scattering and bound states of spinless particles in a mixed vector–scalar smooth step potential

Annals of Physics 324 (2009) 2372–2384

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Annals of Physics

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Scattering and bound states of spinless particles in amixed vector–scalar smooth step potential

M.G. Garcia, A.S. de Castro *

UNESP – Campus de Guaratinguetá, Departamento de Física e Química, 12516-410 Guaratinguetá, SP, Brazil

a r t i c l e i n f o

Article history:Received 5 May 2009Accepted 27 May 2009Available online 3 June 2009

Keywords:Klein–Gordon equationBound statesScattering

0003-4916/$ - see front matter � 2009 Elsevier Indoi:10.1016/j.aop.2009.05.010

* Corresponding author.E-mail address: [email protected] (A.

a b s t r a c t

Scattering and bound states for a spinless particle in the back-ground of a kink-like smooth step potential, added with a scalaruniform background, are considered with a general mixing of vec-tor and scalar Lorentz structures. The problem is mapped into theSchrödinger-like equation with an effective Rosen–Morse poten-tial. It is shown that the scalar uniform background present subtleand trick effects for the scattering states and reveals itself a high-handed element for formation of bound states. In that process, itis shown that the problem of solving a differential equation forthe eigenenergies is transmuted into the simpler and moreefficient problem of solving an irrational algebraic equation.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

The one-dimensional step potential is of certain interest to model the transition between twostructures. In solid state physics, for example, a step-like potential which changes continuously overan interval whose dimensions are of the order of the interatomic distances can be used to model theaverage potential which holds the conduction electrons in metals. In the presence of strong potentials,though, the Schrödinger equation must be replaced by their relativistic counterparts. The scattering ofspin-1/2 particles by a square step potential, considered as a time component of a vector potential, iswell-known and crystallized in textbooks [1]. In that scenario it appears the celebrated Kleins paradox[3]. The analysis of the same problem with the Klein–Gordon (KG) equation was not neglected [2,4].The background of the kink configuration of the /4 model ðtanh cxÞ [5] is of interest in quantum fieldtheory where topological classical backgrounds are responsible for inducing a fractional fermion num-ber on the vacuum. Models of these kinds, known as kink models are obtained in quantum field theory

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S. de Castro).

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M.G. Garcia, A.S. de Castro / Annals of Physics 324 (2009) 2372–2384 2373

as the continuum limit of linear polymer models [6]. In a recent paper the complete set of bound statesof fermions in the presence of this sort of kink-like smooth step potential has been addressed by con-sidering a pseudoscalar coupling in the Dirac equation [7]. A peculiar feature of the kink-like potentialis the absence of bounded solutions in a nonrelativistic theory because it gives rise to an ubiquitousrepulsive potential. Of course this problem neatly reveals that our nonrelativistic preconceptionsare mistaken.

It is well known from the quarkonium phenomenology that the best fit for meson spectroscopy isfound for a convenient mixture of vector and scalar potentials put by hand in the equations (see, e.g.[12]). The same can be said about the treatment of the nuclear phenomena describing the influence ofthe nuclear medium on the nucleons [13]. It happens that when the vector and scalar potentials fulfillthe conditions for spin and pseudospin symmetries, i.e. they have the same magnitude, the energyspectrum does not depend on the spinorial structure, being identical to the spectrum of a spinless par-ticle [14].

In the present work the scattering a spinless particle in the background of a kink-like smooth steppotential, added with a scalar uniform background, is considered with a general mixing of vector andscalar Lorentz structures. Although the scalar potential finds many of their applications in nuclear andparticle physics, it could also simulate an effective mass term in solid state physics and so it could beuseful for modelling transitions between structures such a Josephson junctions [15]. It is often useful,because of simplicity, to approximate the behaviour of relativistic fermions by spinless particles obey-ing the KG equation. It turns out that some results almost do not depend on the spin structure of theparticle, e.g. the onset of scaling in some structure functions in the case of relativistic quark modelsused for studying quark-hadron duality [8], the photoelectron spectra in the strong field laser-inducedionization and recollision process [9], the electric polarizability of the ground state of a particle boundin a strong Coulomb field [10], and the differential scattering cross-section for forward scattering [11].Nevertheless, our purpose is to investigate the basic nature of the phenomena without entering intothe details involving specific applications. In other words, the aim of this paper is to search new solu-tions of a fundamental equation in physics which can be of help to see more clearly what is going oninto the details of a more specialized and complex circumstance. In passing, it is shown that a seriousproblem with the square step potential, overlooked in the literature, does not manifest for the smoothstep potential. Our problem is mapped into an exactly solvable Sturm–Liouville problem of aSchrödinger-like equation with an effective Rosen–Morse potential which been applied in discussingpolyatomic molecular vibrational states [16]. The scalar uniform background makes its influence notonly for the scattering states but reveals itself a high-handed element for formation of bound states. Inthat process, the problem of solving a differential equation for the eigenenergies is transmuted intothe simpler and more efficient problem of solving an irrational algebraic equation. With this method-ology the whole relativistic spectrum is found, if the particle is massless or not. Nevertheless, boundedsolutions do exist only under strict conditions.

