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Scattering solutions of the Klein-Gordon equation for a step potential with hyperbolic tangent potential
Clara Rojas∗ CEIF IVIC Apdo 21827, Caracas 1020A, Venezuela
Received 28 March 2014 We solve the Klein-Gordon equation for a step potential with hyperbolic tangent potential. The scattering solutions are derived in terms of hypergeometric functions. The reflection coefficient R and transmission coefficient T are calculated, we observed superradiance and transmission resonances. Keywords: Hypergeometric functions, Klein-Gordon equation, Scattering theory PACS numbers: 02.30.Gp,03.65.Pm, 03.65.Nk
1. Introduction The study of the scattering solutions of the Klein-Gordon equation1–6 and for the Dirac equation7, 8 with different potentials has been extensively studied in recent years. The phenomenon of superradiance, when the reflection coefficient R is greater than one, has been widely discussed. Manogue9 discussed the superradiance on a potential barrier for Dirac and Klein-Gordon equations. Sauter10 and Cheng11 have studied the same phenomenon for the hyperbolic tangent potential with the Dirac equation. Superradiance for the Klein-Gordon equation with this particular potential has been studied for Cheng11 and Rojas.12 Transmision resonances, when the transmission coefficient T is one, has been observed in the scattering of scalar relativistic particles in Klein-Gordon equation1, 4 and the Dirac equation.7, 8 In this paper we have calculated the scattering solutions of the Klein-Gordon equation in terms of hypergeometric functions for a step potential with hyperbolic tangent potential. The reflection coefficient R and transmission coefficient T are calculated numerically. The behaviour of the reflection R and transmission T coefficients is studied for three different regions of energy. We have observed superradiance11, 13 and transmission resonances. This paper is organized of the following way. Section 2 shows the one-dimensional Klein-Gordon equation. In section 3 the step potential with hyperbolic tangent po∗
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tential is shown. Section 4 shows the scattering solutions and the behaviour of the reflection coefficient R and transmission coefficient T . Finally, in section 5 conclusions are discussed. 2. The Klein-Gordon equation The one-dimensional Klein-Gordon equation to solve is, in natural units ~ = c = 114 o d2 φ(x) n 2 2 + [E − V (x)] − m φ(x) = 0, (1) dx2 where E is the energy, V (x) is the potential and, m is the mass of the particle. 3. Step potential with hyperbolic tangent potential The step potential with hyperbolic tangent potential has the following form −a, for x < x0 , V (x) = a, for x0 < x < x1 , a tanh(b x), for x > x1 ,
(2)
where a represents the height of the potential and b gives the smoothness of the curve. The value of x0 and x1 represents the position of the step. The form of step potential with hyperbolic tangent potential is showed in the Fig. (1). From Fig. 1 we can note that the step potential with hyperbolic tangent potential reduces to two barriers or two wells for b → ∞.
Fig. 1. Step potential with hyperbolic tangent potential for a = 5 with b = 2 (solid line) and b = 50 (dotted line).
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4. Scattering States In order to consider the scattering solutions for x < x0 , we solve the differential equation i d2 φ(x) h 2 2 + (E + a) − m φ(x) = 0, dx2
(3)
φI (x) = b1 e−irx + c1 eirx ,
(4)
Eq. (3) has the general solution
where r =
q
2
(E + a) − m2 .
The scattering solutions for x0 < x < x1 are obtained by solving the differential equation i d2 φ(x) h 2 2 φ(x) = 0, + (E − a) − m dx2
(5)
φII (x) = b2 e−iqx + c2 eiqx ,
(6)
Eq. (5) has the general solution
where q =
q
2
(E − a) − m2 .
Now we consider the solutions for x > x1 o d2 φ(x) n 2 2 φ(x) = 0. + [E − a tanh(bx)] − m dx2
(7)
On making the substitution y = −e2bx , Eq. (7) becomes # " 2 d 1+y dφ(y) 2 4b y E+a y + − m φ(y) = 0. dy dy 1−y 2
(8)
Putting φ( y) = y α (1−y)β f (y), Eq. (8) reduces to the hypergeometric differential equation y(1 − y)f ′′ + [(1 + 2α) − (2α + 2β + 1)y]f ′ − (α + β − γ)(α + β + γ)f = 0,
(9)
where the primes denote derivates with respect to y and the parameters α, β, and γ are
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p
(E + a)2 − m2 , √ 2b b + b2 − 4a2 , β = λ with λ = p 2b (E − a)2 − m2 γ = iµ with µ = . 2b
α = iν with ν =
(10) (11) (12)
Eq. (9) has the general solution in terms of Gauss hypergeometric functions 15 F 2 1 (µ, ν, λ; y) φ(y) = C1 y α (1 − y) + C2 y
