Bound state solutions of the Klein Gordon equation with energy-dependent potentials

TL;DR

This paper derives exact bound state solutions for the Klein-Gordon equation with an energy-dependent Coulomb-like potential, introducing a novel energy-dependent mass function and analyzing various potential configurations.

Contribution

It presents a new approach by considering an energy-dependent mass function and provides analytic solutions for different potential cases using the asymptotic iteration method.

Findings

Derived explicit energy spectra for energy-dependent potentials

Identified relations among tuning parameters affecting solutions

Validated wave functions satisfy boundary conditions

Abstract

In this manuscript, we investigate the exact bound state solution of the Klein-Gordon equation for an energy-dependent Coulomb-like vector plus scalar potential energies. To the best of our knowledge, this problem is examined in literature with a constant and position dependent mass functions. As a novelty, we assume a mass-function that depends on energy and position and revisit the problem with the following cases: First, we examine the case where the mixed vector and scalar potential energy possess equal magnitude and equal sign as well as an opposite sign. Then, we study pure scalar and pure vector cases. In each case, we derive an analytic expression of the energy spectrum by employing the asymptotic iteration method. We obtain a non-trivial relation among the tuning parameters which lead the examined problem to a constant mass one. Finally, we calculate the energy spectrum by the Secant method and show that the corresponding unnormalized wave functions satisfy the boundary conditions. We conclude the manuscript with a comparison of the calculated energy spectra versus tuning parameters.

Tables

Table 1. Table 1: Calculated energy spectrum in the EMES limit of the mixed potential energies. Energy eigenvalues are denoted with two quantum numbers n 𝑛 n and l 𝑙 l as E n ​ l subscript 𝐸 𝑛 𝑙 E_{nl} . The units of δ 𝛿 \delta and λ ​ b 𝜆 𝑏 \lambda b are M ​ e ​ V − 1 𝑀 𝑒 superscript 𝑉 1 MeV^{-1} . The energy eigenvalue is in M ​ e ​ V 𝑀 𝑒 𝑉 MeV unit.

$\delta$	$\lambda b$	$E_{00}$	$E_{10}$	$E_{11}$	$E_{20}$	$E_{21}$	$E_{22}$	$E_{30}$	$E_{31}$	$E_{32}$	$E_{33}$
-0.00300	-0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	127.37813	None	130.83021	132.40021	129.92379	132.36407	133.19980	133.68234
-0.00300	0.00000	-3.03943	None	None	None	None	None	None	None	None	None
		None	106.22197	123.27091	123.27091	128.59284	130.94895	128.59284	130.94895	132.20124	132.94711
-0.00300	0.00300	-127.55222	-132.19801	-133.07406	-133.48904	-133.85027	-134.17528	-134.04496	-134.22986	-134.41219	-134.54598
		29.88598	103.11331	118.71065	120.60345	126.05962	129.21955	126.80482	129.32736	131.01021	132.05808
0.00000	-0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	114.45942	None	123.30931	127.81796	None	127.51480	129.98951	131.35124
0.00000	0.00000	-1.81465	None	None	None	None	None	None	None	None	None
		None	79.81538	107.32122	107.32122	118.69067	124.32234	118.69067	124.32234	127.48761	129.43379
0.00000	0.00300	-129.64146	-133.19720	-133.88866	-134.07683	-134.34733	-134.54808	-134.43223	-134.56603	-134.67698	-134.75200
		22.77861	83.76911	101.24833	106.48183	114.41062	120.62914	117.00422	121.19953	124.78258	127.26083
0.00300	-0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	98.91588	None	112.76280	121.45996	None	120.24283	125.36472	127.98363
0.00300	0.00000	-1.29096	None	None	None	None	None	None	None	None	None
		None	63.05730	92.57467	92.57467	107.70124	116.20530	107.70124	116.20530	121.36605	124.69813
0.00300	0.00300	-132.17136	-134.17013	-134.55301	-134.59518	-134.73627	-134.81915	-134.75456	-134.82187	-134.86714	None
		18.73778	71.69401	87.71693	95.21508	103.47799	111.34457	107.83252	112.63070	117.52188	121.28761

