solutions of dirac equation for a new improved pseudo coulomb ring shaped potential

Owatea, a Theoretical Physics Group, Department of Physics, University of Port Harcourt, Nigeria b Department of Physics, Shahrood University of Technology, Shahrood, Iran Received 9 Aug

Solutions of Dirac equation for a new improved pseudo-Coulomb

ring-shaped potential

A.N Ikota,* , M.C Onyeajua, M.I Ngwuekea, H.P Obonga, I.O Owatea,

a Theoretical Physics Group, Department of Physics, University of Port Harcourt, Nigeria

b Department of Physics, Shahrood University of Technology, Shahrood, Iran Received 9 August 2016; revised 10 November 2016; accepted 10 November 2016

Abstract

We proposed a new solvable novel pseudo-Coulomb ring-shaped potential and investigate its pseudospin symmetry by solving the Dirac equation under the condition of an equal mixing of scalar and vector potentials using the factorization techniques We determine in closed form the energy eigenvalues and eigenfunctions of the bound states of the Dirac equation analytically We also discuss the non-relativistic limits

© 2016 The Authors Production and hosting by Elsevier B.V on behalf of University of Kerbala This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)

PACSnumbers: 03.65Ge; 03.65Pm; 03.65Db

Keywords: Dirac equation; Pseudospin symmetry; Pseudo-Coulomb potential; Bound state; Eigenvalues

1 Introduction

The pseudospin symmetry of the Dirac Hamiltonian

was discovered many years ago, however, this

sym-metry has recently been recognized empirically in

nuclear and hadronic spectroscopic [1] Within the

theory of Dirac equation, the concept of pseudospin

symmetry is used in deformed nuclei,

super-deformation, and effective shell model [2,3] It was

shown that the exact pseudospin symmetry occurs in

the Dirac equation when dSðrÞ

dr ¼ 0, i.e

SðrÞ ¼ VðrÞ þ SðrÞ ¼ const, where VðrÞ; SðrÞ are repulsive and attractive scalar potentials, respectively The pseudospin symmetry usually refers to as a quasi-degeneracy of single nucleon doublets with the non-relativistic quantum number

n; l; j ¼ l þ1

2

 and

n 1; l þ 2; j ¼ l þ3

2

 , where n; l and j are single nucleon radial, orbital and total angular quantum numbers, respectively The total angular momentum is

j¼ ~lþ ~s, where ~l¼ l þ 1 is a pseudo-angular mo-mentum and~s is pseudospin angular momentum The Dirac equation with different potentials in relativistic

* Corresponding author.

E-mail address: ndemikotphysics@gmail.com (A.N Ikot).

Peer review under responsibility of University of Kerbala.

http://dx.doi.org/10.1016/j.kijoms.2016.11.002

2405-609X/© 2016 The Authors Production and hosting by Elsevier B.V on behalf of University of Kerbala This is an open access article under the CC BY-NC-ND license ( http://creativecommons.org/licenses/by-nc-nd/4.0/ ).

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quantum mechanics with pseudospin symmetry has

been investigated in recent years [4e10] In order to

fully understand the origin of the concept of

pseudo-spin symmetry, the motion of the nucleons in a

rela-tivistic mean field theory has to be taken into

consideration which considered the Dirac equation

[11] Zhou et al.[12]have studied the Dirac equation

for Makarov potential with pseudospin symmetry Guo

et al.[13]studied the bound states of relativistic

par-ticles for ring-shaped non-spherical harmonic

oscil-lator potential under pseudospin symmetry The

ring-shaped molecular potentials are non-central potentials

and have many applications in physics and quantum

chemistry Non-central potentials have also been

studied extensively in the literature since it provides a

useful theoretical background for describing the

interaction between the ring-shaped molecules and that

between the deformed nucleus [14e16] In a similar

development, the ring-shaped potentials have found

many useful applications in quantum chemistry and are

also important in nuclear physics to study

ro-vibrational energy level of molecules, atoms and

deformed nucleus and to describe ring-shaped

molec-ular benzene in chemistry [17e20] In recent years,

considerable efforts have been made by many authors

to obtain the exact analytical solutions of the

Schr€odinger equation with ring-shaped potentials

[21e23] Also, different kinds of ring-shaped like

po-tential have been investigated such as the non-spherical

harmonic oscillator (NHO) [24], the ring-shaped

oscillator (RHO) [25], the double ring-shaped

har-monic oscillator (DRHO) [26] among others Dong

et al.[27]studied a ring-shape non-spherical harmonic

(RNHO) potential and obtained the non-relativistic

energy spectra and wave function Berkdemir [28] in

his paper proposed the novel angle-dependent (NAD)

