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J. Math. Phys. 47, 102104-9 (2006)dma[imecc]UniCamp ΣδΛ

QUATERNIONIC DIFFUSION BY A POTENTIAL STEP

Stefano De Leo1 and Gisele C. Ducati2

1 Department of Applied Mathematics, University of CampinasPO Box 6065, SP 13083-970, Campinas, Brazildeleo@ime.unicamp.br

2 Department of Mathematics, University of ParanaPO Box 19081, PR 81531-970, Curitiba, Brazilducati@mat.ufpr.br

Submitted: August, 2006. Revised: September, 2006.

Abstract. In looking for qualitative differences between quaternionic and complex for-mulations of quantum physical theories, we provide a detailed discussion of the behaviorof a wave packet in presence of a quaternionic time-independent potential step. In thispaper, we restrict our attention to diffusion phenomena. For the group velocity of thewave packet moving in the potential region and for the reflection and transmission times,the study shows a striking difference between the complex and quaternionic formulationswhich could be matter of further theoretical discussions and could represent the startingpoint for a possible experimental investigation.

PACS. 03.65.Fd – 03.65.Nk

I. INTRODUCTION

Despite much research on quaternionic quantum mechanics, reviewed in its mathematical and physicalaspects in the excellent book of Adler [1], there have been few breakthroughs on the most naturalquestion about the effect that quaternionic potentials play in the dynamics of elementary particles[2–5] and, as consequence of it, about the possibility to look for an experimental proposal [7–9]. Inthis paper, by using the new mathematical tools developed in the analytic resolution of eigenvalueproblems [10, 11] and differential equations [6, 12–14], we analyze in detail the diffusion of a wavepacket by a quaternionic potential step.

The fundamental (mathematical) point in our discussion is the use of real quaternionic innerproducts and wave functions. The use of a division algebra is needed to guarantee that amplitudes ofprobability satisfy the requirement that, in the absence of quantum interference effects, the probabilityamplitude superposition reduces to the probability superposition. The associative law of multiplication(which for example fails for octonions) is needed to satisfy the completeness formula and to guaranteethe invariance of the inner product for anti-self-adjoint operators [1]. This does not mean that wecannot formulate a consistent quantum theory based on the use of complexified quaternions, Cliffordalgebras or octonionions. The requirement of an associative division algebra only applies to innerproducts. For example, in litterature we find interesting formulation of quantum theory based onthe use of wave functions defined in the complexified quaternionic algebra [15–21], in the space-timealgebra [22–26] and in the octonionic algebra [27, 28] but with inner products projected over thecomplex field. Nevertheless, from our point of view the choice of quaternionic inner products seemsto be the best choice in investigating deviations from standard quantum theories [29–32].

For the convenience of the reader and to facilitate access to the individual topics, this work isrendered as self-contained as possible. In Section II, we set up notation and terminology and proceedwith the study of diffusion by quaternionic potentials. This section contains the (analytic) plane wavesolution of the quaternionic Schrodinger equation in the presence of a potential step. This representsa fundamental mathematical tool in the discussion of the quaternionic stationary phase method (see

2 Stefano De Leo, Gisele C. Ducati: quaternionic diffusion by a potential step

Section III). We will touch only a few aspects of the theory of quaternionic integral transformsand restrict our attention to the diffusion of quaternionic wave packets with a peaked convolutionfunction. The advantage of using the stationary phase method lies in the fact that, in the presenceof a potential step, the motion of the wave packet can be correctly estimated by analyzing the phasederivative calculated at the maximum of the convolution function [33–35]. For a different shape ofpotentials, see for example the barrier, the stationary phase method, depending on the width of thepotential and on the group velocity of the incoming particle, could break down. There is a rich numberof articles leading with this problem in standard quantum mechanics [36–42].

The results of this paper (conclusion and outlooks are drawn in Section IV) shed some new lighton the properties of quaternionic potentials. In particular, it is explicitly shown how the presence ofa quaternionic perturbation modifies the momentum of the non-relativistic incoming particle and itsreflection (transmission) time. The study presented in this article represents a starting point in viewof a complete understanding of the behavior of wave packets impinging on quaternionic potentials. Adetailed analysis of this topic could be fundamental in looking for experiments in which deviationsfrom the complex quantum theory could be really seen. It is worth pointing out that the question offinding the best experimental proposal to prove the existence of quaternionic potentials is, at present,far from being solved and this paper aims to contribute to this debate.