Table 2. Table 2: Calculated energy spectrum in the EMOS limit of the mixed potential energies. Energy eigenvalues are denoted with two quantum numbers n 𝑛 n and l 𝑙 l as E n ​ l subscript 𝐸 𝑛 𝑙 E_{nl} . The units of δ 𝛿 \delta and λ ​ b 𝜆 𝑏 \lambda b are M ​ e ​ V − 1 𝑀 𝑒 superscript 𝑉 1 MeV^{-1} . The energy eigenvalue is in M ​ e ​ V 𝑀 𝑒 𝑉 MeV unit.

$\delta$	$\lambda b$	$E_{00}$	$E_{10}$	$E_{11}$	$E_{20}$	$E_{21}$	$E_{22}$	$E_{30}$	$E_{31}$	$E_{32}$	$E_{33}$
-0.00300	-0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	None	None	None	None	None	None	None	None
-0.00300	0.00000	None	None	None	None	None	None	None	None	None	None
		None	None	None	None	None	None	None	None	None	None
-0.00300	0.00300	None	None	None	None	None	None	None	None	None	None
		None	133.47228	134.50966	134.39701	134.71651	134.81349	134.67191	134.81129	134.86369	134.89421
0.00000	-0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	None	None	None	None	None	None	None	None
0.00000	0.00000	None	None	None	None	None	None	None	None	None	None
		None	None	None	None	None	None	None	None	None	None
0.00000	0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	133.55348	None	134.19336	134.50388	None	134.48332	134.64992	134.74036
0.00300	-0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	None	None	None	None	None	None	None	None
0.00300	0.00000	None	None	None	None	None	None	None	None	None	None
		None	None	None	None	None	None	None	None	None	None
0.00300	0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	131.74999	None	133.23676	134.00740	None	133.89988	134.30896	134.50141

Table 3. Table 3: Calculated energy spectrum in the pure vector limit of the mixed potential energies. Energy eigenvalues are denoted with two quantum numbers n 𝑛 n and l 𝑙 l as E n ​ l subscript 𝐸 𝑛 𝑙 E_{nl} . The units of δ 𝛿 \delta and λ ​ b 𝜆 𝑏 \lambda b are M ​ e ​ V − 1 𝑀 𝑒 superscript 𝑉 1 MeV^{-1} . The energy eigenvalue is in M ​ e ​ V 𝑀 𝑒 𝑉 MeV unit.

$\delta$	$\lambda b$	$E_{00}$	$E_{10}$	$E_{11}$	$E_{20}$	$E_{21}$	$E_{22}$	$E_{30}$	$E_{31}$	$E_{32}$	$E_{33}$
-0.00300	-0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	133.95087	None	134.40744	134.62193	None	134.61559	134.73115	134.79771
-0.00300	0.00000	None	None	None	None	None	None	None	None	None	None
		None	None	132.00638	None	133.34195	133.96617	None	133.94393	134.27837	134.46865
-0.00300	0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	129.06586	None	131.74488	132.97652	None	132.93977	133.59648	133.97227
0.00000	-0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	131.78623	None	133.23041	133.94417	None	133.88182	134.26363	134.46326
0.00000	0.00000	None	None	None	None	None	None	None	None	None	None
		None	None	125.79630	None	129.94127	132.04770	None	131.82295	132.95215	133.52334
0.00000	0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	118.10358	None	125.52115	129.31889	None	128.98354	131.04809	132.13809
0.00300	-0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	127.37315	None	130.89006	132.84476	None	132.47080	133.50836	133.94133
0.00300	0.00000	None	None	None	None	None	None	None	None	None	None
		None	None	113.82811	None	122.99727	129.00451	None	127.50826	130.83289	132.06453
0.00300	0.00300	None	None	None	None	None	None	None	None	None	None
		None	None	102.72718	None	115.57032	123.97104	None	122.35322	127.21930	129.41673

Table 4. Table 4: Calculated energy spectrum in the pure scalar limit of the mixed potential energies. Energy eigenvalues are denoted with two quantum numbers n 𝑛 n and l 𝑙 l as E n ​ l subscript 𝐸 𝑛 𝑙 E_{nl} . The units of δ 𝛿 \delta and λ ​ b 𝜆 𝑏 \lambda b are M ​ e ​ V − 1 𝑀 𝑒 superscript 𝑉 1 MeV^{-1} . The energy eigenvalue is in M ​ e ​ V 𝑀 𝑒 𝑉 MeV unit.