2. The KG equation with vector and scalar potentials

The (1 + 1)-dimensional KG equation for a free particle of rest mass m corresponding to the relativ-istic energy-momentum relation E2 ¼ c2p2 þm2c4, where the energy E and the momentum p aresubstituted by operators, i�ho=ot and � i�ho=ox, respectively, acting on the wave function Uðx; tÞ. Here,c is the speed of light and �h is the Planck constant ð�h ¼ h=ð2pÞÞ. In the presence of external potentialsthe energy-momentum relation becomes

E� Vtð Þ2 ¼ c2 p� Vsp

c

� �2

þ mc2 þ Vs� �2 ð1Þ

where the subscripts for the potentials denote their properties under a Lorentz transformation: t andsp for the time and space components of a vector potential, and s for the scalar potential. A continuityequation for the KG equation

oqotþ oJ

ox¼ 0 ð2Þ

2374 M.G. Garcia, A.S. de Castro / Annals of Physics 324 (2009) 2372–2384

is satisfied with q and J defined as

q ¼ i�h2mc2 U�

oUot� oU�

otU

� �� Vt

mc2 Uj j2

J ¼ �h2im

U�oUox� oU�

oxU

� �� Vsp

mcUj j2

ð3Þ

Note that the KG equation is covariant under the charge-conjugation operation, meaning that the KGequation remains invariant under the simultaneous transformations U! �U�, E! �E, p! �p,Vsp ! �Vsp and Vt ! �Vt . In other words, if U is a solution for a particle (antiparticle) with energyE and momentum p for the potentials Vt , Vsp and Vs, then �U� is a solution for a antiparticle (particle)with energy �E and momentum �p for the potentials �Vt , �Vsp and Vs. Note also thatq! �q and J ! �J. These last results are of particular importance to interpret q and J as charge den-sity and charge current density, respectively, and to recognize that the vector potential couples withthe charge of the particle/antiparticle whereas the scalar potential couples with the mass, as one couldsuspect from the appearance of Vs in (1) and from the absence of Vs in (3). Furthermore, the changeE! �E and related change i�ho=ot ! � i�ho=ot permit us to conclude that if the particle travels forwardin time then the antiparticle travels backward in time.

For time-independent potentials the KG equation admits solutions in the form

Uðx; tÞ ¼ uðxÞe i�hcK xð Þ e�

i�hEt ð4Þ

where u satisfies the time-independent KG equation

� �h2

2mu00 þ Veffu ¼ Eeff u ð5Þ

with

Eeff ¼E2 � mc2

� �2

2mc2 ; Veff ¼V2

s � V2t

2mc2 þ Vs þE

mc2 Vt ð6Þ

and

K xð Þ ¼Z x

dgVsp gð Þ ð7Þ

The density and flux corresponding to (4) are then

q ¼ E� Vt

mc2 uj j2; J ¼ �h2im

u�ouox� ou�

oxu

� �ð8Þ

Since q and J are independent of time, u is said to describe a stationary state. Notice that the densitybecomes negative in regions of space where Vt > E, so that the KG wave function must be normalizedas
Z þ1
�1dx

E� VtðxÞmc2 uðxÞj j2 ¼ �1 ð9Þ

where the ± sign must be used for

E ?

Rþ1�1 dxVtðxÞ uðxÞj j2R þ1�1 dx uðxÞj j2

ð10Þ

Meanwhile, in the nonrelativistic approximation (potential energies small compared tomc2 and E ’ mc2) Eq. (5) becomes

� �h2

2md2

dx2 þ Vt þ Vs

!u ¼ E�mc2

� �u ð11Þ


so that u obeys the Schrödinger equation with binding energy equal to E�mc2 without distinguishingthe contributions of vector and scalar potentials. Furthermore, the density and current (and the nor-malization condition with the plus sign too) reduce precisely to the corresponding values of the non-relativistic theory.