II. REFLECTION AND TRANSMISSION COEFFICIENTS

The quaternionic Schrodinger equation in the presence of a constant potential is given by

i~

2mΨxx(x, t) − i V1 + j V2 + k V3

~Ψ(x, t) = Ψt(x, t) , (1)

where i V1+j V2+k V3 represents the quaternionic generalization of the anti-hermitian complex poten-tial i V1. For a complete discussion see ref. [1]. This partial differential equation, by the substitution

Ψ(x, t) = Φ(x) exp[− i E t/~] , (2)

can be reduced to the following ordinary second order differential equation with constant quaternioniccoefficients,

i~

2

2mΦ′′(x) − ( i V1 + j V2 + k V3 ) Φ(x) = −Φ(x)E i . (3)

The solution of the Schrodinger equation in the presence of constant quaternionic potential has beenmatter of study in the recent years [4–6]. New mathematical techniques, essentially based on the righteigenvalue problem for quaternionic operators [10, 12], allow to obtain the solution without the needto translate the quaternionic problem in its complex counterpart [2, 3]. In particular, in the presenceof a potential step and for the diffusion case

E >√

V 2

1+ V 2

2+ V 2

3,

the quaternionic plane wave solutions (for a detailed derivation see ref. [6]) are:

[I] free region (x < 0) : ΦI(x) = exp[ i ǫ x ] + r exp[− i ǫ x ] + j r exp[ ǫ x ] ,

[II] potential region (x > 0) : ΦII(x) = (1 + j w) t exp[ i ρ− x ] + (w + j) t exp[− ρ+ x ] ,(4)

where

ǫ =

√

2m

~2 E , ρ± =

√

2m

~2

(

√

E2 − V 2

2− V 2

3± V1

)

, w = − iV2 − iV3

E +√

E2 − V 2

2− V 2

3

,

and

t =2ǫ

ǫ + ρ−

[

1 − |w|2 ǫ − iρ−

ǫ + ρ−

ǫ + iρ+

ǫ + ρ+

]− 1

, t =iρ− − ǫ

ǫ + ρ+

w t ,

r =ǫ − ρ−

2ǫ

[

1 − |w|2 ǫ − iρ−

ǫ − ρ−

ǫ − iρ+

ǫ + ρ+

]

t , r =iρ− + ρ+

ǫ + ρ+

w t .

Stefano De Leo, Gisele C. Ducati: quaternionic diffusion by a potential step 3

From the current conservation

[

Ψ(x, t)Ψ(x, t)]

t=

~

2 m

[

Ψ(x, t) i Ψxx(x, t) + h.c]

, (5)

by recalling that we are considering stationary solutions of the Schrodinger equation, we obtain

Φ(x) i Φ′′(x) + h.c. = 0 .

This implies that the current density,

J(x) = Φ(x) i Φ′(x) + h.c , (6)

is a quantity independent of x. Due to the continuity of the wave function and its derivative, thecurrent density has to satisfy the following constraint

JI(0) = JII(0) . (7)

By using the explicit form of the plane wave solutions given in Eqs.(4), and the condition (7), astraightforward calculation conduces to

R + T = 1 (8)

whereR = |r|2 and T =

ρ−

ǫ(1 − |w|2) |t|2 .

Similarly to the predictions of complex quantum mechanics, the incident particle has a non-zeroprobability of turning back. Nevertheless, we know that in standard quantum mechanics no phaseis created by such reflection [35]. The situation drastically changes in the presence of a quaternionicperturbation. We shall come back to this point in Section III.

II.A REFLECTION AND TRANSMISSION PHASES

From the stationary wave functions given in Eqs.(4), we shall construct, by linear superposition, wavepackets and we shall study their time evolution (see Section III). In this spirit, it is convenient torewrite the reflection and transmission coefficients in terms of their modulus and phases. By simplealgebraic manipulations, we find

r =

√

[(ǫ − ρ−)(ǫ + ρ+) − |w|2 (ǫ2 − ρ−ρ+)]2 + |w|4ǫ2(ρ− + ρ+)2

[(ǫ + ρ−)(ǫ + ρ+) − |w|2 (ǫ2 + ρ−ρ+)]2 + |w|4ǫ2(ρ− − ρ+)2exp[ i θr ] ,

t =2 ǫ ( ǫ + ρ+ )