−α
(1 −
β
2 F1 (α + β − γ, α + β + γ, 1 + 2α; y) β y) 2 F1 (−α + β − γ, −α + β + γ, 1 − 2α; y) .
(13)
In terms of variable x Eq. (13) becomes φ(x) = C1 −e2bx + C2 −e2bx
iν
1 + e2bx
−iν
λ
1 + e2bx
2 F1
λ
2 F1
iν + λ − iµ, iν + λ + iµ, 1 + 2iν; −e2bx
−iν + λ + iµ, −iν + λ − iµ, 1 − 2iν; −e2bx . (14)
From Eq. (14) the reflected and incident waves are φref (x) = b3 1 + e2bx
λ
φinc (x) = c3 1 + e2bx We define
e−2ibνx 2 F1 −iν + λ + iµ, −iν + λ − iµ, 1 − 2iν; −e2bx . (15)
λ
e2ibνx 2 F1 iν + λ − iµ, iν + λ + iµ, 1 + 2iν; −e2bx . (16)
φIII (x) = b3 φref (x) + c3 φinc (x)
(17)
Using the relation15
2 F1 (a, b, c; z)
Γ(c)Γ(b − a) (−z)(−a) 2 F1 (a, 1 − c + a, 1 − b + a; z −1 ) Γ(b)Γ(c − a) Γ(c)Γ(a − b) + (−z)(−b) 2 F1 (b, 1 − c + b, 1 − a + b; z −1 ), Γ(a)Γ(c − b) =
(18)
with Eqs. (15), (16) and keeping the solution that asymptotically corresponds to a transmitted particle moved from left to right, the transmited wave becomes
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φtrans (x) = c4 e−2bλx 1 + e2bx We define
λ
e2ibµx 2 F1 iν + λ − iµ, −iν + λ − iµ, 1 − 2iµ; −e−2bx . (19)
φIV (x) = φtrans (x).
(20)
When x → ±∞ the V → ±a and the asymptotic behaviour of Eqs. (4) and, (20) are plane waves with the relations of dispersion r, q, ν and, µ φI (x) = b1 e−irx + c1 eirx , φIV (x) = c4 e
2ibµx
,
(21) (22)
In order to find R and T , we used the definition of the electrical current density for the one-dimensional Klein-Gordon equation (1) ~ − φ∇φ ~ ∗ ~j = i φ∗ ∇φ 2
(23)
The current as x → −∞ can be descomposed as jL = jI , where jI correspond to the incident current jinc minus the reflected current jref . Analogously we have that, on ther right side, as x → ∞ the current is jR = jIV , where jIV is the transmitted current jtras .1 The reflection coefficient R, and the trasmission coefficient T , in terms of the reflected jinc , jref , and jtrans currents are jref |b1 |2 . = jinc |c1 |2
(24)
jtrans 2bµ |c4 |2 = . jinc r |c1 |2
(25)
R=
T =
The reflection coefficient R, and the transmission coefficient T satisfy the unitary relation T + R = 1 and are expresses in terms of the coefficients b1 , c1 and, c4 . In order tor obtain R and T we proceed to equate at x = x1 , x = x0 and, x = 0 the wave functions and their first derivatives. From the matching condition we derive a system of equations governing the dependende of coefficients b1 , b2 , c1 , c2 , b3 and, c3 on c4 that can solve numerically with the Software Wolfram Mathematica 9. The dispersion relations r and µ must be positive because it correspond to an incident particle moved from left to right and, their sign depends on the group velocity, defined by16
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r′ dE = ≥ 0. ′ dr E+a
(26)
dE µ′ = ≥ 0. dµ′ E−a
(27)
For the step potential with hyperbolic tangent potential we have studied three different regions. In the region a − m > E > m, µ′ < 0 and, r′ > 0 we have that R > 1, so superradiance occurs. In the region a + m > E > a − m the dispersion relations µ and r are imaginary pure and the transmited wave is attenuated, so R = 1. In the region E > a + m we observed transmission resonances. Figs. 2(a) and 2(b) shows the reflection R and transmission T coefficients for E > m, a = 5, b = 2, x0 = −4 and, x1 = −2. Fig. 3(a) and 3(b) shows the reflection R and transmission T coefficients R for E > m, a = 5, b = 50, x0 = −4 and, x1 = −2. Figs. 4(a) and 4(b) shows the reflection R and transmission T coefficients for E > m, a = 5, b = 2, x0 = −6 and, x1 = −1. Fig. 5(a) and 5(b) shows the reflection R and transmission T coefficients R for E > m, a = 5, b = 50, x0 = −6 and, x1 = −1. We observed in Figs. 2(a), 3(a), 4(a) and, 5(a) that in the region a − m > E > m the reflection coefficient R is bigger than one whereas the coefficient of transmission T becomes negative, so we observed superradiance.11, 13 In Figs. 2(b), 3(b), 4(b) and, 5(b) we find transmission resonances in the region E > a + m. On the other hand, the change in the values of x0 and x1 affects both the phenomenon of superradiance and transmission resonances: the position, height and numbers of the peaks depends on x0 and x1 .
