Relativistic Solutions of Dirac Equation with the Molecular Hua Potential in the Spin Symmetry Limit

Introduction

The solutions of wave equations are known to be vital in quantum mechanics and related areas of physics. The reason is because the solutions have all the relevant parameters required to evaluate the associated properties of a physical system under consideration. Reports on the nonrelativistic rotational vibrational energies of molecules obtained from solutions of Schrodinger equation in various potential models have been presented [1, 2, 3, 4, 5]. It has been established that relativistic interactions are essential for an accurate determination of the rotation-vibration energy spectra of molecules by using quantum mechanical techniques [6]. Recently, by solving Dirac equation with General molecular potential, Improved Tietz potential and Improved Rosen-Morse potential, some authors investigated the relativistic rotation-vibrational energies for $5^1\Delta_g$ state of Na$_2$ molecule, the $X^2 \sum^+$ state of the CP molecule and $3^3 \sum^-$ state of the Cs$_2$ molecule, and observed that nonrelativistic energies decreases as a result of relativistic effects [7, 8, 9].

In this paper, we attempt to investigate the solutions of Dirac equation with the Hua potential energy model. We also explore the relativistic effects of rotational vibrational energies for some molecules.

Recently, Hassanabadi H, et al. [10] studied the Schrodinger equation with Hua potential using the super symmetry quantum mechanics. Also, a similar form of Hua potential has been reported by Hua W, et al. [11], to study the rotation-vibration spectrum of different molecules. The Hua potential can be used in describing the energy levels of diatomic molecules, hence motivation for this work. The Hua potential is expressed as [12, 13].

$$U_H = V_0 \left( \frac{1 - e^{-b_h(t-r_h)}}{1 - qe^{-b_h(t-r_h)}} \right)^2, \quad b_h = \beta (1 - c_h),$$ where V0 , re, q and β are respectively the potential depth, bond length, deformation parameter and Morse constant. The work is drafted as: The Formula method is presented in Section 5, Section 6 is a review of Dirac equation under spin symmetry [14]. The bound state solutions are given in Section 7. Discussion comes in Section 9. Finally, conclusion is presented in Section 10.

Formula Method

The Formula method is applied by considering the equation.

α α ψ ψ α ξ ξ ξ ψ α $$ \begin{array}{l} \frac {d ^ {2} \psi (s)}{d s ^ {2}} + \frac {\alpha_ {1} - \alpha_ {2} s}{s \left(1 - \alpha_ {3} s\right)} \frac {d \psi (s)}{d s} \\ + \frac {\xi_ {1} s ^ {2} + \xi_ {2} s + \xi_ {3}}{s ^ {2} \left(1 - \alpha_ {3} s\right) ^ {2}} \psi (s) = 0. \tag {2} \\ \end{array} $$

The energy and the wave function are derived respectively,

from the equations

2 1 2 2 2 3 1 2 3 2 2 3

s d s d s s s ds dss s s s s ( ) ( ) (1 )

( ) 0. (1 )

2 2 2 2 2 4 5 2 3 2 1 2 3 2 5 3 2

$$ \left[ \frac {\alpha_ {4} ^ {2} - \alpha_ {5} ^ {2} - \left[ \frac {1 - 2 n}{2} - \frac {1}{2 \alpha_ {3} ^ {2}} \left(\alpha_ {2} - \sqrt {\left(\alpha_ {3} - \alpha_ {2}\right) ^ {2} - 4 \xi_ {1}}\right) \right] ^ {2}}{2 \left[ \frac {1 - 2 n}{2} - \frac {1}{2 \alpha_ {3} ^ {2}} \left(\alpha_ {2} - \sqrt {\left(\alpha_ {3} - \alpha_ {2}\right) ^ {2} - 4 \xi_ {1}}\right) \right]} ^ {2} - \alpha_ {5} ^ {2} = 0, \alpha_ {3} \neq 0 \tag {3} $$ ( )