$\delta$	$\lambda b$	$E_{00}$	$E_{10}$	$E_{11}$	$E_{20}$	$E_{21}$	$E_{22}$	$E_{30}$	$E_{31}$	$E_{32}$	$E_{33}$
-0.00300	-0.00300	-112.14923	-127.07890	-130.31170	-130.97926	-132.26062	-133.14380	-132.56605	-133.20127	-133.69216	-134.01646
		127.33455	132.98452	134.02566	134.07068	134.44112	134.63118	134.46021	134.63349	134.73678	134.80009
-0.00300	0.00000	-97.84172	-119.53740	-124.15598	-126.39841	-128.40409	-130.24408	-129.50566	-130.55972	-131.61308	-132.39564
		117.23009	129.98704	132.36231	132.63318	133.49989	134.01007	133.61571	134.02737	134.30500	134.47994
-0.00300	0.00300	-87.73377	-112.80110	-117.75088	-121.77591	-124.07708	-126.64236	-126.17297	-127.44184	-128.95466	-130.21501
		107.10831	126.36678	130.01599	130.78578	132.16035	133.09651	132.49330	133.15825	133.66895	134.00335
0.00000	-0.00300	-119.14454	-130.18354	-132.42563	-132.68033	-133.52057	-134.01920	-133.63229	-134.03603	-134.30943	-134.48239
		119.14454	130.18354	132.42563	132.68033	133.52057	134.01920	133.63229	134.03603	134.30943	134.48239
0.00000	0.00000	-105.71706	-124.55269	-128.49478	-129.60941	-131.19239	-132.38590	-131.70348	-132.49571	-133.15986	-133.61163
		105.71706	124.55269	128.49478	129.60941	131.19239	132.38590	131.70348	132.49571	133.15986	133.61163
0.00000	0.00300	-95.22484	-118.90859	-123.76368	-126.17664	-128.25226	-130.15812	-129.40852	-130.48890	-131.56907	-132.36880
		95.22484	118.90859	123.76368	126.17664	128.25226	130.15812	129.40852	130.48890	131.56907	132.36880
0.00300	-0.00300	-127.33455	-132.98452	-134.02566	-134.07068	-134.44112	-134.63118	-134.46021	-134.63349	-134.73678	-134.80009
		112.14923	127.07890	130.31170	130.97926	132.26062	133.14380	132.56605	133.20127	133.69216	134.01646
0.00300	0.00000	-117.23009	-129.98704	-132.36231	-132.63318	-133.49989	-134.01007	-133.61571	-134.02737	-134.30500	-134.47994
		97.84172	119.53740	124.15598	126.39841	128.40409	130.24408	129.50566	130.55972	131.61308	132.39564
0.00300	0.00300	-107.10831	-126.36678	-130.01599	-130.78578	-132.16035	-133.09651	-132.49330	-133.15825	-133.66895	-134.00335
		87.73377	112.80110	117.75088	121.77591	124.07708	126.64236	126.17297	127.44184	128.95466	130.21501

Equations

$$\displaystyle\Bigg{[}\hbar^{2}c^{2}\nabla^{2}+\bigg{(}\Big{(}E-V_{v}(r)\Big{)}^{2}-\Big{(}m(r,E)c^{2}+V_{s}(r)\Big{)}^{2}\bigg{)}\Bigg{]}\phi(r)$$

$$\displaystyle\Bigg{[}\hbar^{2}c^{2}\bigg{(}\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}\bigg{)}+\bigg{(}\Big{(}E-V_{v}(r)\Big{)}^{2}-\Big{(}m(r,E)c^{2}+V_{s}(r)\Big{)}^{2}\bigg{)}\Bigg{]}u(r)$$

