Bound states of relativistic particles in the generalized symmetrical double-well potential

doi:10.1016/j.physleta.2005.01.062

Physics Letters A

Volume 337, Issue 3, 4 April 2005, Pages 189-196

https://doi.org/10.1016/j.physleta.2005.01.062 Get rights and content

Abstract

Solving the Klein–Gordon and Dirac equations with equal generalized symmetrical double-well scalar and vector potentials by the method of the supersymmetric quantum mechanics, shape invariance approach and the alternative method, we obtain the exact energy equations and unnormalized wavefunctions for the s-wave bound states. Some interesting results including the standard symmetrical double-well potential, reflectionless-type potential and $V_{0} \tanh^{2} (r / d)$ potential are also discussed.

Introduction

When a particle is in a strong potential field, the relativistic effect must be considered, which yields the correction for non-relativistic quantum mechanics [1]. Taking the relativistic effects into account, a particle in a potential should be described with the Klein–Gordon equation and Dirac equation. There has been much interest in finding exact solutions of the Klein–Gordon equation and Dirac equation [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21]. Within the framework of the relativistic quantum mechanics, in order to solve problems which have been solved in the non-relativistic quantum mechanics, some authors have assumed that the scalar potential is equal to the vector potential and solved the Klein–Gordon equation and Dirac equation for some potentials, such as the Hulthen potential [13], [14], Wood–Saxon potential [15], reflectionless-type potential [16], pseudoharmonic oscillator [17], ring-shaped harmonic oscillator [18], $V_{0} \tanh^{2} (r / d)$ potential [19], five-parameter exponential-type potential [20], and Rosen–Morse-type potential [21], etc.

In 1997, Büyükkilic et al. [22] proposed a symmetrical double-well potential, which represents the vibrations of polyatomic molecules. They obtained the bound state energy spectra and the corresponding wavefunctions for the symmetrical double-well potential by using the Nikiforov–Uvarov method. Applying the symmetrical double-well potential model, they also obtained the concrete energy spectra and the corresponding wavefunctions for the ammonia molecule (NH₃). In 2003, Yang [23] used the deformed hyperbolic functions [24] to generalize the symmetrical double-well potential as the generalized symmetrical double-well potential. With the framework of non-relativistic quantum mechanics, the energy spectra of the generalized symmetrical double-well potential are determined by using the supersymmetry WKB method [23].

In the present work, we solve the s-wave Klein–Gordon and Dirac equations with equal scalar and vector generalized symmetrical double-well potentials by using the method of supersymmetric quantum mechanics [25], [26], [27], shape invariance approach [28], [29], and the alternative method [30]. We have obtained the exact energy equations and unnormalized wavefunctions for the generalized symmetrical double-well potentials within the framework of the relativistic quantum mechanics.

Section snippets

Bound states of the Klein–Gordon equation with scalar and vector generalized symmetrical double-well potentials

According to Ref. [2], the s-wave Klein–Gordon equation with scalar and vector potentials is $(ℏ = c = 1)$ ${\frac{d^{2}}{d r^{2}} + {[E - V (r)]}^{2} - {[M + S (r)]}^{2}} u (r) = 0,$ where E denotes the energy, M denotes the mass, and the radial wavefunction is $R (r) = u (r) / r$ . We consider the generalized symmetrical double-well potential model, which is expressed as [23] $V (r) = V_{1} \tanh_{q}^{2} α r - \frac{V_{2}}{\cosh_{q}^{2} α r},$ where the range of real parameter q is $q > 0$ . When $q = 1$ , the potential given in Eq. (2) turns to the standard symmetrical double-well potential

Bound states of the Dirac equation with scalar and vector generalized symmetrical double-well potentials