√

[(ǫ + ρ−)(ǫ + ρ+) − |w|2 (ǫ2 + ρ−ρ+)]2 + |w|4ǫ2(ρ− − ρ+)2exp[ i θt ] , (9)

where

θr = arctan

[

ǫ (ρ+ + ρ−) |w|2(ǫ − ρ−)(ǫ + ρ+) − |w|2 (ǫ2 − ρ−ρ+)

]

+ θt ,

θt = arctan

[

ǫ (ρ+ − ρ−) |w|2(ǫ + ρ−)(ǫ + ρ+) − |w|2 (ǫ2 + ρ−ρ+)

]

. (10)

The important point to be noted here is the dependence on the energy, E, and the complex imaginarypart of the potential, V1, as expected from the standard quantum case, and the new dependence onthe modulus of the pure quaternionic part of the potential, |V2 + i V3|. This last result means thatonce fixed the modulus of the quaternionic perturbation any rotation in the plane (V2 , V3) does notmodify the reflection and transmission coefficients. The quaternionic rotation invariance is due to thechoice of i as the imaginary unit in the anti-hermitian momentum operator, i~ ∂xx/2m, which appearsin the quaternionic Scrhodinger equation (1).

4 Stefano De Leo, Gisele C. Ducati: quaternionic diffusion by a potential step

II.B THE COMPLEX LIMIT

The standard (complex) quantum results can be obtained by taking a simple limit case, i.e. V2,3 → 0.In fact, by observing that

V2,3 → 0 ⇒

w → 0 ,

ρ− → σ =

√

2m

~2 (E − V1) ,

we findR = |r|2 and T =

σ

ǫ|t|2 ,

where

r =ǫ − σ

ǫ + σand t =

2 ǫ

ǫ + σ. (11)

As expected, the reflection and transmission coefficients (r and t) are real (θr = θt = 0) and thisimplies that there is no phase created by reflection or transmission [35].

II.C THE PURE QUATERNIONIC LIMIT

It is interesting to consider a second limit, i.e. V1 → 0. This represents the case of a pure quaternionicpotential. Noting that

V1 → 0 ⇒

ρ± → ρ =

√

2m

~2

√

E2 − V 2

2 − V 2

3 ,

|w|2 → ǫ2 − ρ2

ǫ2 + ρ2,

we obtain

R = |r|2 and T =2 ρ3

ǫ (ǫ2 + ρ2)|t|2 ,

where

r =ǫ − ρ√ǫ2 + ρ2

exp

[

i arctan

(

ǫ

ρ

)]

and t =ǫ

ρ. (12)

In this limit, the symmetry between reflection and transmission times is broken down. For a purequaternionic potential step, we find an instantaneous transmission but not an instantaneous reflection(we shall discuss in detail this point in Section III).

III. STATIONARY PHASE METHOD

Until now, we have been concerned only with plane waves. In this Section, we are going to studythe time evolution of quaternionic wave packets and deducing from them several important proper-ties. The principle of superposition guarantees that every real linear combination of the plane wavesΦI(x) exp[−iEt/~] and ΦII(x) exp[−iEt/~] will satisfy the Schrodinger equation in the presence of aquaternionic potential step.

Let g(ǫ) be a real convolution function with a maximum in ǫ0. In the free region (x < 0), thesuperposition can be written as follows

ΩI(x, t) =

∫

∞

ǫmin

dǫ g(ǫ) exp[ i ǫ x ] + r exp[− i ǫ x ] + j r exp[ ǫ x ] exp[− i ǫ2~ t/ 2m ] , (13)

where

ǫmin =

√

2m

~2

√

V 2

1 + V 2

2 + V 2

3 .

Stefano De Leo, Gisele C. Ducati: quaternionic diffusion by a potential step 5

The first term in Eq.(13) represents the incident wave, the second term the reflected wave and thethird term an evanescent wave. The phases for the incoming and reflected waves are

θinc[ǫ; x, t] = ǫ x − ǫ2~ t

2m,

θref[ǫ; x, t] = − ǫ x − ǫ2~ t

2m+ θr . (14)

The stationary phase condition (the derivative with respect to ǫ of the argument calculated in ǫ0

equal to zero) enables us to calculate the position of the maximum of the incident and reflected wavepackets:

xmax

inc(t) =

~ ǫ0

mt ,

xmax

ref (t) = −~ ǫ0

mt +

[

dθr

dǫ

]

0

. (15)

The maximum of the incident wave packet arrives at the step discontinuity at time t = 0 (as ithappens in the complex case). During a certain interval of time, the wave packet is localized in theregion x ∼ 0. For large times the incident wave packet has practically disappeared and we only findthe reflected wave packet. It is important to observe that contrary to the predictions of complexquantum mechanics (θr = 0), the maximum of the reflected wave packets is found at x = 0 at timet = (m/~ǫ0) [dθr/dǫ]0. This means that in the presence of a quaternionic perturbation we do nothave an instantaneous reflection: for large times the maximum of the reflected wave packet is not at−~ ǫ0 t/m but is shifted with respect to this value by a quantity equal to [dθr/dǫ]0.