1 2 1 4 2 2 0, 0 1 2 1 2 4 2 2

n α α α α α ξ α α α ( )

n α α α ξ α

$$ F _ {1} \left(- n, n + 2 \left(\alpha_ {4} + \alpha_ {5}\right) + \frac {\alpha_ {2}}{\alpha_ {3}} - 1; 2 \alpha_ {4} + \alpha_ {1}; \alpha_ {3} s\right), \tag {4} $$ α α α α α α α ( )

2 4 1 1 $$ = \frac {\left(1 - \alpha_ {1}\right) + \sqrt {\left(1 - \alpha_ {1}\right) ^ {2} - 4}}{2} $$ ξ α α α   

2 $$ = \frac {1}{2} + \frac {\alpha_ {1}}{2} - \frac {\alpha_ {2}}{2 \alpha} + \sqrt {\left(\frac {1}{2} + \frac {\alpha_ {1}}{2} - \right.} $$ $$ \left. \frac {1}{4}\right) ^ {2} - \left(\frac {\xi_ {1}}{\alpha_ {3} ^ {2}} + \frac {\xi_ {2}}{\alpha_ {3}} + \varphi\right) $$      

2 2 2 1 2 2 2 1

ξ α ξ α ξ α α α α α α α  

 

$$ \alpha_ {1} = 1, \alpha_ {2} = \alpha_ {3} = q, $$

, we put forward a

simplified energy equation from Equation (3) ( ) ( )            

2 2 2 2 ξ ξ ξ ξ ξ ξ $$ \sqrt {- \xi_ {3}} \left(q + \sqrt {q ^ {2} - 4 \left(\xi_ {1} + q \xi_ {2} + q ^ {2} \xi_ {3}\right)} + 2 q n\right) = q ^ {2} \xi_ {3} - \xi_ {1} - q ^ {2} n (n + 1 $$

4 2 1 3 1 2 3 3 1 2 1 4 4 1 2 3 1 2 3 2 2

q q q q q qn q q n n ( ) ( )

q qn q q q q q q

2 2 2 2 ξ ξ ξ ξ ξ ξ $$ - q n \sqrt {q ^ {2} - 4 \left(\xi_ {1} + q \xi_ {2} + q ^ {2} \xi_ {3}\right)} - \left(\frac {q}{2} + \frac {1}{2} \sqrt {q ^ {2} - 4 \left(\xi_ {1} + q \xi_ {2} + q ^ {2}\right)}\right) $$ (6)

Dirac Equation

The Dirac equation with scalar, S(r) and vector potential, V(r) is given as $$ \begin{array}{l} \hat {H} _ {D} \Psi (r) = E _ {v k} \Psi (r) \\ \hat {H} _ {D} = c \alpha . \overrightarrow {p} + \beta \left(\mu c ^ {2} + S (r)\right) + V \\ \end{array} $$     

2 ( ) ( ) H r E r D vk H c p c S r V r α β µ . ( ( )) ( )

D

, (7) $$ \mathrm {w h e r e} \mu , E _ {v k}, \overrightarrow {p} = - i \hbar \overrightarrow {\nabla} $$

are the reduced mass, relativistic energy, and momentum operator respectively. β α, are the

4×4 Dirac matrices given by

$$ = \left( \begin{array}{c c} 0 & \sigma_ {i} \\ \sigma_ {i} & 0 \end{array} \right) $$ i σ σ α   

0 β I   

$$ \gamma = \left( \begin{array}{c c} I & 0 \\ 0 & - I \end{array} \right) $$

   

0 i , , (8) Where I is the 2×2 matrix and i σ is the Pauli matrices given as

1 0 $$ = \left( \begin{array}{c c} 0 & - i \\ i & 0 \end{array} \right) $$

$$ = \left( \begin{array}{c c} 1 & 0 \\ 0 & - 1 \end{array} \right) $$

2 i i σ   

     

$$ = \left( \begin{array}{c c} 0 & 1 \\ 1 & 0 \end{array} \right) $$

     

1 σ

3 σ , , (9) The spinor, ) (r Ψ can be written as              

l n jm l n jm θ φ ( ) ( , ) 1 ( ) ( ) ( , )

