† Corresponding author. E-mail:
We obtain the quantized momentum eigenvalues Pn together with space-like coherent eigenstates for the space-like counterpart of the Schrödinger equation, the Feinberg–Horodecki equation, with a combined Kratzer potential plus screened coulomb potential which is constructed by temporal counterpart of the spatial form of these potentials. The present work is illustrated with two special cases of the general form: the time-dependent modified Kratzer potential and the time-dependent screened Coulomb potential.
Any physical phenomenon in nature is usually characterized by solving differential equations. A good example is the time-dependent Schrödinger equation that describes quantum-mechanical phenomena, in which it dictates the dynamics of a quantum system. Solving this differential equation by means of any method results in the eigenvalues and eigenfunctions of that Schrödinger quantum system. However, solving the time-dependent Schrödinger equation analytically is not straightforward except in some cases where the time-dependent potentials are constant, linear and quadratic functions of the coordinates.[1–4]
The Feinberg–Horodecki (FH) equation is a space-like counterpart of the Schrödinger equation derived by Horodecki[5] from the relativistic Feinberg equation.[6] This equation has been demonstrated in a possibility of describing biological systems[7,8] in terms of the time-like supersymmetric quantum mechanics.[9] The space-like solutions to the Feinberg–Horodecki (FH) equation can be employed to test its relevance in different areas of science including physics, biology and medicine.[7,8]
The space-like quantum systems with the Feinberg–Horodecki equation have attracted a great deal of attention from many scientists especially in some branches of physics, such as extended special relativity and extended quantum mechanics.[7,10–12] For example, they are used to explain the force between electric charges, the electric charge source, and the mass in a stable particle.[7,13] Among these studies, Molski has also constructed the space-like coherent states of a time-dependent Morse potential with the Feinberg–Horodecki equation and showed that the obtained results for space-like coherent states can be used for Gompertzian systems.[7]
Molski constructed the space-like coherent states of a time-dependent Morse oscillator on the basis of the FH quantal equation for minimizing the time-energy uncertainty relation and showed that the results are useful for interpreting the formation of the specific growth patterns during crystallization process and biological growth.[7] In addition, Molski obtained the FH equation to demonstrate a possibility of describing the biological systems in terms of the space-like quantum supersymmetry for an-harmonic oscillators.[8] Hamzavi et al.[14] obtained the exact bound state solutions to the FH equation with a rotating time-dependent Deng–Fan oscillator potential by means of the parametric NU method. Eshghi et al.[15] solved the FH equation for a time-dependent mass distribution (TDM) harmonic oscillator quantum system with a certain interaction applied to a mass distribution m(t) to provide a particular spectrum of stationary energies. In addition, the spectrum of harmonic oscillator potential V(t) acting on a TDM m(t) oscillator was obtained.
In the non-relativistic level, the Nikiforov–Uvarov (NU) method was used to obtain the bound state solutions to the arbitrary angular momentum Schrödinger equation with a modified Kratzer potential.[16] The factorization method was also used to obtain the solution of the non-central modified Kratzer potential for the diatomic molecules.[17] The exact solutions to the Schrödinger equation with modified Kratzer and corrected Morse potentials for position-dependent mass were also found.[18] The coherent states for a particle in Kratzer-type potentials are constructed by solving Feynman’s path integral.[19] Furthermore, the exact solution to the Schrödinger equation for the modified Kratzer potential plus a ring-shaped potential was solved.[20]
On the other hand, in the relativistic level, approximate solutions to the D-dimensional Klein–Gordon equation are obtained for the scalar and vector general Kratzer potential for any l by using the ansatz method and the solutions of the Dirac equation with equal scalar and vector ring-shaped modified Kratzer potential were found by means of the Nikiforov–Uvarov method.[21,22]
At the level of applications, some authors have studied the modified Morse–Kratzer potential for alkali hydrides,[23] and the effect of the modified Kratzer potential on the confinement of an exciton in a quantum dot.[24] Moreover, there have been analyses of applications of the modified Kratzer potential, the bound states of two limiting cases of interest of the interactions and hence this approximation to obtain the solution to the Schrödinger equation for the Morse potential.[25]
Very recently, a superposition of modified Kratzer potential plus screened Coulomb potential was suggested to study diatomic molecules.[26] Edet et al. have obtained an approximate solution to the Schrödinger equation for the modified Kratzer potential plus the screened Coulomb potential model, within the framework of the Nikiforov–Uvarov method. They obtained bound state energy eigenvalues for N2, CO, NO, and CH diatomic molecules for various vibrational and rotational quantum numbers. Special cases were considered when the potential parameters were altered, resulting in modified Kratzer potential, screened Coulomb potential, and standard Coulomb potential.[26] Further, Okorie et al. have solved the Schrödinger equation with the modified Kratzer plus a screened Coulomb potential using the modified factorization method. They have also employed both the Greene–Aldrich approximation scheme and a suitable transformation scheme to obtain the energy eigenvalue equation and its corresponding energy eigenfunctions for CO, NO, and N2 diatomic molecules. They have used the energy eigenvalues of the modified Kratzer plus the screened Coulomb potential to obtain the vibrational partition functions and other thermodynamic functions for the selected diatomic molecules.[27]
However, recently some authors used the improved Rosen–Morse potential and improved Tietz potential to represent the internal vibrations of diatomic molecules, including NO,[28–30] N2,[31] and CO,[32–34] and some triatomic molecules,[35–37] and successfully predicted the vibrational partition functions and important thermodynamic properties for some pure substances.