potential where the potential VðrÞ contains a Coulomb

potential or a harmonic potential in addition to NAD

potential Zhang et al [29] extended the work of

Berkdemir and proposed the harmonic novel angle

dependent (HNAD) potential by simply replacing sin q

by cos q in the numerator of NAD and obtain HNAD

Another form of the ring shaped potential investigated

is the double ring-shaped oscillator (DRSO) [30]

which reduces on special cases to the ring-shaped

oscillator (RSO) and spherical oscillator (SO),

respectively In addition, Zhang [31] proposed new

RNHO and study the energy spectra and wave function

for the potential Very recently, Chen et al [32,33]

studied the ring-shaped potentials using the universal

associated Legendre polynomials and they went further

to discussed the super-universal associated Legendre

polynomials Even though solutions of the Dirac equation with different potential model have attracted a great attention in recent years, the exact solutions of the Dirac equation are only possible for a few simple systems such as harmonic and hydrogen atom[34,35] Various analytical techniques have been employed to find the bound state solution of the Schrodinger, KleineGordon and Dirac equations such as

Nikifor-oveUvarov method (NU) and formula method

[36e38], supersymmetric quantum mechanics

Consequently, the recent advances in the search for the solutions of Dirac equation with physical motivated potential models will lead to the discovery of a new phenomenon in addition to the spin and pseudospin symmetry discover many years ago in the nuclei of atom in the Dirac theory The investigated potentials include but not limited to Coulomb-like potentials

[43], ManningeRosen potential [44], DengeFan po-tential[45], Mobius potential[6], shifted Hulthen po-tential[46]and others[47e49]

Motivated by the study of the ring-shaped-like po-tential [50] we proposed the novel pseudo-Coulomb ring-shaped potential of the form,

Vðr; qÞ ¼A

r þB

r2þ C þ1

r2

D cos2qþ t sin q cos q

2

þ1

r2

b sin2qþ g cos2qþ l sin q cos q

2

where A; B and C are the potential depths and b; g; l; t and D are the five dimensionless ring-shaped parame-ters The plots of the behaviour of this potential are illustrated in Figs 1e3 The purpose of the present paper is to investigate the above potential under the

Fig 1 The plot of the behaviour of novel pseudo-Coulomb potential for a fixed r and various values of q ¼ 30; 60 and 90 respectively.

pseudospin symmetry limit and study the quantum

behaviour arising from it

The organization of the paper is as follows In

Section 2, we review the Dirac theory under

pseudo-spin symmetry limit Section3is devoted to the exact

solution of the Dirac equation with pseudo-Coulomb

ring-shaped potential Finally, we give a brief

conclu-sion in Section4

2 Theory of Dirac equation

The Dirac equation for spin- 1

2 particles moving in

an attractive scalar potential SðrÞ and repulsive vector

potential VðrÞ in the relativistic unit ðZ ¼ c ¼ 1Þ is

½ a!$ p!þ bðM þ SðrÞÞjðrÞ ¼ ½E  VðrÞjðrÞ; ð2Þ

where E is the relativistic energy of the system, p

! ¼ iV! is the three-dimensional momentum oper-ator and M is the mass of the fermionic particle a!;b are the 4 4 Dirac matrices given as

a

! ¼
0 !si

s

!

