The scattering and bound states in relativistic and non-relativistic quantum mechanics with external potentials have received attention from theorist in recent years^[1–3] and have assisted in the description of the behaviour of particles, atoms, and molecules in Physics. They find their applications in atomic and molecular Physics^[4–17] and in condensed matter physics.^[18–26] The Dirac equation covers the anti-particle scattering as well as the particle scattering. The scattering states have continuum wave functions and its energy is within the neighbourhood of E ≥ 0 whereas the bound states with normalizable wave functions have energy E < 0. For a free Dirac particle, there exist energy gaps E ≤ |m| (where m is the mass of the particle) that separate the positive and negative energy continuum states. The positive states correspond to the particle states and the negative energy states describe the anti-particle states. The energy gap becomes distorted on the introduction of a potential V(r) and bound states now occur between E = −m and E = m, which can be described as the band gap energy in condensed matter Physics.

The relativistic and non-relativistic symmetries have been investigated under a wide range of potentials.^[27–31] The Pöschl-Teller potential (PTP) in this regards has attracted the interest of many researchers in recent years owing to its applications in molecular and nuclear physics.^[32–40] The relativistic solutions to the PTP have been obtained with the Duffin-Kemmer-Petiau,^[1] Dirac,^[32–34] Klein-Gordon,^[35] and the Schrödinger wave equation.^[36–37] For instance, Jia et al.^[32] obtained the analytical solutions of the Dirac equation with the generalized Pöschl-Teller potential including the pseudo-centrifugal term while the relativistic symmetries with the trigonometric Pöschl-Teller potential plus Coulomb-like tensor interaction have also been obtained by Falaye and Ikhdair.^[33] The bound and scattering state of the q-hyperbolic Pöschl-Teller (qHPT) has also been investigated for the Duffin-Kemmer-Petiau equation.^[1] The scattering and bound states of a spinless Klein-Gordon equation with the generalized PTP has been studied^[35] and the author calculated the eigenvalues, normalized wave functions, and the scattering phase shift respectively. In Ref. [34] the spin symmetry for the Dirac equation with modified qHPT in D dimensions were also solved and the relativistic energy spectrum was obtained by using the Nikiforov-Uvarov method. Also in Ref. [36] the bound state solutions of the Schröinger wave equation with the generalized Pöschl-Teller potential in D spatial dimension was obtained. The Pöschl-Teller potential is a typical diatomic molecular potential that has related applications in relativistic and nonrelativistic cases for real diatomic molecules.^[37–38] For instance, Jia et al.^[37] used the Improved Pöschl-Teller potential energy model in fitting the experimental RKR potential curves over a large range of internuclear distances for six molecules and was found to fit better than the Morse potential. Other molecular potential models of interest that have been improved upon in this regard include the Tietz,^[39] Manning-Rosen,^[40–41] and the Rosen-Morse.^[42] the Dirac equation has been used to study the deformation of nuclei,^[38] which plays a major role in understanding the deformation potentials in quantum dots (QDs).^[38–40] A study made by some authors have shown that in the absence of mass term, the Dirac equation can be used to obtain the bound states of confined graphene QDs.^[18–26]

Motivated by the diverse applications of the PTP model in condensed matter and molecular Physics, the scattering and bound states solution of the qHPT potential will be studied using the Dirac equation.

Accordingly, the q-parameter hyperbolic Pöschl-Teller potential (qHPT) is given by^[1]

The organization of this paper consists of four sections: In Sec. 2, we review the basic Dirac equations with the qHPT potential and sought for the scattering states in terms of the hypergeometric function. In Sec. 3 the solutions of the bound states were also calculated and finally the conclusion in Sec. 4.

The basic theories and equations governing the Dirac particles are given in Refs. [27–34].

Let us recall that the Dirac equation with the scalar potential S(x) and vector potential V(x) in one-dimensional is given by^[27]

2.1 The Case of Σ(x) = constant

First of all, let us consider the case for which Σ(x) = C_p = constant so that Δ(x) has the asymmetric qHPT potential (V(x)) given in Eq. (1).

In solving for the scattering states, we study the wave functions for x < 0 and then the Dirac equation with the q-parameter hyperbolic Pöschl-Teller Potential (qHPT) is given by inserting Eq. (1) into Eq. (7) to obtain

Applying the variable Z_L = (1 + (1/q)e^2αx)⁻¹, we obtain the following equation:

where,

We sought for the solution at the region x > 0 by insert Eq. (1) into Eq. (7) to obtain

On changing variable we obtain the following form of the hypergeometric equation

s where

2.2 Transmission and Reflection Coefficients for the Case Σ(x) = constant

Equations (9) and (12) have singularities at z = 0, z = 1, and z = ∞, we may, therefore, define the following trial wave functions as

which turns into a hypergeometric differential equation^[27]

We sought for the scattering states by looking at the trial wave functions G^p(z) for the region (left) x < 0. The solution to Eq. (15) is the second type of hypergeometric function^[27]

with the parameters a^p, b^p, and c^p given as

Finally, from Eqs. (14) and (16), we obtain

Equation (18) has the form of a hypergeometric equation and thus, by comparison, we obtain

and from the upper term of Eq. (4) we have that

Now we sought for the physical interpretation of the problem under investigation so as to obtain the desired result, the solutions so far obtained must be used with appropriate boundary conditions as x → − ∞ and x → + ∞. By applying the asymptotic behaviour of the wave function in Eq. (18) for x → − ∞, z_L → 0, and (1 − z)^υ → 1, we have

The right-side solution is written as

At this region we find a plane wave traveling from left to right (no reflection occurs) so that R₃ = 0 and Eq. (22) reduces to

Now we consider the asymptotic behaviour of the right for which x > 0 and in the limit x → ∞, z_R → 0, and Eq. (23) becomes

Therefore from Eqs. (21) and (24), we may write

from the upper component of Eq. (3) we have that

So that the upper component of the wave function in the infinity limit is

Matching the two solutions G^pL(x = 0) and G^pR(x = 0) are done by applying the continuity of the wave function and its derivatives at x = 0, i.e. G^pL (x = 0) = G^pR (x = 0) and G^pL′ (x = 0) = G^pR′ (x = 0), which respectively give

where

In arriving at Eq. (29) we use the formula of the hypergeometric function, i.e.

Recall that the probability current density for the Dirac equation is given by

from Eqs. (31) and (33). We can compute the current density in the asymptotic regions,

where the incident, reflected and transmitted fluxes are

The continuity conditions on the current density give

In order to find the bound state solution for the Dirac particle with qHPT, we map 4λ(λ − 1) → − V₀ and the potential assume a square well form. Accordingly Eq. (1) takes the form,

We have solved the exact solution of a relativistic one-dimensional Dirac equation for the asymmetric q-parameter hyperbolic Pöschl-Teller potential and have obtained in terms of hypergeometric functions the scattering states as well as transmission and reflection coefficient using the continuity conditions of the wave function and its derivatives. The bound state solution is obtained by vanishing the determinant of the coefficients of the wave function for the pHPT potential. This study can find its applications to physics especially condensed matter Physics in view of the recent development in grapheme QD materials.