In the present work, we study the solutions to the Feinberg–Horodecki equation and extend the subject of coherent states to the space-like coherent states for the temporal counterpart of the Kratzer plus the screened Coulomb potential. The motivation of the present work is to obtain the eigen-solution to the Feinberg–Horodecki equation with a time-dependent modified Kratzer plus a screened Coulomb potential by means of the NU method. The rest of this work is organized as follows: the NU method is briefly introduced in Section
It is necessary to note that as the potential V = V(t) depends on t,
The NU method[38] is being applied to find the approximate solutions to the FH equation for the Kratzer plus a screened Coulomb potential, then the eigenvalues and eigenfunctions of two special cases are obtained from the results.
The time-dependent Kratzer potential is given by[39–41]
Using the Greene–Aldrich approximation[26] defined as
![](cpb_29_6_060303/cpb_29_6_060303_ieqn1.gif)
![](cpb_29_6_060303/cpb_29_6_060303_ieqn2.gif)
![](cpb_29_6_060303/cpb_29_6_060303_ieqn3.gif)
Due to the NU method used to obtain the eigenvalues, the polynomial solutions of the hypergeometric function yn(s) depend on the weight function ρ(s), which can be determined by solving Eq. (
![](cpb_29_6_060303/cpb_29_6_060303_ieqn4.gif)
![](cpb_29_6_060303/cpb_29_6_060303_ieqn5.gif)
To obtain the modified Kratzer potential from the general form, D, A and α are set to zero, q = 1, B = 2teDe and
On the other hand, to determine the eigenfunctions associated with the modified Kratzer potential, substituting the same parameters in Eq. (
Further by setting the values of B, C, and D to zero and A to –Ze2 we have the screened coulomb potential. By substituting these values in Eq. (
Setting α = 0 in the screened Coulomb potential, we can obtain the quantized eigenvalues of the FH equation with the Coulomb potential as follows:
We compute the momentum eigenvalues of the time-dependent Kratzer plus screened Coulomb potential for some diatomic molecules like CO, NO, O2 and I2. This was done using the spectroscopic parameters displayed in Table
![]() | Table 1. Spectroscopic parameters of the various diatomic molecules. . |
Its worth noting that when we solve the FH equation in the absence of the interaction potential, i.e., V(t) = 0, the quantized momentum eigenvalues are negative. Further, in the presence of Coulomb interaction potential V(t) = –ze2/t, the quantized momentum still stands negative as shown by Eq. (
![]() | Table 2. The FH quantized momentum eigenvalues (in units of eV/c) with the Kratzer plus the screened Coulomb potential for the diatomic molecules (α = 0.001, De, te and μ as defined in Table |
![]() | Table 3. The FH quantized momentum eigenvalues (in units of eV/c) with the Kratzer plus the screened Coulomb potential for the diatomic molecules (α = 0.05, De, te and μ as defined in Table |
![]() | Table 4. The FH quantized momentum eigenvalues (in units of eV/c) for the Kratzer plus the screened Coulomb potential for the diatomic molecules (α = 0.1, De, te and μ as defined in Table |
In Fig.
In Fig.
![]() | Fig. 3. FH quantized momentum eigenvalues for the Kratzer plus the screened Coulomb potential for the diatomic molecules. |
In Fig.
![]() | Fig. 4. FH quantized momentum eigenvalues for the Kratzer plus the screened Coulomb potential for the diatomic molecules. The graphs are plotted vs De for α = 0.005. |
Figure
![]() | Fig. 5. FH quantized momentum eigenvalues with the modified Kratzer potential for the diatomic molecules. |
We have solved the Feinberg–Horodecki equation for the time-dependent general form of the Kratzer potential via the Nikiforov–Uvarov method, and obtain the approximate quantized momentum eigenvalues solution to the FH equation. It is worth mentioning that the method is elegant and powerful. Our results can be applied in biophysics and other branches of physics. In this paper, we have applied our result for the modified Kratzer and screened coulomb potentials, as special cases of the used potential, for quantized momentum eigenvalues.
[1] | |
[2] | |
[3] | |
[4] | |
[5] | |
[6] | |
[7] | |
[8] | |
[9] | |
[10] | |
[11] | |
[12] | |
[13] | |
[14] | |
[15] | |
[16] | |
[17] | |
[18] | |
[19] | |
[20] | |
[21] | |
[22] | |
[23] | |
[24] | |
[25] | |
[26] | |
[27] | |
[28] | |
[29] | |
[30] | |
[31] | |
[32] | |
[33] | |
[34] | |
[35] | |
[36] | |
[37] | |
[38] | |
[39] | |
[40] | |
[41] |