It is remarkable that the KG equation with a scalar potential, or a vector potential contaminatedwith some scalar coupling, is not invariant under V ! V þ const., this is so because the vector poten-tial couples to the charge of the particle, whereas the scalar potential couples to the mass of the par-ticle. Therefore, if there is any scalar coupling the absolute values of the energy will have physicalsignificance and the freedom to choose a zero-energy will be lost. As we will see explicitly in this work,a constant added to the scalar potential is undoubtedly physically relevant. As a matter of fact, it canplay a crucial role to ensure the existence of bound-state solutions even though the bound states arenot present in the nonrelativistic limit of the theory. It is well known that a binding potential in thenonrelativistic approach is not binding in the relativistic approach when it is considered as a Lorentzvector. It is not immediately obvious that relativistic binding potentials may only result in scatteringstates in the nonrelativistic approach. The secret lies in the fact that vector and scalar potentials cou-ple differently in the KG equation whereas there is no such distinction among them in the Schrödingerequation. This observation permit us to conclude that even a ‘‘repulsive” potential might be a bona fidebinding potential.

3. The mixed vector–scalar kink-like potential and the effective Rosen–Morse potential

Now let us focus our attention on scalar and vector potentials in the form of smooth steps:

Vt ¼ gtV ; Vs ¼ gs Vþ Vð Þ

V ¼ const; V ¼ V0

21þ tanh

x2L

� � ð12Þ

where the dimensionless coupling constants, gt and gs, are real numbers constrained by gt þ gs ¼ 1.The positive parameter L is related to the range of the interaction which makes V to change noticeablyin the interval �2L < x < 2L, and as L! 0 the potential approximates the square step potential. V0 > 0is the height of the potential at x ¼ þ1. The uniform background makes the height of the scalar po-tential at x ¼ �1 to be gsV. The reason for including a scalar uniform background will be clear later –it makes possible the existence of bound-state solutions.

Before proceeding, it is useful to note that the effective potential corresponding to (12) is recog-nized as the exactly solvable Rosen–Morse potential [16,17]

Veff ¼ �V1sech2 x2Lþ V2 tanh

x2Lþ V3 ð13Þ

where the following abbreviations have been used:

V1 ¼ 2gs � 1ð Þ V20

8mc2

V2 ¼ 2V1 þV0

2mc2 gsMc2 þ gtE� �

V3 ¼ V2 þ Eeff �E2 � Mc2

� �2

2mc2

ð14Þ

and

M ¼ mþ gsV

c2 ð15Þ

The Rosen–Morse potential approaches V3 � V2 as x! �1, and has an extremum when jV2j < 2jV1j at

xm ¼ L ln2V1 � V2

2V1 þ V2

� �ð16Þ


given by

Veff xmð Þ ¼ V3 � V1 1þ V2

2V1

� �2" #

ð17Þ

The second derivative of Veff at xm is given by

V 00eff xmð Þ ¼1

2V1ð Þ32V1ð Þ2 � V2

2

2L

" #2

ð18Þ

in such a way that the extremum is a local minimum (maximum) only if V1 > 0 ðV1 < 0Þ. Note that theextremum, if it exists at all, is a minimum (maximum) only if gs > 1=2 ðgs < 1=2Þ. In particular, thesymmetric Rosen–Morse potential is that one with V2 ¼ 0 which can be obtained with

g2s V ¼ � gs mc2 þ gt Eþ 2gs � 1ð ÞV0

2

� ð19Þ

As a matter of fact, potential-well structures can be achieved when jV2j < 2jV1j with V1 > 0 and apressing need for bound-state solutions implies that Eeff defined in (6) must satisfyVeff ðxmÞ < Eeff < Veff ð�1Þ. From the condition Eeff < Veff ð�1Þ one concludes that