$$m(r,E)$$

$$m(r,E)c^2$$

$$V(r,E)\equiv V_v(r,E)=V_s(r,E)=-\frac{A(1+\delta E)}{r}$$

$$\displaystyle\Bigg{[}\Bigg{(}\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}\Bigg{)}+\frac{1}{\hbar^{2}c^{2}}\Bigg{(}E^{2}-m(r,E)^{2}c^{4}-2V(r,E)\Big{(}E+m(r,E)c^{2}\Big{)}\Bigg{)}\Bigg{]}u(r)$$

$$\displaystyle u^{\prime\prime}(r)+\Bigg{[}\frac{E^{2}-m_{0}^{2}c^{4}}{\hbar^{2}c^{2}}+\Bigg{(}\frac{2\lambda bAm_{0}^{2}c^{4}}{\hbar^{2}c^{2}}+\frac{2A(E+m_{0}c^{2})}{\hbar^{2}c^{2}}\Bigg{)}\frac{(1+\delta E)}{r}$$

$$\displaystyle-\Bigg{(}\frac{m_{0}^{2}c^{4}\lambda^{2}b^{2}A^{2}}{\hbar^{2}c^{2}}+\frac{2\lambda bA^{2}m_{0}c^{2}}{\hbar^{2}c^{2}}+\frac{l(l+1)}{(1+\delta E)^{2}}\Bigg{)}\frac{(1+\delta E)^{2}}{r^{2}}\Bigg{]}u(r)=0.$$

$$\displaystyle\frac{d^{2}v}{dz^{2}}+\Bigg{(}-\tau^{2}+\frac{\beta^{2}}{z}-\frac{\eta(\eta+1)}{z^{2}}\Bigg{)}v=0.$$

$${\tau }^2$$

$${\beta }^2$$

$$\eta (\eta +1)$$

$$v(z)$$

$$f^{\prime \prime }(z)$$

$${\lambda }_0(z)$$

$$s_0(z)$$

$$\frac{s_0}{{\lambda }_0}$$

$$\frac{s_1}{{\lambda }_1}$$

$$\frac{s_2}{{\lambda }_2}$$

$${\tau }_n$$

$$\sqrt{\frac{m_0^2{c}^4-E_{nl}^2}{(1+\delta E_{nl}{)}^2}}$$

$$V(r,E)\equiv V_v(r,E)=-V_s(r,E)=-\frac{A(1+\delta E)}{r}$$

$$\displaystyle\Bigg{[}\Bigg{(}\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}\Bigg{)}+\frac{1}{\hbar^{2}c^{2}}\Bigg{(}E^{2}-m(r,E)^{2}c^{4}-2V(r,E)\Big{(}E-m(r,E)c^{2}\Big{)}\Bigg{)}\Bigg{]}u(r)$$

$${\tau }^2$$

$${\beta }^2$$

$$\eta (\eta +1)$$

$$\sqrt{\frac{m_0^2{c}^4-E_{nl}^2}{(1+\delta E_{nl}{)}^2}}$$

$$V_s(r)$$

$$V(r,E)\equiv V_v(r,E)$$

$$\displaystyle\Bigg{[}\Bigg{(}\frac{d^{2}}{dr^{2}}-\frac{l(l+1)}{r^{2}}\Bigg{)}+\frac{1}{\hbar^{2}c^{2}}\Bigg{(}E^{2}-m(r,E)^{2}c^{4}-V(r,E)\Big{(}2E-V(r,E)\Big{)}\Bigg{)}\Bigg{]}u(r)$$

$${\tau }^2$$

$${\beta }^2$$

$$\eta (\eta +1)$$

$$\sqrt{\frac{m_0^2{c}^4-E_{nl}^2}{(1+\delta E_{nl}{)}^2}}$$

$$V_v(r)$$

$$V(r,E)\equiv V_s(r,E)$$

Full text

Klein-Gordon equation, bound state solution, position-energy dependent mass, energy dependent Coulomb-like potential energy, asymptotic iteration method

I Introduction

One of the important goals in relativistic or non-relativistic quantum mechanics is to obtain an exact solution of a physical problem by employing potential energy Dirac_Book ; Landau_Book ; Greiner_Book . It is found that among the analytically solvable potential energies, only a few of them are very appropriate to represent the physical systems. For example, Coulomb potential energy is used to describe the Hydrogen atom. Bound state solution of the Coulomb-like potential energy is investigated in details in the relativistic Greiner_Book and non-relativistic Schiff_Book equations.