i 0



; b ¼

0 I



where I is 2 2 unitary matrix and s!i are the Pauli three-vector matrices:

s1¼

0 I

I 0



; s2¼

0 i

i 0



; s3¼

0 I



and I is the 2 2 unitary matrix In addition, we can write the Dirac wave function as

jðrÞ ¼

4ð r!Þ

cð r!Þ



ð5Þ

where 4ð r!Þ;cð r!Þ represent the upper and lower components of the Dirac wavefunctions and

s

!$ p!cð r!Þ ¼ ½ε  M  SðrÞ4ð r!Þ ð6Þ s

!$ p!4ð r!Þ ¼ ½ε þ M  DðrÞcð r!Þ ð7Þ where

SðrÞ ¼ VðrÞ þ SðrÞ ¼ Cps; DðrÞ ¼ VðrÞ  SðrÞ ¼ Cs

In the limiting case, that is under the condition of pseudospin symmetry, Eqs.(6) and (7)become

s

whereεsM i.e only real negative energy states exist when Cps ¼ 0 (exact pseudospin symmetry) and the energy eigenvaluesε depends on the quantum numbers

n and L and also on the pseudo-orbital angular mo-mentum quantum number ~l

Combining Eqs.(7) and (8)and taking DðrÞ as the pseudo-Coulomb potential, we obtain the Schr €odinger-like equation for the lower component as,



V2ε2 M2

 ðε  MÞ



A

r þB

r2þ C

þ 1

r2

D cos2qþ t sin q cos q

2

þ1

r2

b sin2qþ g cos2qþ l sin q cos q

2)#

cnlmðr; q; fÞ ¼ 0

ð10Þ

Fig 2 The plot of the novel pseudo-Coulomb potential with various

values of r ranging from r ¼ 5; 25 and 40 with fixed q.

Fig 3 The 3D plot of novel pseudo-Coulomb ring shaped potential

as a function of r and q Here we have choose A ¼ B ¼ C ¼ b ¼ g ¼

l ¼ 1; D ¼ 4 and t ¼ 3.

V2¼1

r2



v

vr

r2v

vr



sin q

v vq

sin q v vq



sin2q

v2

vf2

ð11Þ Now since the Eq.(10) has a decoupling of

pseu-dospin and pseudo-orbital momentum, then the lower

component of the spinor wave function has spin up or

spin down i.e.,

1 0

 or

0 1

 multiplying by the component of the spherical co-ordinate, thus the lower

component of the wave function can be written as,

cn~ ðr; q; fÞ ¼Rn~ lðrÞH~lmðqÞFmðfÞ

where m¼ ±1

2and cmis the spin up or spin down

two-component spinors

Now after substituting Eqs.(11) and (12) into Eq

(10)and making a separation of variable, we obtain the

following sets of second order Schr€odinger-like

dif-ferential equations:

d2Rn~lðrÞ

dr2 þ

ε2 M2

 ðε  MÞC

þAðε  MÞ

r Lþ Bðε  MÞ

r2

Rn~lðrÞ ¼ 0

ð13Þ

d2H~lmðqÞ

dq2 þcotqdH~lmðqÞ

"

L m2 sin2qðεMÞ

D cos2qþt

sin q cos q

2

ðεMÞ

b sin2qþgcos2qþl

sin q cos q

2#

H~lmðqÞ¼0;

ð14Þ

d2FðfÞ

where L and m2 are the separation constants The

so-lutions of Eq (15) satisfy the boundary

Fðf þ 2pÞ ¼ FðfÞ whose solution is given as,

FðfÞ ¼ 1ffiffiffiffiffiffi

2p

In the following subsection, we will give the solu-tion of the radial part (Eq.(13)) and polar angular part

of Eq.(14)