Ej j < jMjc2; jE� gtV0j < Mc2 þ gsV0

ð20Þ

Writing VeffðxmÞ as

Veff xmð Þ ¼ V2 � V1 1þ V2

2V1

� �2" #

þ Eeff �E2 � Mc2

� �2

2mc2 ð21Þ

one can see that the condition Eeff > Veff ðxmÞ turns into

E – � gt

gsMc2 ð22Þ

According to this last result, there is no crossing between energy levels with E > �gtMc2=gs (to beassociated with the particle levels) and those ones with E < �gtMc2=gs (to be associated with the par-ticle levels). This fact implies that there is no channel for spontaneous particle–antiparticle creation, sothat the single-particle interpretation of the KG equation is preserved. In particular, null energies arenot permissible for bound-state solutions in the case of a pure scalar coupling ðgs ¼ 1Þ, when the en-ergy levels are disposed symmetrically about E ¼ 0. Furthermore, the coefficient V2 in (13) can also beexpressed as

V2 ¼ �2V1 þV0

2mc2 gsMc2 þ gtEþ 2gs � 1ð ÞV0

h ið23Þ

From this and from the definition of V2 in (14), it follows that

gsMc2 þ gtE < 0 ð24Þ
and
gsMc2 þ gtEþ 2gs � 1ð ÞV0 > 0 ð25Þ
Combining these two inequalities one concludes that an additional sine qua non-condition for theexistence of bounded solutions is that Mc2 must be into the limits
� 2gs � 1gs

V0 þgt

gsE

� �< Mc2 < � gt

gsE ð26Þ

To acknowledge that the effective potential for the mixing given by (12) is a Rosen–Morse potentialcan help you to see more clearly how a kink-like smooth step potential might furnish bound-statesolutions. After all, we shall not use the knowledge about the exact analytical solution for the Ro-sen–Morse potential.


3.1. The asymptotic solutions

As jxj � L the effective potential is practically constant (the main transition region occurs injxj < 2L) and the solutions for the KG equation can be approximate by those ones for a free particle.Furthermore, the asymptotic behaviour will show itself suitable to impose the appropriate boundaryconditions to the complete solution to the problem.

The vector and scalar potentials approach to zero and to gsV as x! �1. Hence the solution for theKG equation can be written as

u �1ð Þ ¼ Aþeþikx þ A�e�ikx ð27Þ

where

�hck ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � Mc2

� �2r

ð28Þ

For j E j> jMjc2, the solution expressed by (27) describes plane waves propagating on both directionsof the X-axis with group velocity vg ¼ ðdE=dkÞ=�h equal to the classical velocity. If we consider that par-ticles are incident on the potential ðE > jMjc2Þ, Aþ expðþikxÞ will describe particles coming to the po-tential region from �1 ðvg ¼ þc2�hk=E > 0Þ, whereas A� expð�ikxÞ will describe reflected particlesvg ¼ �c2�hk=E < 0� �

. The flux corresponding to u given by (27), is expressed as

Jð�1Þ ¼ Jinc � Jref ð29Þ

where

Jinc ¼�hkm

Aþj j2; Jref ¼�hkm

A�j j2 ð30Þ

Note that the relation J ¼ qvg maintains for the incident and reflected waves, since

q�ð�1Þ ¼E

mc2 A�j j2 ð31Þ

On the other hand, the vector and scalar potentials approach to V0 and to gs (vþ V0) as x! þ1. In thisasymptotic region one should have vg P 0 in such a way that the solution in this region of space de-scribes an evanescent wave or a progressive wave running away from the potential potential region.The general solution has the form

u þ1ð Þ ¼ Bþeþijx þ B�e�ijx ð32Þ

where

�hcj ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE� gtV0ð Þ2 � Mc2 þ gsV0

� �2r

ð33Þ

Due to the twofold possibility of signs for the energy of a stationary state, the solution involvingB� cannot be ruled out a priori. As a matter of fact, this term may describe a progressive wave withnegative charge density and phase velocity vph ¼ jEj=ð�hjÞ > 0. It is true that if j 2 R the solutiondescribing a plane wave propagating in the positive direction of the X-axis with group velocityvg ¼ �c2�hj=ðE� gtV0Þ > 0 is possible only if E ? gtV0 with B� ¼ 0. In this case the density and the fluxcorresponding to u given by (32) are expressed as

qðþ1Þ ¼ E� gtV0

mc2 B�j j2; Jðþ1Þ ¼ Jtrans ¼ ��hjm

B�j j2 ð34Þ

If j is imaginary one can write j ¼ �iQ with Q 2 R, and (32) with B� ¼ 0 describes an evanescent waveðvg ¼ 0Þ. The condition B� ¼ 0 is necessary for furnishing a finite charge density as x!1. In this case

qðþ1Þ ¼ E� gtV0

mc2 B�j j2e�2Qx; Jðþ1Þ ¼ 0 ð35Þ

When j 2 R, a bizarre circumstance occurs as long as E < gtV0 since both qðþ1Þ and Jðþ1Þ are neg-ative quantities. The maintenance of the relation J ¼ qvg , though, is a license to interpret the solution