Klein-Gordon (KG) equation is one of the relativistic equation that is introduced to describe the dynamics of the bosonic particles. Although it was first introduced about a century ago Klein_1926 , it still attracts the scientists’ interest. Especially, researchers focus on the solutions of the KG equation with equal and unequal vector and scalar potential energies. Among a large number of articles, we would like to underline the investigations that employ Woods-Saxon Lutfuoglu_et_al_2018_LK , Hulthén dominguez_1989 , Morse Chen_2005 , Rosen-Morse Yi_et_al_2004 , Eckart Olgar_et_al_2006 , Manning-Rosen Jia_et_al_2013 , Pöschl-Teller Chen_et_al_2009 , Kratzer Saad_et_al_2008 , Hylleraas Ikot_2012 , multiparameter Diao_et_al_2004 , exponential-type molecule Ikot_et_al_2016 , pseudoharmonic oscillator Ikhdair_et_al_2008 , Hartmann Chen_et_2006 , double ring-shaped oscillator Fa-lin_et_2004 potential energies.

In the last decades, position-dependent mass (PDM) are gained popularity to interpret the properties of several quantum systems such as quantum liquids Arias_et_al_1994 , quantum wells Serra_et_al_1997 , quantum rings Li_et_al_2003 , quantum dots Peter_et_al_2008 , etc. More applications about quantum semiconductor structures can be found in the book of Weisbuch and Vinter Weisbuch_Book .

On the other hand, energy-dependent potential (EDP) energies are being used to describe the physical systems for a while Synder_1940 ; Schiff_1940 ; Green_1962 . It is worth noting that these applications cover a wide area of physics. For example, in mathematical physics Znojil_2004 ; Znojil_et_al_2004 ; Gunther_et_al_2007 ; Schulze_2013 , quantum electronics Milanovic_et_al_1996 ; Belyavski_et_al_2007 , nuclear physics Niyogi_et_al_1981 ; McKellar_1983 , quantum chromodynamics Martynenko_2008 ; Sanctis_2009 etc… In recent times, the number of studies considering several EDP energies in both relativistic and non-relativistic wave equations is increased Formanek_et_al_2004 ; Lombard_et_al_2007 ; Martinez_2009 ; HH1 ; HH2 ; Gupta_et_al_2012 ; Ikot_Hassanabadi_2013 ; Benchikha_et_al_2013 ; Ikot_2 ; Benchikha_et_al_2014 ; Benchikha_et_al_2017_1 ; Benchikha_et_al_2017_2 ; Boumali_et_al_2017 ; Salti ; Yesiltas_2 ; Boumali_et_al_2018 ; Lutfuoglu_Ikot_et_al_2019 ; Langueur_et_al_2019 ; Yesiltas_2019 ; Borrego_et_al_2020 .

In 2009, Ikhidair investigated the KG equation in $3+1$ dimension with PDM Ikhdair_2009 . Later, Hassanabadi et al. explored the exact solutions of the D-dimensional KG equation with Coloumb-like EDP energy with a constant mass in 2012 HH2 . To the best of our knowledge, the KG equation with an energy PDM under an EDP energy has not examined. Therefore, In this manuscript, we are motivated to investigate the bound state solutions of the Coulomb-like EDP energy by considering mixed vector and scalar potential energy in the presence of energy and PDM. We explore the bound state energy eigenvalues and corresponding wave functions for the KG particle in terms of following cases: mixed vector and scalar potential energies with equal magnitude and equal sign (EMES), mixed vector and scalar potential energies with equal magnitude and opposite sign (EMOS), pure vector (PV) potential energy, pure scalar (PS) potential energy and constant mass-energy limits.