3 Solutions of the Dirac equation with the novel pseudo-Coulomb ring-shaped potential

3.1 Solution of the angular part

In order to obtain the energy eigenvalues and the corresponding wave function for the angular part of Dirac equation under the pseudospin symmetry limit,

we take a new variable transformation of the form, x¼ cos2q in Eq.(13)and after a little algebra we obtain,

xð1xÞd2H~lmðxÞ

dx2 þ

13x 2



dH~lmðxÞ

xð1xÞ



u1x2þu2xu3



H~lmðxÞ¼0

ð17Þ

where,

u1¼Lþ ðε  MÞ



D2þ ðg þ bÞ2

u2¼Lm22DtðεMÞþ2bðεMÞðbgþlÞ

u3¼ðε  MÞ



ðb þ lÞ2þ t2

We take the physically acceptable ansatz for the wave function as,

H~lmðxÞ ¼ xpð1  xÞq

and substituting it into Eq.(17)yields,

xð1  xÞf00

ðxÞ þ

 2pþ1

2

2pþ 2q þ3

2

 x

f0ðxÞ



pþ q þ1

4 s



pþ q þ1

4þ s



fðxÞ ¼ 0 ð22Þ

p¼1

4

1þqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4ðε  MÞðb þ lÞ2þ t2

;

q¼1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðε  MÞD2þ ðg þ bÞ2

þ ðε  MÞðb þ lÞ2þ t2

þ m2þ 2Dtðε  MÞ  2bðε  MÞðb  g þ lÞ

q

;

s ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1

16þLþ ðε  MÞ



D2þ ðg þ bÞ2 4

Now comparing Eq.(22)with the standard form of

the second order differential of hypergeometric

func-tion [51]

zð1  zÞ400

ðzÞ þ ½c  ða þ b þ 1Þx40ðzÞ  ab4ðzÞ ¼ 0

ð24Þ

we get the parameters a; b; c as follows,

a¼ p þ q þ1

4 s;

b¼ p þ q þ1

4þ s;

c¼1

2þ 2p

ð25Þ

The solutions of Eq.(22)can be expressed in terms

of the Gauss's hypergeometric function as

H~lmðqÞ ¼cos2qp

sin2qq

2F1

 nr; nrþ 2p þ 2q þ1

2; 2p þ1

2; cos2q

 ð26Þ However, when pþ q þ1

4þ s ¼ nr or

pþ q þ1

4 s ¼ nr for nr¼ 0; 1; 2:::, then the

hy-pergeometric function in Eq (26) reduces to a

poly-nomial of degree nr In order to obtain the relationship

between the separation constant L and the

non-negative integer nr, we used the quantization

condi-tion pþ q þ1

4 s ¼ nr with Eq.(24), we obtain

Eq (26)is the contribution of the angle-dependent

part of pseudo-Coulomb plus ring-shaped like

poten-tial However, when the ring-shaped term potential

vanishes, that is g¼ b ¼ l ¼ D ¼ t ¼ 0, then the

constant of separation becomes L¼ ~lð~lþ 1Þ, where ~l

is the orbital angular momentum is defined as

~l¼ 2nrþ 1 þ jmj; m ¼ 0; 1; 2::: It can be observed that the angular part of the Pseudo-Coulomb ring-shaped potential has singularities at q¼ tpðt ¼ 0; 1; 2:::Þ when ðD; t; b; g; lÞ are taken as positive values and also at very small and very large values of r However, it is known that the angular wave function HlmðqÞ exist as

an odd and even function [52e55] So in order to remove this singularities then the combination of the ring shaped parameters ðD; t; b; g; lÞ ¼ nð4n  2Þ or ðD; t; b; g; lÞ ¼ nð4n þ 2Þ For instance when

g¼ nð4n  2Þ and similarly for other terms, the rest of the parameters being zero would reduced the HlmðqÞ to the associated legendre polynomials Pm

lðqÞ which is an odd or an even function [52e55] thus removing the singularities The complete angular wave function can

be written as,

H~lmðqÞ ¼ Nn

 cos2qp

sin2qq

2F1

 nr; nrþ 2p þ 2q

þ1

2; 2p þ1

2; cos2q



ð28Þ where Nn is the normalization constant of the angular wave function HlmðqÞ In order to determine the normalized angular wave function, we used the following normalization conditions[51e54],

Z2p 0

sinðqÞjH~lmðqÞj2

dq¼ 2

Z1 0

jH~lmðqÞj2

Z1 0

zg1ð1  zÞs g½2F1ð  n; n þ s; g; zÞ2

dz

¼ n!