B�e�ijx as describing the propagation, in the positive direction of the X-axis, of particles with chargesof opposite sign to the incident particles. This interpretation is consistent if the particles moving inthis asymptotic region have energy �E and are under the influence of a potential �gtV0. It means that,in fact, the progressive wave describes the propagation of antiparticles in the positive direction of theX-axis. If j is imaginary, though, the solution with E > gtV0 ðE < gtV0Þ describes an evanescent waveassociated with particles (antiparticles).

Defining

e ¼ gt

1� gtjMjc2; Vc ¼

EþMc2

2gt�1 ; for gt >12

1; for gt 612

(ð36Þ

and the following cases:

� Case I. M > 0.� Case II. M < 0 with gt 6 1=2.� Case III. M < 0 with 1=2 < gt < 1 and E > e.� Case IV. M < 0 with 1=2 < gt < 1 and E < e.

One can readily envisage that the segregation between j real and j imaginary allows us to identifythree distinct class of scattering solutions for particles depending on V0:

� Class A. j is real with V0 < E�Mc2 for the cases I and III, and V0 < Vc for the case IV.� Class B. j is imaginary with E�Mc2 < V0 < Vc for the cases I, II, III, and Vc < V0 < E�Mc2 for the

case IV.� Class C. j is real with V0 > Vc for the cases I and III, and V0 > E�Mc2 for the case IV.

Now we focus attention on the calculation of the reflection ðRÞ and transmission ðTÞ coefficients.The reflection (transmission) coefficient is defined as the ratio of the reflected (transmitted) flux tothe incident flux. Since oq=ot ¼ 0 for stationary states, one has that J is independent of x. This fact im-plies that

R ¼ jA�j2

jAþj2ð37Þ

T ¼ � jkjB�j2

jAþj2; for j 2 R; B� ¼ 0

0; for j ¼ �iQ

(ð38Þ

For all the cases one should have Rþ T ¼ 1, as expected for a conserved quantity. This fact is easilyverified for j imaginary. For j real one has to wait for the complete solution of the problem whoseasymptotic behaviour allows one to calculate the amplitudes of all waves relative to amplitude ofthe incident wave. Is is instructive to note that the case with Bþ ¼ 0 presents R > 1, the alluded Kleinsparadox, implying that more particles are reflected from the potential region than those incoming.Note that for V > �mc2=gs ðM > 0Þ the threshold for pair production is equal to 2mc2 for gt ¼ 1and greater than 2mc2 for 1=2 < gt < 1. For V < �mc2=gs ðM < 0; gt < 1Þ, though, one has that thethreshold is equal to 2jMjc2, and that the threshold tends to zero as V tends to �mc2=gs. In this lastcircumstance, pair production occurs for every V0, however small.

It is worthwhile to note that the asymptotic solutions with k ¼ �iq, where q 2 R, and j ¼ �iQ ,might describe the possible existence of bound states realized beforehand in the previous section,since Jð�1Þ ¼ 0. In this case, one has to impose that A� ¼ 0 for k ¼ �iq.

3.2. The complete solutions

Armed with the knowledge about asymptotic solutions and with the definitions of the reflectionand transmission coefficients we proceed for searching solutions on the entire region of space.


Changing the independent variable x in (5) to

y ¼ 12

1� tanhx

2L

� �ð39Þ

the KG equation is transformed into the differential equation for uðyÞ:
y 1� yð Þu00 þ 1� 2yð Þu0 þHu ¼ 0 ð40Þ
where

H ¼ L�hc

� �2 E� gtV0 1� yð Þ½ �2 � mc2 þ gs Vþ V0 1� yð Þ½ �� 2

y 1� yð Þ ð41Þ

Introducing a new function wðyÞ through the relation

uðyÞ ¼ ym 1� yð ÞlwðyÞ ð42Þ
and defining
a ¼ lþ mþ 1�x2

; b ¼ lþ mþ 1þx2

; C ¼ 2mþ 1

l2 ¼ � kLð Þ2; m2 ¼ � jLð Þ2; x2 ¼ 1þ 2gs � 1ð Þ 2LV0

�hc

� �2 ð43Þ

Eq. (40) becomes the hypergeometric differential equation [18]

y 1� yð Þw00 þ C � aþ bþ 1ð Þy½ �w0 � abw ¼ 0 ð44Þ
whose general solution can be written in terms of the Gauss hypergeometric series
2F1 a; b;C; yð Þ ¼ C Cð ÞC að ÞC bð Þ