We prepared the manuscript as follows. In sect. II we introduced the radial KG equation with mixed vector and scalar equation. Then, in subsection II.1, we declared the position and energy dependent mass function. Then, in its subsections, we solved the radial KG equation by employing the AIM method in different limits. In sect. II.2, we obtained the wave function solution by using the boundary conditions in terms of confluent hypergeometric functions. Then, in sect. III, we employed the numerical methods and calculated the energy spectrum and corresponding wave functions for a neutral pion particle. Before we concluded the manuscript in sect. IV, we analyzed the role of the tuning parameters on the energy spectrum in the pure scalar limits.

II Klein-Gordon equation

In $(3+1)$ dimensions, in the presence of spherical symmetric vector, $V_{v}(r)$, and scalar, $V_{s}(r)$, potential energies the time independent KG equation with a PDM is given as Ikhdair_2009

[TABLE]

Here, $\hbar$ and $c$ represent the reduced Planck constant and the speed of light, respectively. $E$ is the energy of the bosonic particle. The mass term, $m(r,E)$, depends on the energy in addition to the spatial coordinate. The decomposition of the spatial wave function into radial wave function, $R(r)$ and angular-dependent spherical harmonics, $Y_{m}^{l}(\theta,\varphi)$, ends up with

[TABLE]

where $u(r)\equiv rR(r)$.

II.1 Bound state solution

In this manuscript, we investigate bound state solutions of the KG equation with a position and energy dependent mass function that is defined by

[TABLE]

Here, $\lambda$ and $m_{0}$ are the reduced Compton wavelength and the rest mass of the spin-0 particle, respectively. $b$ is the coupling constant. The term $\frac{-A(1+\delta E)}{r}$ is a Coulomb-like energy-related potential energy function that will be called ”vector potential energy” in the rest of the article. $A$ represents the strength of the potential well, where $\delta$ determines the energy contribution. On the other hand, $b$ and $\delta$ can be seen as two tuning parameters that allow projecting the obtained analytic solutions on various physical problems. For instances, $(b=0)$, $(b=0,\,\,\delta=0)$, and $(b\neq 0,\,\,\delta=0)$ cases represent the constant mass with energy dependent Coulomb-like potential energy, Coulomb-like potential, and the position dependent as well as energy independent solutions, respectively.

Note that, we can express the investigated mass energy term as follows:

[TABLE]

II.1.1 Mixed vector and scalar potential energies with equal magnitude and equal sign case

In this subsection, we use attractive scalar potential energy that has EMES with the vector potential energy.

[TABLE]

The EMES potential energies condition turns Eq. (2) into

[TABLE]

Then, we employ Eqs. (3) and (5) in Eq. (6) and we obtain

[TABLE]

We introduce a new coordinate transformation, $z=(1+\delta E)r$, hence, $u(r)\rightarrow v(z)$. We get

[TABLE]

where

[TABLE]

We put forward an ansatz by examine the asymptotic behaviours

[TABLE]

and we find

[TABLE]

In order to obtain the energy eigenvalues, we employ the Asymptotic Iteration Method (AIM) which can be found in details in Ref.Ciftci_et_al_2003 . According to AIM, we firstly find the homogeneous linear second-order differential equation of the form $\varphi^{\prime\prime}(z)=\lambda_{0}(z)\varphi^{\prime}(z)+s_{0}(z)\varphi(z)$. Comparing this equation with Eq. (13), we get

[TABLE]

By using the iteration formula $\lambda_{n}(z)=\lambda_{n-1}^{\prime}(z)+\lambda_{0}(x)\lambda_{n-1}(z)+s_{n-1}(z)$ and $s_{n}(z)=s_{n-1}^{\prime}(z)+s_{0}(z)\lambda_{n-1}(z)$ as well as the quantization condition $\frac{s_{n}(z)}{\lambda_{n}(z)}=\frac{s_{n-1}(z)}{\lambda_{n-1}(z)}$, we obtain,

[TABLE]

We can easily found a general formula for a given $n$ quantum number as follows,

[TABLE]

We substitute $\tau$, $\beta$ and $\eta$ parameters into Eq. (19) and find

[TABLE]

where $n=0,1,2,3,\cdots$.