According to Ref. [13], the Dirac equation with scalar and vector potentials is $(ℏ = c = 1)$ ${C • P + D [M + S (r)]} Ψ (r) = [E - V (r)] Ψ (r) .$ In the relativistic quantum mechanics, the complete set of the conservative quantities for a particle in a central field can be taken as $(H, K, J^{2}, J_{z})$ , the eigenfunctions of which are given by [36] $Ψ = \frac{1}{r} [\begin{matrix} F (r) ϕ_{j m_{j}}^{A} \\ i G (r) ϕ_{j m_{j}}^{B} \end{matrix}] (k = j + 1 / 2),$ $Ψ = \frac{1}{r} [\begin{matrix} F (r) ϕ_{j m_{j}}^{B} \\ i G (r) ϕ_{j m_{j}}^{A} \end{matrix}] (k = - j - 1 / 2),$ where $ϕ_{j m_{j}}^{A} = \frac{1}{\sqrt{2 l + 1}} [\begin{matrix} \sqrt{l + m + 1} Y_{l, m} \\ \sqrt{l - m} Y_{l, m + 1} \end{matrix}],$ $ϕ_{j m_{j}}^{B} = \frac{1}{\sqrt{2 l + 3}} [\begin{matrix} - \sqrt{l - m + 1} Y_{l + 1, m} \\ \sqrt{l + m + 1} Y_{l + 1, m + 1} \end{matrix}] .$ Substituting Eq. (25a) or (25b)

Discussion

In this section, within the framework of the Klein–Gordon and Dirac theory with equally mixed potentials, we obtain the energy equations and corresponding wavefunctions for the standard symmetrical double-well potential, reflectionless-type potential and $V_{0} \tanh^{2} (r / d)$ potential by choosing appropriate parameters in the generalized symmetrical double-well potential model.

Conclusion

In this work, we may conclude that the Klein–Gordon and Dirac equations for equal scalar and vector generalized symmetrical double-well potentials can be solved exactly for s-wave bound states with the help of the method of supersymmetric quantum mechanics, shape invariance approach and the alternative method, so the relativistic mass-energy corrections can be obtained in a non-perturbative way. There exist mathematically identical energy equations for the generalized symmetrical double-well

Acknowledgment

The authors wish to thank the referee for his helpful comments and suggestions.

References (38)

F. Domínguez-Adame
Phys. Lett. A
(1989)
F. Domínguez-Adame et al.
Phys. Lett. A
(1995)
N.A. Rao et al.
Phys. Lett. A
(2002)
S.H. Dong et al.
Phys. Lett. A
(2003)
A.D. Alhaidari
Phys. Lett. A
(2004)
Y.F. Diao et al.
Phys. Lett. A
(2004)
L.Z. Yi et al.
Phys. Lett. A
(2004)
A. Arai
J. Math. Anal. Appl.
(1991)
E. Witten
Nucl. Phys. B
(1981)
F. Cooper et al.
Ann. Phys.
(1983)

F. Cooper et al.

Phys. Rep.

(1995)

C.S. Jia et al.

Phys. Lett. A

(2002)

C.S. Jia et al.

Phys. Lett. A

(2003)

I.C. Wang et al.

Phys. Rev. D

(1988)

V.M. Villalba et al.

J. Math. Phys.

(2002)

B.F. Samsonov et al.

Eur. J. Phys.

(2003)

O. Mustafa

J. Phys. A: Math. Gen.

(2003)

A.D. Alhaidari

Phys. Rev. A

(2002)

A.D. Alhaidari

J Phys. A: Math. Gen.

(2004)

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^☆: Work supported by the Sichuan Province Foundation of China for Fundamental Research Projects (04JY029-062-2).

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Bound states of relativistic particles in the generalized symmetrical double-well potential☆

Abstract

Introduction

Section snippets

Bound states of the Klein–Gordon equation with scalar and vector generalized symmetrical double-well potentials

Bound states of the Dirac equation with scalar and vector generalized symmetrical double-well potentials

Discussion

Conclusion

Acknowledgment

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

Phys. Lett. A

J. Math. Anal. Appl.

Nucl. Phys. B

Ann. Phys.

Phys. Rep.

Phys. Lett. A

Phys. Lett. A

Phys. Rev. D

J. Math. Phys.

Eur. J. Phys.

J. Phys. A: Math. Gen.

Phys. Rev. A

J Phys. A: Math. Gen.