X1n¼0

C aþ nð ÞC bþ nð ÞC C þ nð Þ

yn

n!ð45Þ

in the form [18]

w ¼ A2F1 a; b; C; yð Þ þ By�2m2F1 aþ 1� C; bþ 1� C;2� C; yð Þ ð46Þ

in such a way that

u ¼ Aym 1� yð Þl2F1 a; b; C; yð Þ þ By�m 1� yð Þl2F1 aþ 1� C; bþ 1� C;2� C; yð Þ ð47Þ

with the constants A and B to be fitted by the asymptotic behaviour analyzed in the previousdiscussion.

As x! þ1 (that is, as y! 0), one has that y ’ expð�x=LÞ and (47), because 2F1ða; b;C;0Þ ¼ 1, re-duces to

u þ1ð Þ ’ Ae�mx=L þ Bemx=L ð48Þ

so the asymptotic behaviour requires that B ¼ 0, and A ¼ B� for m ¼ �ijL. The asymptotic behaviour asx! �1 ðy! 1Þ can be found by using the relation for passing over from y to 1� y:

2F1 a; b;C; yð Þ ¼ c�2F1 a; b; aþ b� C þ 1;1� yð Þ

þ cþ2F1 C � a;C � b;C � a� bþ 1;1� yð Þð1� yÞC�a�b ð49Þ

where cþ and c� are expressed in terms of the gamma function as

c� ¼C Cð ÞC C � a� bð ÞC C � að ÞC C � bð Þ ; cþ ¼

C Cð ÞC aþ b� Cð ÞC að ÞC bð Þ ð50Þ

which can also be written as

c� ¼C 2mþ 1ð ÞC �2lð Þ

C 1þx2 þ m� l

� �C 1�x

2 þ m� l� �

cþ ¼C 2mþ 1ð ÞC 2lð Þ

C 1þx2 þ mþ l

� �C 1�x

2 þ mþ l� � ð51Þ


Now, as x! �1, 1� y ’ expðþx=LÞ. This time, (47) tends to

u �1ð Þ ’ Acþe�lx=L þ Ac�eþlx=L ð52Þ

so that Ac� ¼ A� for l ¼ �ikL, in accordance with the previous analysis for very large negative valuesof x.

Therefore, the asymptotic behaviour of the general solution dictates that B ¼ 0 and establishes con-ditions on l and m, but not on x. The reflection (37) and transmission (38) coefficients can now beexpressed as

R ¼ jc�j2

jcþj2 ¼

C 1þx2 þ mþ l

� �C 1�x

2 þ mþ l� � 2

C 1þx2 þ m� l

� �C 1�x

2 þ m� l� � 2 C �2lð Þj j2

C 2lð Þj j2ð53Þ

T ¼ml

1jcþj

2 ¼ ml

C 1þx2 þmþlð ÞC 1�x

2 þmþlð Þj j2C 2mþ1ð ÞC 2lð Þj j2

; for m ¼ �ijL

0; for m 2 R

8<: ð54Þ

In the numerical evaluation of R and T one has not only to distinguish the sign of the imaginary part ofm but also if or not x is real. The following identities involving the gamma function [18]:

C z�ð Þ ¼ C� zð Þ; C zð ÞC 1� zð Þ ¼ psin pzð Þ ð55Þ

are sufficient enough to show that

C uþ ivð ÞC 1� uþ ivð Þj j2 ¼ 2p2

cosh 2pvð Þ � cos 2puð Þ ð56Þ

where u and v are the real and imaginary parts of z. Furthermore, the following identities will be useful[18]:

C ivð Þj j2 ¼ pv sinh pvð Þ ; C 1þ ivð Þj j2 ¼ pv

sinh pvð Þ ð57Þ

Hence, one can find for l ¼ �ikL and m ¼ �ijL:

R ¼cosh 2p k�jð ÞL½ ��cos p 1þxð Þ½ �cosh 2p k�jð ÞL½ ��cos p 1þxð Þ½ � 7 1; for x 2 R

cosh p k�jþNð ÞL½ � cosh p k�j�Nð ÞL½ �cosh p k�j�Nð ÞL½ � cosh p k�jþNð ÞL½ � 7 1; for x ¼ 2iNL; N 2 R

8<: ð58Þ

T ¼� 2 sinh 2pkLð Þ sinh 2pjLð Þ

cosh 2p k�jð ÞL½ ��cos p 1þxð Þ½ � ? 0; for x 2 R

� sinh 2pkLð Þ sinh 2pjLð Þcosh p k�j�Nð ÞL½ � cosh p k�jþNð ÞL½ � ? 0; for x ¼ 2iNL; N 2 R

8<: ð59Þ

whereas for l ¼ �ikL and m 2 R (j pure imaginary) one has T ¼ 0 and

R ¼C 1þx

2 þ m� ikL� �

C 1�x2 þ m� ikL

� � 2C 1þx

2 þ mþ ikL� �

C 1�x2 þ mþ ikL

� � 2 ¼ 1 8 x ð60Þ

At any circumstance, from the hyperbolic trigonometric identities involving coshðz1 þ z2Þ, one caneasily show that Rþ T ¼ 1 and that as x! 1 one finds:

R ¼k�jk�j

� �2; for l ¼ �ikL; m ¼ �ijL

1; for l ¼ �ikL; m 2 R

(ð61Þ

T ¼� 4kj

k�jð Þ2; for l ¼ �ikL; m ¼ �ijL

0; for l ¼ �ikL; m 2 R

(ð62Þ


Note that (61) and (62) reduce to the results for the square step potential [2,4], as they should be sincex! 1 as L! 0. Now, R and T blow up for j ¼ k (Class C), i.e. when (19) is satisfied. Of course, thiscrisis never mentioned in the literature does not mean that the KG theory fails. It only means thatthe calculations lose their validity for discontinuous potentials.

3.3. Bound states

As we have said, the solution expressed by (47) with l and m as real quantities (k and j as imag-inary numbers), viz.

l ¼ L�hc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMc2� �2

� E2

r

m ¼ L�hc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMc2 þ gsV0

� �2� E� gtV0ð Þ2

r ð63Þ

might describe the possible existence of bound states by imposing that B ¼ 0 and cþ ¼ 0. In view of(50) one has to locate the singular points of CðaÞCðbÞ. It happens that CðzÞ has simple poles only onthe real axis at z ¼ �n ðn ¼ 0;1;2; . . .Þ, and invoking the expression of a and b in terms of l,m and x given by (43) one concludes that x has got to be a real number. Evidently LV0 must be chosensuch that

LV0 <1; for gs P 1=2

�hc2ffiffiffiffiffiffiffiffiffiffi1�2gs

p ; for gs < 1=2

(ð64Þ

The quantization condition is thus given by a ¼ �n for x > 0 and b ¼ �n for x < 0. Since 2F1ða; b;C; yÞis invariant under exchange of a and b, one obtains a quantization condition independent of the sign ofx:

lþ mþ 1� jxj2

¼ �n; n ¼ 0;1;2; . . . ð65Þ

Eq. (65) can also be written in the form
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMc2� �2
� E2

rþ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiMc2 þ gsV0

� �2� E� gtV0ð Þ2

r

¼ �hc2L

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2gs � 1ð Þ 2LV0

�hc

� �2s

� 2nþ 1ð Þ

24

35; n ¼ 0;1;2; . . . ð66Þ

Now we have an irrational algebraic equation to be solved numerically, but there are still some ques-tions that one ought to get answered. Does it furnish proper solutions for the KG equation? Evidently uas in (47) is a square-integrable function, and l and m must be positive in order to furnish a wavefunction vanishing at x ¼ �1. Hence, the following supplementary conditions must be imposed:

Ej j < jMjc2; jE� gtV0j < Mc2 þ gsV0

ð67Þ

as given by (20), and

gs > 1=2; n ¼ 0;1;2; . . . < s ð68Þ

where

s ¼ 12�1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 2gs � 1ð Þ 2LV0

�hc

� �2s2

435 ð69Þ

The first pair of supplementary conditions ensures that l and m are positive. The second pair is nec-essary to make positive the right-hand side of (66). Note that this last pair of supplementary condi-tions imposes an additional restriction on the product LV0 beyond that one which makes x a real


number as given by (64). As a matter of fact, those conditions kill all the possibilities for bound states ifthe scalar coupling does not exceeds the vector coupling. At any rate, the possible solutions of (66)constitute a finite set of solutions. According to the second line of (68) and (69) one has

