transmission coefficients**

To simplify our notation let us introduce the a dimensional quantity a and explicitly rewrite the quaternionic wave function in region I,

expexp,

and region II,

,

where

The continuity equations at z=0 lead to the following system

,

(14)

After algebraic manipulations, we find

,

(15)

In the complex limit, W_{0} → 0, we recover the standard reflection
and transmission rates [2], ,

And

(17)

**The Current Probability Density**

The relation between the reflection and transmission coefficients can be obtained by using the continuity equation

(18)

Where and

Observing that is independent of time and the spatial dependence of Ψ is only on the z-coordinate, we find

The continuity of the current density between the free region I and the potential region II implies

.

For the previous equation, we find

. (19)

In the complex limit, W_{0} → 0, we recover the well know continuity
equation

(20)

For we have and . Eq.(20) then implies
a reflection rate <1 (diffusion zone). For , we
have . Consequently, and Eq.(20) implies a reflection
rate=1. Finally, for , we have and , we find
a reflection rate>1suggesting pair production [2,34-38]. In **Figure 1**, we
plot the reflection rates for different value of quaternionic potentials
as a function of E/m. In the Klein zone, the reflection rate increases
by increasing the quaternionic potential. In this energy zone, the
quaternionic perturbation thus contributes to the phenomenon of pair
production. This is confirmed by the transmission rates in **Figure 2**.

**Figure 1:** The reflection rates as a function of the incoming energy for a
fixed complex potential V0=3m (blue continuous line) and for quaternionic
perturbation |W0|=(1, 2, 3)m (cyan/magenta/red dashed and dotted lines). By
increasing the quaternionic part of the potential, the tunneling zone decreases.
We also observe that in presence of quaternionic perturbation, due to the
additional diffusion solution, we have not total reflection in the tunneling zone.

**Figure 2:** The transmission rates are plotted as function of the incoming
energy for a fixed complex potential and different quaternionic perturbation. As
observed in the caption of Figure 1, in presence of Quaternionic potentials total
reflection breaks down. This is clear from the plot in (b) which confirms The
presence of additional diffusion solutions.

It is important to observe that for complex tunneling only evanescent
waves, , appears in the potential region and this is seen
by observing (**Figure 1a**) that the reflection rate is 1. The presence of a
quaternionic potential breaks total reflection and this is explained by
the fact that together evanescent wave, , also oscillatory
waves, , are present in the potential region. The
numerical calculation for the combined reflection and transmission
coefficients is shown in **Figure 3** and confirms the continuity
equation (19). From **Figure 3**, we also observe a shift in the starting
point of the diffusion zone and more important we have a decreasing
energy zone for the evanescent waves by increasing the quaternionic
potential. The last observation clearly suggest that oscillatory waves
kill evanescent waves in presence of great quaternionic potentials.
Finally, we also find that the reflection and transmission rates and are very small compared to the reflection rates and , thus as
a first approximation we can use these last coefficient rates to study
quaternionic perturbations of complex quantum scattering problems.

**Figure 3:** The reflection and transmission rates satisfy the probability
conservation. In the Klein zone, we find a negative transmission flux and
consequently a reflection rate greater than one. This effect is explained in terms
of pair production. The quaternionic perturbation amplifies the pair production.

The incoming complex plane wave has the standard oscillatory exponential, exp[i( pz − Et)], and consequently its group velocity is given by

(21)

The transmitted quaternionic plane waves contain, in the diffusion and Klein zones, the oscillatory exponentials , which determine the following group velocities

Recalling that

We immediately obtain

,

and consequently

(22)

The additional quaternionic spinor is then characterized by a
velocity v_{+} which is a typical velocity of diffusion **Figure 4a**. Indeed,
we have not tunneling or Klein zones as happens for the velocity v_{−} **Figure 4b**. By increasing the quaternionic part of the potential we see
decreasing the velocity of the additional quaternionic spinor and by

increasing the incoming energy we obtain an increasing value of the
velocity. These are typical effects of diffusion. As observed before,
the velocity v_{−} presents three different energy zones (see **Figure 4b**),
the diffusion zone for where the potential acts
decelerating the particles, the Klein zone for where
the negative velocities solutions generates a flux from the right to the
left which gives a reflection rates greater than one and consequently
interpreted in terms of pair production and finally the intermediate
tunneling zone where we have not propagation due to the presence of
evanescent solutions. It is interesting to observe that in the diffusion
zone, we have two propagation regimes and consequently two different
velocities. These different velocities permit to decouple the two
quaternionic solutions. In the pure quaternionic limit (V_{0} → 0), as we
shall discuss in our conclusions, these velocities tend to the same value
and this value is p/E the free propagation velocity.

Our analysis shows a surprising effect for pure quaternionic potentials, V0 = 0. In this case,

,,and ,

and the continuity equation leads to the following system for the reflection and transmission coefficients,

and . (23)

The solutions are immediately obtained by showing total transmission,

(24)

The transmitted wave-function for the pure quaternionic potential is then given by

(25)

and propagates with group velocity p/E. Due to the fact that the
transmitted quaternionic spinor satisfies = , and moves with
the same velocity of the free incoming particle seems impossible to see
the presence of pure quaternionic potentials. Nevertheless the problem
of an apparent invisible presence of pure quaternionic potential can
be solved by taking the mean value of operators. For example, the z
component spin mean value for the complex wave function Ψ_{c,inc} is
given by

, (26)

which in the non-relativistic limit, p << m, reproduces the well-known result . In thepresence of a pure quaternionic potential, we should find

(27)

Which implies a z-dependence on the mean value of spin similar to what happens for circular polarized light [39,40].

In this article we have studied the scattering of a Dirac particle by
a quaternionic potential step. Our analysis shows the presence of an
additional quaternionic solution characterized by a group diffusion
velocity v_{+} . The quaternionic solution which generalizes the standard
one is characterized by a group velocity v_{−} and presents three energy
zones, diffusion-tunneling-Klein zones. For diffusion the two solutions
are, in general, different and could be decouple the two solutions in the
potential region showing the presence of a quaternionic potential. By
increasing the quaternionic part of the potential we increase the Klein
zone and decrease the tunneling zone, (**Figure 4b**). The most interesting
effect is seen for a pure quaternionic potential where total transmission
and a ȥ dependence on the spin operators appear. This suggests a
possible investigation in quaternionic optics. Before to do it, due to the
fact that we have to introduce Gaussian laser, in a forthcoming, we aim
to extend the plane wave analysis done in this article to quaternionic
wave packets.

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