ðs þ 2nÞ

GðgÞ2

Gðn þ s  g þ 1Þ

L¼

0

B

B

B

1

2

1þqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4ðε  MÞðb þ lÞ2þ t2

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðε  MÞD2þ ðg þ bÞ2

þ ðε  MÞðb þ lÞ2þ t2

þm2þ 2Dtðε  MÞ  2bðε  MÞðb  g þ lÞ

v u t

þ1

2þ 2nr

1 C C C

2

ðε  MÞD2þ ðg þ bÞ2

1 4

ð27Þ

Now substituting Eq.(28)into Eq.(29), we obtain,

2N2

n

Z1

0

xpð1xÞq

2F1

nr;nrþ2pþ2qþ1

2;2pþ1

2;x

2dx

¼1

ð31Þ Thus, we obtain the normalization constant as,

Nn¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð2nrþpþqþ1Þ

2nr!

Gðnrþpþqþ1ÞGðpþ2qþ2Þ

Gð1þpÞ2

Gðnrþqþ1Þ s

ð32Þ 3.2 Solutions of the radial part

Now for the radial equation, we let the following

parameters

k¼pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiε2 M2

; L ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1

4þ L þ Bðε  MÞ

r

1

2;

q¼A

2

ffiffiffiffiffiffiffiffiffiffiffiffi

ε  M

ε þ M

r

; h2¼1

4ðε þ MÞ

ð33Þ

and using the new coordinate transformationr ¼ 2kr,

Eq.(13)becomes

d2RðrÞ

dr2 þ



 h2þqrLðL þ 1Þr2

where L is given in Eq.(27) Based on the behaviour of

the wave function at the origin and at infinity, we define

RðrÞ ¼ rLþ 1

By substituting Eq.(35)into Eq.(33), we find that

the wave function 4ðrÞ satisfies the following second

order differential equation,

r400 ðrÞ þ ½2ðL þ 1Þ  2hr40ðrÞ þ ðq  2hðL þ 1ÞÞ4ðrÞ

¼ 0

ð36Þ

If we let z¼ 2hr in Eq.(36), we obtain

z400ðzÞ þ ½2ðL þ 1Þ  z40

Lþ 1  q 2h

 4ðzÞ ¼ 0

ð37Þ The solution of Eq.(37)is nothing but the confluent hypergeometric function F

Lþ 1  q

2h; 2ðL þ 1Þ; z

 However, for the bound states the confluent function are terminated by a polynomial[52]such that,

Lþ 1  q

where n is the node of the radial wave function Using

Eq.(33), we obtain the energy eigenvalues as,

½ðε þ MÞ  C2

ðε2 M2Þ ¼

4A2

where n0¼ 1 þ n þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

4þ L þ Bðε  MÞ

q

As a special case in the non-relativistic limit, if we set ε þ M/  Enl; ε  M/2m

Z 2; C ¼ B ¼ 0, we obtain the eigenvalues for the Coulomb potential plus a ring shaped potential as,

En~l¼  8mA2

where L0¼1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 4L0

p

, L¼ ~l0ð~l0þ 1Þ with ~l0 repre-senting the non-relativistic orbital angular momentum quantum number and