LV0 > �hc

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin nþ 1ð Þ2gs � 1

sð70Þ

This means that the number of allowed bound states increase with LV0, and there is at least one solu-tion, no matter how small is LV0. Now the Gauss hypergeometric series 2F1ða; b;C; yÞ reduces to noth-ing but a polynomial of degree n in y when a or b is equal to �n: Jacobis polynomial of index a and b.Indeed, for a ¼ �n one has [18]

2F1 a; b;C; yð Þ ¼ 2F1 �n;aþ 1þ bþ n;aþ 1; yð Þ ¼ n!

aþ 1ð ÞnP a;bð Þ

n nð Þ ð71Þ

where

a ¼ 2m; b ¼ 2l; n ¼ 1� 2y ¼ tanhx

2Lð72Þ

and ðaÞn ¼ aðaþ 1Þðaþ 2Þ ðaþ n� 1Þ with ðaÞ0 ¼ 1. Jacobis polynomials Pða;bÞn ðnÞ are orthogonalwith respect to the weighting function wða;bÞðnÞ ¼ ð1� nÞað1þ nÞb on the interval ½�1;þ1�, and canbe standardized as

P a;bð Þn 1ð Þ ¼ aþ 1ð Þn

n!ð73Þ

so that
Z þ1
�1dn w a;bð ÞðnÞP a;bð Þ

n nð ÞP a;bð Þn0 nð Þ ¼ dnn0h

a;bð Þn ð74Þ

where hða;bÞn is given by [18]

h a;bð Þn ¼ 2aþbþ1

2nþ aþ bþ 1C nþ aþ 1ð ÞC nþ bþ 1ð Þ

n!C nþ aþ bþ 1ð Þ ð75Þ

Hence the KG wave function can be written as (see Ref. [17]):

un nð Þ ¼ Nn 1� nð Þa=2 1þ nð Þb=2P a;bð Þn nð Þ ð76Þ

where Nn is given by

Nn ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ab

2aþbL aþ bð ÞC aþ bþ nþ 1ð ÞC nþ 1ð ÞC aþ nþ 1ð ÞC bþ nþ 1ð Þ

sð77Þ

in such a manner that
Z þ1
�1dx unðxÞj j2 ¼ 1 ð78Þ

and
Z þ1
�1dxVtðxÞ unðxÞj j2 ¼ V0

bbþ a

ð79Þ

Nevertheless, using (9) and (10) one can show that the normalized KG wave function must be writtenas

u Nð Þn ðnÞ ¼ N�unðnÞ ð80Þ

with

N� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�mc2 aþ b

aþ bð ÞE� gtV0b

s; � for E ? gtV0

bbþ a

ð81Þ


It is worthwhile to note that invoking the property Pða;bÞn ð�nÞ ¼ ð�1ÞnPðb;aÞn ðnÞ, one can conclude that theKG wave functions with definite parities (putting a ¼ b), associated with an even potential, are ob-tained on the condition that the uniform background satisfies (19), i.e. the effective potential is thesymmetric Rosen–Morse potential.

4. Conclusions

We have explored the influence of a scalar uniform background added to a mixed vector–scalarsmooth step potential. We have verified that the background presents drastic effects on bothscattering and bound-state solutions. The background increases the threshold for the pair productionwhen the vector coupling exceeds the scalar coupling and V > �mc2=gs. On the other hand, ifV < �mc2=gs the background decreases the threshold and pair production may occur for a potentialbarrier arbitrarily small. When the scalar coupling exceeds the vector coupling there appears thepossibility of a finite set of bound-state solutions. It is curious that the smooth step potential mighthold bound states in spite of the fact that the potential given by (12) is everywhere repulsive, so thatone cannot expect bound states in the nonrelativistic limit. Of course, the scalar uniform backgroundplays a peremptory role for the actual occurrence of bound states but no nonrelativistic limit can ex-pected since the background has to be in a range of values which do not acquiesce a nonrelativisticlimit.

Acknowledgment

This work was supported in part through funds provided by CNPq.

References

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scattering and bound states of spinless particles in a mixed vector–scalar smooth step potential

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