II.1.2 Mixed vector and scalar potential energies with equal magnitude and opposite sign case

In this subsection, we use repulsive scalar potential energy that possesses equal magnitude with the vector potential energy but with an opposite sign.

[TABLE]

We employ the EMOS condition in Eq. (2) and we derive

[TABLE]

We follow similar steps and find Eq. (8) with different parameters

[TABLE]

First, we employ the same ansatz and then, apply the AIM method, we get the quantization condition as follows

[TABLE]

where $n$ is an integer.

II.1.3 Pure vector potential energy case

In this subsection, we examine the case where the scalar potential energy is equal to zero.

[TABLE]

We would like to remind you that since the variable mass has vector potential energy coupling, even in the absence of scalar potential energy, the KG equation is not in the minimal coupling form. We use the pure vector condition in Eq. (6) and we find

[TABLE]

We go through the same procedure given in details above and we obtain Eq. (8) with different parameters as

[TABLE]

After, we use the same ansatz and the several iterations in the AIM method, we derive the quantization condition as given

[TABLE]

Here, the quantum number $n$ is an integer.

II.1.4 Pure scalar potential energy case

In this subsection, we investigate the case where the vector potential energy is equal to zero. Consequently, the mass term does not depend on either position or energy. On the other hand, the mass term couples with an attractive scalar potential energy.

[TABLE]

In this case the KG equation given in Eq. (2) turns to be

[TABLE]

After basic algebra which is given in details in the previous subsections, we find the quantization condition as follows:

[TABLE]

II.1.5 Constant mass energy limit

In this section, we discuss a limit where the vector and scalar potential energy couplings to the mass term vanish. Basically this limit is obtained by employing the definition of the non constant mass term, which is given in Eq. (4), in the KG equation, which is expressed in Eq. (2). We find

[TABLE]

Since $b$ is an arbitrary parameter, it is always possible to obtain

[TABLE]

via $b=-\frac{V_{s}(r,E)}{\hbar V_{v}(r,E)}$. For instance if we assume an attractive vector and a repulsive scalar potential energies with different magnitudes

[TABLE]

with position and energy dependence. Then, we find

[TABLE]

Note that, the tuning parameter $b$ does not need to be a positive number.

Before we conclude this subsection, we strongly advice to read papers that examine some classes of exactly-solvable KG equations within $V_{v}(r)\equiv V_{0}+\beta V_{s}(r)$ relation where $V_{0}$ and $\beta$ are arbitrary constants Chen_2005 ; Chen_et_al_2006 ; Dutra_et_al_2006 .

II.2 Wave function solution

In this section we obtain the wave function solution. We start with a new transformation, $y=2\tau z$, in Eq. (13). We find

[TABLE]

This equation is similar to the confluent hypergeometric equation Abramowitz_et_al_Book

[TABLE]

which has the solution

[TABLE]

Here, $\,{}_{1}F_{1}(a,c,x)$ and $\,{}_{1}U_{1}(a,c,x)$ are the first and second kind confluent hypergeometric functions, respectively. We obtain the coefficients with the match of the Eq. (43) and Eq. (44)

[TABLE]

and therefore the solution of Eq. (13)

[TABLE]

Then the wave function $u(r)$ is found to be

[TABLE]

Note that second kind confluent hypergeometric function does not satisfy the boundary conditions. Hence $N_{2}$ is chosen to be zero.