L0¼

0

B

B

B

B

B

@

1

2 1þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 4

2m

Z 2



ðb þ lÞ2þ t2

þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2m

Z 2



D2þ ðg þ bÞ2

þ2m

Z 2



ðb þ lÞ2þ t2

þm2þ4m

Z 2Dt4mb

Z 2ðb  g þ lÞ

v u u

þ1

2þ 2nr

1 C C C C C A

2

2m

Z2



D2þ ðg þ bÞ2

1 4

ð40bÞ

One can observe that the angular orbital momentum

quantum in the non-relativistic limits differ from the

relativistic case by the factorε  M/2m

Z 2 This is a new result which has not been reported

before to the best of our knowledge However, if we

choose D¼ t ¼ B ¼ C ¼ b ¼ l ¼ 0 then the

poten-tial (1) turns to the Coulomb potenpoten-tial plus a new

ring-shaped potential [53],

Vðr; qÞ ¼A

r þ a cos2q

where a¼ g2: Making the corresponding replacement

of parameters in Eq.(40), we obtain the energy spectra

and the corresponding wave function for the Coulomb

plus new ring-shaped potential in the non-relativistic

limit as,

Where l0¼ 2nrþ3

2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ma

Z 2 þ m2

q

 ffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ma

Z 2 þ1 4

q

and when a¼ 0, then the angular momentum quantum

number becomes l0¼ 2nrþ 1 þ jmj This result is

consistent with that given in Ref [53] when

m¼ Z ¼ 1

The corresponding radial eigen functions are

expressed as,

RnLðrÞ¼BnLð2krÞLþ1e2khrF

Lþ1q 2h;2ðLþ1Þ;2hkr

 ð43Þ

where BnL is the normalization constant Using the orthogonality of radial wave function,

Z∞ 0

jRnLj2

and the following known relations[53,54],

Z∞ 0

zmezLmnðzÞLm

n 0ðzÞ ¼Gðm þ n þ 1Þ

LmnðzÞ ¼Gðn þ m þ 1Þ

n!Gðm þ 1Þ Fðn; m þ 1; zÞ ð46Þ and we obtain the normalized radial wave function as,

Rn;LðrÞ ¼ 2hn!

ðn0Þ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n0þ L þ 1 p

!1
4khr

n0

L þ1

e 4khrn0 L2Lþ1n

4khr

n0

Finally, using Eqs.(5) and (8), we obtain the spinor wave function as,

1þ n þ1

2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ 42nrþ3

2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ma

Z 2 þ m2

q

þ ffiffiffiffiffiffiffiffiffiffiffiffiffi2ma

Z 2 þ1 4

q

2nrþ3

2þ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ma

Z 2 þ m2

q

 ffiffiffiffiffiffiffiffiffiffiffiffiffi2ma

Z 2 þ1 4

q

jð r!Þ ¼ 1ffiffiffiffiffiffi

2p

p

!$ p!s

ε  M~cm

 1 r

2hn!

ðn0Þ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n0þ L þ 1 p

!1
4khr

n0

L þ1

e 4khrn0 L2Lþ1n

4khr

n0



 Nn



cos2q1

sin2qm 2

2F1

 nr;1 2

1þ m þ

l0þ1 2



þ1

4þ s;3

2; cos2q



eim4

ð42bÞ

jð r!Þ ¼ 1ffiffiffiffiffiffi

2p

p

!$ p!s

ε  M~cm

 1 r

2hn!

ðn0Þ2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

n0þ L þ 1 p

!1
4khr

n0

L þ1

e 4khrn0 L2Lþ1n

4khr

n0



 Nn



cos2qp

sin2qq

2F1

 nr; p þ q þ1

4þ s; 2p þ1

2; cos2q



eim4

ð48Þ

The positive energy solutions for the spin symmetry

can be obtain directly from the pseudospin symmetric

solutions by using the following mapping,

~l/l; Vðr; qÞ/  Vðr; qÞ; Cps/  Cs; ε/  ~ε

4 Conclusions

In this article, we have proposed a novel

Pseudo-Coulomb ring-shaped potential and investigate its

exact solutions completely In the limit of the

pseu-dospin symmetry, we solved the Dirac equation and

obtain the energy eigenvalues and eigenfunction using

the traditional factorization method Special cases of

this potential have been discussed Now rescaling the

potential parameter of our ring shaped like parameter

as follows: a/b2; b/g2; c/l2; p/2bg; 2bl/

q; 2gl/f ; a0¼ q þ c; b0¼ f þ c, then the novel

pseudo-Coulomb plus ring shaped like potential

becomes,

Vðr; qÞ ¼ A

rþB

r2þ C þ Z2

2Mr2

ða sin2

qþ a0Þ cos2q

þðb cos2qþ b0Þ

sin2q þ p



ð49Þ

Many useful ring-shaped potentials can be deduced

from Eq.(49)as special cases It is important for us to

point out at this point that these results will have many

applications in nuclear physics and quantum chemistry

[56]

Acknowledgement

The authors wish to thank the kind reviewers for

their positive comments on the manuscript

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