An analogous discussion for the transmitted wave packet (x > 0),

ΩII(x, t) =

∫

dǫ g(ǫ)

t exp[ i ρ− x ] + w t exp[− ρ+ x ]

exp[− i ǫ2~ t/ 2m ] +

j

∫

dǫ g(ǫ)

w t exp[ i ρ− x ] + t exp[− ρ+ x ]

exp[− i ǫ2~ t/ 2m ] , (16)

where the phases to be considered are

θ(1, i)

tra [ǫ; x, t] = ρ− x − ǫ2~ t

2m+ θt

θ(j, k)

tra[ǫ; x, t] = ρ− x − ǫ2

~ t

2m+ θt + arctan

[

V2

V3

]

, (17)

leads to a similar conclusion for the transmitted time. Contrary to what happens in the standard(complex) quantum mechanics where there is an instantaneous transmission, in the presence of aquaternionic potential step, the maximum of the transmitted wave packet,

xmax

tra (t) =

~ ǫ0

mt −[

dθt

dǫ

]

0

/[

dρ−

dǫ

]

0

, (18)

is found at x = 0 at time t = (m/~ǫ0) [dθt/dǫ]0. At first glance, it could appear a logical consequenceof the result obtained for the reflection time. Nevertheless, it is important to note that θr 6= θt

and consequently the symmetry between reflection and transmission times is always broken down.For example, as it was explicitly shown in the previous section, instantaneous transmission does notnecessarily imply instantaneous reflection.

In order to simplify the discussion about the results obtained in our study, let us introduce thefollowing notation

V0 =√

V 2

1 + V 2

2 + V 2

3 ,

6 Stefano De Leo, Gisele C. Ducati: quaternionic diffusion by a potential step

and rewrite the maximum of the incident, reflected and transmitted wave packets in terms of E0 (themaximum value of the energy spectrum of the incoming particles)

xmax

inc (t) =

√

2E0

mt ,

xmax

ref(t) = −

√

2E0

mt +

~√2mV0

dθr

d√

EV0

0

,

xmax

tra(t) =

√

2E0

mt − ~√

2mV0

dθt

d√

EV0

0

/[

dρ−

dǫ

]

0

. (19)

The incident and reflected wave packets propagate, respectively, with velocities of v0 and −v0,

v0 =√

2E0/m . (20)

This is the standard result obtained in complex quantum mechanics. For the transmitted wave packetthe velocity is given by

vtra = v0/[

dρ−

dǫ

]

0

. (21)

Due to the fact that the quantity ρ− has an additional dependence on |V2 + iV3| with respect to thestandard dependence on V1, the complex and quaternionic formulations give different predictions. Forexample, of particular interest, it is the comparison between the group velocity of the transmittedwave packet for the complex case, V0 = V1,

v(i)

tra= v0/

(

√

ǫ2 − 2m

~2 V0

)

ǫ

= v0

√

1 − V0

E0

, (22)

and that one for the pure quaternionic case, V0 = |V2 + iV3|,

v(j, k)

tra = v0/

[

ǫ4 −(

2m

~2 V0

)2 ]1/4

ǫ

= v0

[

1 −(

V0

E0

)2 ]3/4

. (23)

A first important observation is that whereas v(i)tra

is greater or smaller than v0 depending on the signof V1, v(j, k)

trais always smaller than the group velocity in the fre region. For incident particles with

an energy spectrum peaked in E0, with E0 ≫ V0, the group velocities of the wave packet travellingin the potential region, (22) and (23), can be approximated by taking the first terms in their Taylorexpansions,

v(i)tra

= v0

[

1 − 1

2

V0

E0

− 1

8

(

V0

E0

)2 ]

+ O[(

V0

E0

)3 ]

,

v(j, k)tra = v0

[

1 − 3

4

(

V0

E0

)2 ]

+ O[(

V0

E0

)4 ]

.