3
1
0
T
R
2
-1
1
-2
0m
a-m
a+m
E
(a)
2a
-3m
a-m
a+m
2a
E
(b)
Fig. 2. The reflection R and transmission T coefficients varying energy E for the step potential with hyperbolic tangent potential for m = 1, a = 5, b = 2, x0 = −4 and, x1 = −2.
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1
0
T
R
2
-1
1
-2
0 m a-m a+m 2a
4a
-3 m a-m a+m 2a
6a
4a
6a
E
E
(a)
(b)
Fig. 3. The reflection R and transmission T coefficients varying energy E for the step potential with hyperbolic tangent potential for m = 1, a = 5, b = 50, x0 = −4 and, x1 = −2. 3
1
0
T
R
2
-1
1
-2
0m
a-m
a+m
-3m
2a
a-m
a+m
2a
E
E
(a)
(b)
Fig. 4. The reflection R and transmission T coefficients varying energy E for the step potential with hyperbolic tangent potential for m = 1, a = 5, b = 2, x0 = −6 and, x1 = −1. 3
1
0
T
R
2
-1
1
-2
0 m a-m a+m 2a
4a
E
(a)
6a
-3 m a-m a+m 2a
4a
6a
E
(b)
Fig. 5. The reflection R and transmission T coefficients varying energy E for the step potential with hyperbolic tangent potential for m = 1, a = 5, b = 50, x0 = −6 and, x1 = −1.
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5. Conclusion In this paper we have discussed the scattering solutions of the Klein-Gordon equation for a step potential with hyperbolic tangent potential. The solutions are determined in terms of hypergeometric functions. The calculation of the reflection coefficient R and transmission coefficient T is shown. For the region where a − m > E > m, the phenomenon of superradiance occurs. In the case b = 2 transmission resonances are observed for a + m < E < 2a in the two cases considered, while in the case where b = 50 transmission resonances appears for E > 4a for x0 = −4, x1 = −2 and E > 2a for x0 = −6, x1 = −1. References 1. C. Rojas and V. M. Villalba. Scattering of a Klein-Gordon particle by a Woods-Saxon potential. Phys. Rev. A, 71:052101, 2005. 2. C. Rojas and V. M. Villalba. The Klein-Gordon equation with the Woods-Saxon potential well. Rev. Mex. Fis, 52:127, 2006. 3. V. M. Villalba and C. Rojas. Bound states of the Klein-Gordon equation in the presence of short range potentials. Int. J. Mod. Phys. A, 21:313, 2006. 4. V. M. Villalba and C. Rojas. Scattering of a relativistic scalar particle by a cusp potential. Phys. Lett. A, 362:21, 2007. 5. O. Aydogdu, A. Arda and, R. Sever. Scattering and bound state solutions of the asymmetric Hulth´en potential. Phys. Scr., 84:025004, 2011. 6. O. Aydogdu, S. Alpdogan and, A. Havare. Relativistic spinless particles in the generalized asymmetric Woods-Saxon potential. J. Phys. A: Math. Theor., 46:015301, 2013. 7. P. Kennedy. The Woods-Saxon potential in the Dirac equation. J. Phys. A.: Math. Gen., 35:689, 2002. 8. V. M. Villalba and W. Greiner. Transmission resonances and supercritical states in a one-dimensional cusp potential. Phys. Rev. A, 67:052707, 2003. 9. C. A. Manogue. The Klein paradox and superradiance. Ann. Phys, 181:261, 1988. 10. F. Sauter. Zum “Kleinschen paradoxon”. Z. Phys, 73:547, 1931. 11. Q. Su, T. Cheng, M. R. Ware and R. Grobe. Pair creation rates for one-dimensional fermionic and bosonic vacua. Phys. Rev. A, 80:062105, 2009. 12. C. Rojas. Scattering of a relativistic particle by a hyperbolic tangent potential. Sent to publish, 2014. 13. Q. Su, R. E. Wagner, M. R. Ware and R. Grobe. Bosonic analog of the Klein paradox. Phys. Rev. A, 81:024101, 2010. 14. W. Greiner. Relativistic Quantum Mechanics. Wave equations. Springer, 1987. 15. M. Abramowitz and I. A. Stegun. Handbook of Mathematical Functions. Dover, New York, 1965. 16. A. Calogeracos and N. Dombey. History and physics of the Klein paradox. Contemp. Phys., 40:313, 1999.