III Results and Discussions

In this section, in order to get quantitative analysis, we illustrate the energy eigenvalues of a neutral pion. The rest mass energy and Compton wavelength of the neutral pion are $134.977$ $MeV$ and $1.462$ $fm$, respectively. We assume the strength of the potential energy parameter, $A$, as $200$ $MeV\cdot fm$. We assign negative and positive values to the tuning parameters $\delta$ and $b$ in addition to their vanishing values. First, we use Eq. (20) to calculate the energy spectra in the EMES limit. We tabulate the obtained energy spectra in Table 1. We plot the unnormalized wave functions, the vector potential energy and the mass function in Fig. 1. Note that, when $b$ is zero, $\lambda b$ becomes zero and the mass loses both energy and position dependence. However, energy dependency on the potential energy continues. To comprehend the effect of energy dependence, we use three values in which the tuning parameter $\delta$ has negative, $-0.003$ $MeV^{-1}$, and positive $0.003$ $MeV^{-1}$, values in addition to zero. In this case, the spectra consist only of the upper or lower energy eigenvalues. During the increase of delta parameter from the negative value to the positive value, we observe that lower energy eigenvalues increase and upper energy eigenvalues decrease. When $b$ has negative value, we realize that lower energy eigenvalues do not occur. In this case, the spectra consist only of the upper energy eigenvalues, and as the delta increases, the values of the energy eigenvalues decrease. This decrease is relatively smaller as quantum numbers increase. When $b$ has a positive value, the upper and lower energy eigenvalues exist in the energy spectra in Table 1.

Then, we examine the energy spectra of the EMOS limit by solving Eq. (26). We find that the energy spectra persist non zero values only in positive values of the tuning parameter $b$. We tabulate the calculated energy spectra in Table 2. One of the authors of this study, BCL, in a study published in 2018, examined the generalized symmetric Woods-Saxon potential energy with a constant mass term in the KG equation and showed that an energy spectrum at the EMOS limit cannot be obtained Lutfuoglu_2018 . The results obtained in this study are in agreement with the results given by BCL. In addition to those results, we show that a spectrum can be calculated with upper energy eigenvalues for a specific value of the effective mass.

Then, we calculate the energy spectra in the PV limit, where $V_{s}(r)=0$. We present the energy spectra in Tab. 3. We observe that the ground state eigenvalue does not exist, instead the lowest eigenvalue is $E_{11}$. In all cases, the eigenvalues have positive values. The increase of $\delta$ parameter with the negative value of $b$ parameter creates a decrease in the value of the lowest eigenvalue. Furthermore, these decreases have an increment when $b$ becomes positive.

We study the pure scalar limit and tabulate the energy spectra in Tab. 4. In each particular sub-case, two energy eigenvalues, namely upper and lower eigenvalues are obtained. Similar to the pure vector case, a decrease between the energy eigenvalues is observed in the increasing values of $\delta$ for a fixed $b$ value. To demonstrate it, we plot Fig. 2. There, we illustrate the variation of particular upper and lower energy eigenvalues of the spectra via $\delta$ parameter for the fixed negative value of $b$. When the potential energy is independent of the energy and the mass has a constant value, the energy eigenvalues become symmetrical as expected. In addition, when the mass and potential energy are independent of energy, a symmetry of upper and lower energy eigenvalues exist. Finally, in Fig. 3, we demonstrate the upper and lower energy eigenvalues of the spectra versus $\lambda b$ values for a negative constant value of the tuning parameter $\delta$. We see a higher differentiation of the eigenvalues for lower quantum numbers. We observe a decrease in upper and an increase in lower eigenvalues for small quantum numbers. As the quantum numbers for a fixed $\lambda b$ increase, the difference between the two adjacent energy levels decreases.

IV Conclusion

In this paper, we obtained the bound state solution of a Klein Gordon particle whose mass varies via position and energy. We took into account an energy-dependent Coulomb-like potential energy and employed the asymptotic iteration method. We examined the solution in five different limits, i.e. mixed vector and scalar potential energies with equal magnitudes with equal/opposite signs, pure vector, pure scalar, and constant mass. Except for the trivial constant mass limit, in each cases, we derived a transcendental equation to calculate the energy spectrum. We showed that the radial wave functions are occurring in terms of the confluent hypergeometric functions. Next, we considered a neutral pion particle and used the Secant numerical methods to obtain an energy spectrum in each limit. Although the potential energy parameters were chosen at random, we have set various tuning parameters to investigate all critical situations. We verified the tabulated energy eigenvalues by examining their corresponding wave function behavior which fulfills the boundary conditions. Finally, we discussed the role of the tuning parameters in the pure scalar limit. We believe that the results obtained in this article would have applications in physics.