This means that a clear difference between the complex and the (pure) quaternionic case is expectedfor the group velocity of a wave packet travelling in a region in which a small pertrbation is turnedon. In this spirit, it is also interesting to compare the reflection and transmission times,

√2mV0

~xmax

ref(0) =

dθr

d√

EV0

0

,

√2mV0

~xmax

tra(0) = −

dθt

d√

EV0

0

/[

dρ−

dǫ

]

0

. (24)

Stefano De Leo, Gisele C. Ducati: quaternionic diffusion by a potential step 7

Standard quantum mechanics predicts instanteneous reflection and transmission, i.e.

x(i) max

ref(0) = x(i) max

tra(0) = 0 .

For a pure quaternionic potential, the transmission, in analogy to the complex case, is instantaneous,

x(j, k) max

tra(0) = 0 ,

but the reflection time is different from zero (breaking down the instantaneity),

√2mV0

~x(j, k) max

ref (0) = −

2

√

E0

V0

E0

V0

+

√

(

E0

V0

)2

− 1

[(

E0

V0

)2

− 1

]3/4

−1

.

This predicts, for large times, that the maximum of the reflected wave packet should be found atthe left of the position predicted by standard quantum mechanics, i.e. −~ ǫ0 t/m. For E0 ≫ V0, thedifference between the complex and (pure) quaternionic case is only manifest at the third order inV0/E0, √

2mV0

~xmax

ref (0) = − 1

4

(

V0

E0

)3

+ O[(

V0

E0

)4 ]

,

and consequently, for small perturbations, we practically find an instantaneous reflection. It is im-portant to note that the shift in the position of the maximum of the reflected wave packet becomesimportant when E0 approaches to V0, this implies x(j, k) max

ref (0) → ∞. Nevertheless, for incident wavepackets peaked in E0 ∼ V0, a more careful analysis is needed. In fact, in this limit new effectshave to be considered and these effects cannot be obtained by simply using the stationary phasemethod [37, 39, 40].

IV. CONCLUSIONS AND OUTLOOKS

The study presented in this paper, and based on the use of wave packets, represents, from our point ofview, a first important attempt to discuss deviations from the standard (complex) quantum mechanicsin the presence of quaternionic potentials. The wave packet formalism, with respect to the previousanalysis, essentially based on the plane wave solutions, surely gives a more ”physical” focus. Forexample, this formalism allows to explicitly show the effect that quaternionic perturbations play inthe momentum distribution of elementary particles and, in the particular case of a potential step, tocalculate the new reflection and transmission times due to quaternionic interference phenomena. Toemphasize the main differences beteween the complex and the quaternionic formulation of quantummechanics for diffusion phenomena by a potential step, we have given, in the previous section, adetailed discussion based on the analytic study of the group velocities in the potential region and ofthe reflection time for complex and (pure) quaternionic potentials.

Now, let us return to the discussion for the general case, i.e. a complex potential in presence ofa quaternionic perturbation. In Fig.1-a, fixed the value of V0 and varying its complex component V1,we draw

vtra − v(i)tra

v(i)tra

(25)

as a function of E0/V0. The continuous line represents the case of a small complex component inthe quaternionic potential, consequently such a curve approximates the case of a pure quaternionicpotential,

v(j, k)tra

− v(i)tra

v(i)tra

=

(

1 − V0

E0

)1/4(

1 +V0

E0

)3/4

− 1−→

E0 ≫ V0

1

2

V0

E0

− 3

8

(

V0

E0

)2

+ O[(

V0

E0

)3 ]

.

It is also interesting to observe that the maximum of (v(j, k)tra − v(i)

tra)/v(i)tra is found at E0 = 2 V0.

The plots in Fig.1-b and Fig.1-c respectively show the behavior of the transmission and reflectiontimes as a function of E0/V0. Let us list some results coming out from our analysis. The quaternionicinterefernce phenomena at the step discontinuity produce an new interesting effect in the reflected and

8 Stefano De Leo, Gisele C. Ducati: quaternionic diffusion by a potential step

transmitted wave packets: the maxima of such packets are found at x = 0 before that the incident wavepacket reaches the potential step discontinuity. The symmetry between reflection and transmissiontime is broken down (see the amplification in Fig.1-b and Fig.1-c).

Evidently, all the physical consequences of our analysis regardless of whether we use a complex ora quaternionic potential in the Schrodinger equation deserve further investigation. Nevertheless, wethink that the discussion presented in this paper and based on the use of the wave packet formalismrepresents the starting point for further theoretical studies and a fundamental tool in looking forpossible experimental deviations from standard (complex) quantum mechanics.

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31. S.L. Adler, ”Quaternionic quantum field theory”, Phys. Rev. Lett. 55, 783786 (1985).32. S.L. Adler, ”Quaternionic quantum field theory”, Comm. Math. Phys. 104, 611656 (1986).33. P. T. Matthews, Introduction to quantum mechanics, (New York: McGraw-Hill, 1963).34. E. Merzbacher, Quantum mechanics, (New York: John Wiley & Sons, 1970).35. C. Choen-Tannoudji, B. Diu and F. Laloe, Quantum mechanics, (New York: John Wiley & Sons, 1977).36. T.E. Hartman, “Tunnelling of a wave packet”, J. Appl. Phys. 33, 3427-3432 (1962).37. A. Anderson, “Multiple scattering approach to one-dimensional potential problem”, Am. J. Phys. 57,

230-235 (1989).38. A. Pablo, L. Barbero, H.E. Hernandez-Figueroa, and E. Recami, ”Propagation speed of evanescent

modes”, Phys. E 62, 8628-8635 (2000).39. A. Bernardini, S. De Leo and P. Rotelli, “Above barrier potential diffusion”, Mod. Phys. Lett. A 19,

2717-2725 (2004).40. V.S. Olkhovsky, E. Recami, and J, Jakiel, “Unified time analysis of photon and particle tunnelling”, Phys.

Rep. 398, 133-178 (2004).41. S. De Leo and P. Rotelli, “Above barrier Dirac multiple scattering and resonances”, Eur. Phys. J. C 46,

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10 Stefano De Leo, Gisele C. Ducati: quaternionic diffusion by a potential step

0

0.1

0.2

0.3

0.4

0.5

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0.9

1.0

-0.1

-0.2

-0.3

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-0.5

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-0.8

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-1.0

1.0 2.0 3.0 4.0 5][1 2.0 3.0 4.0 5.0

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Fig. 1-aFig. 1-aFig. 1-aFig. 1-aFig. 1-aFig. 1-aFig. 1-aFig. 1-aFig. 1-aFig. 1-aFig. 1-aFig. 1-aFig. 1-aFig. 1-a Fig. 1-bFig. 1-bFig. 1-bFig. 1-bFig. 1-bFig. 1-bFig. 1-bFig. 1-bFig. 1-bFig. 1-bFig. 1-bFig. 1-bFig. 1-bFig. 1-b

Fig. 1-cFig. 1-cFig. 1-cFig. 1-cFig. 1-cFig. 1-cFig. 1-cFig. 1-cFig. 1-cFig. 1-cFig. 1-cFig. 1-cFig. 1-cFig. 1-c

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

vtra−v(i)tra

v(i)tra

vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

tra (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

√2mV0

~xmax

ref (0) vs.E0

V0

Amp: 5Amp: 5Amp: 5Amp: 5Amp: 5Amp: 5Amp: 5Amp: 5Amp: 5Amp: 5Amp: 5Amp: 5Amp: 5Amp: 5

Amp: 20Amp: 20Amp: 20Amp: 20Amp: 20Amp: 20Amp: 20Amp: 20Amp: 20Amp: 20Amp: 20Amp: 20Amp: 20Amp: 20

Amp: 1Amp: 1Amp: 1Amp: 1Amp: 1Amp: 1Amp: 1Amp: 1Amp: 1Amp: 1Amp: 1Amp: 1Amp: 1Amp: 1V1/V0 =V1/V0 =V1/V0 =V1/V0 =V1/V0 =V1/V0 =V1/V0 =V1/V0 =V1/V0 =V1/V0 =V1/V0 =V1/V0 =V1/V0 =V1/V0 = 0.10.10.10.10.10.10.10.10.10.10.10.10.10.1

0.30.30.30.30.30.30.30.30.30.30.30.30.30.3

0.50.50.50.50.50.50.50.50.50.50.50.50.50.5

0.70.70.70.70.70.70.70.70.70.70.70.70.70.7

0.90.90.90.90.90.90.90.90.90.90.90.90.90.9

Fig. 1. Fixed the value of V0 = |iV1 + jV2 +kV3| and varying its complex component V1, the group velocity ofthe transmitted wave packet, Fig.1-a, and the transmission/reflection times, Fig.1-b and Fig.1-c, are plottedas a function of E0/V0, where E0 is the maximum of the energy spectrum of the incident wave packet. Theanalysis is done for diffusion phenomena (E0 > V0).