F r Y

κ $$ \Psi (r) = \frac {1}{r} \left| i G _ {n \kappa} (r) Y _ {i m} ^ {\tilde {l}} \right. $$ θ φ r iG r Y r κ , (10) with ) (r Fnκ and ) (r Gnκ as the upper and lower components of the Dirac spinors. ( , ) jml Y θ φ is the spherical harmonic of the spin component and ( , ) jml Y θ φ  is the spherical harmonic of the pseudo spin component. l and l~ are the orbital and pseudo-orbital quantum numbers, while κ and m are the spin-orbit coupling operator and projection on z-axis. If the spinor in Equation (10) is used, we deduce the following coupled radial differential equations from the Dirac equation            −                 +       

2 µ $$ 0 = \frac {\left(\mu c ^ {2} - E _ {v k} + \Sigma\right)}{h c} $$

( ) ( ) ( )

c r E G r F r c c r E F r G r c vk d n n dr r κ κ κ η

2 + −∆ = µ ( ) ( ) ( )

vk d n n dr r κ κ κ η

(11)

If ) (r Fnκ is eliminated in favor of ) (r Gnκ , we derive two uncoupled differential equations of the form

2 2 2 µ µ κ κ νκ νκ − + Σ + −∆ + − −

( ( ))( ( )) ( ) ( 1) ( ) ( )

r r d F r c c E E F r F r dr r c κ κ κ n n n

2 2 2 2 η   ∆ +     + = + −∆

κ ( ) ( ) d r d F r dr dr r κ n

0, ( ( )) µ νκ

2 r c E (12)

2 2 2 µ µ κ κ νκ νκ − + Σ + −∆ − − −

( ( ))( ( )) ( ) ( 1) ( ) ( )

r r d G r c c E E G r G r dr r c κ κ κ n n n

2 2 2 2 η   Σ −     − = − + Σ

κ ( ) ( ) d r d G r dr dr r κ n

0, ( ( )) µ νκ

2 r c E (13)

 κ κ κ κ + = + − = + ∆

( 1) ( 1), ( 1) ( 1), ( )

l l l l r .

where = − Σ = +

( ) ( ), ( ) ( ) ( )

V r S r r V r S r

Bound State Solutions

∆ = ∆ = constant. In the non- In spin symmetry, ( ) 0; ( ) d r r C dr relativistic limit, Equation (12) reduces to a Schrodinger form with 2_V(r) in light of the exact symmetry _V(r) = S(r). Adopting a proposal by Alhaidari AD, et al. [15], and setting S(r) = UH, V(r) = Cs + UH(r), expansion of Equation (12) gives     + − +   + − + −          

2 2 κ κ µ κ κ νκ ( ) ( ) ( 1) ( ) 2

n n H d F r c C C E s s F r U dr c r

2 2 2 2 η ( )

  − + −     = −

2

2 4 2 µ µ νκ νκ κ C c c E E s F r n c ( ).

2 2 η (14) To solve Equation (14), we adopt an approximation                           − − − − ( ) 2 ( ) b r r b r r e e h h

1 1 e e D D D r r e qe qe ≈ + + − −

− − − −

0 1 2 2 2 ( ) ( )

b r re h b r r h e

1 1 (15) The approximation is only valid for 1 b re h qe ≥ . We can set er er r x er h b − = = , α in Equation (14) and expand up to x_2 term, we have  ) 3 ( ) 1( 3 1 1 _q q q D       + − − − + =   

0 α α ) 1( 3 ) 2 ( ) 1( 2

      − − + − =

2 1 q q q D

 α α   

3 ) 1( ) 1( 3 ) 1(

q q q D       + − − − =

2 α α  (16) Substituting Equation (15) in Equation (14) and rearranging gives           − −                   − −                         − − − −          − −        − −                             ( )

2 b r r h e C s − − + +

1 2 1 ( ) e V

ε ε ( )

1 0 2 b r r h e           

− qe d F r n dr e e D D D r qe qe

2 κ ( ) ( )

= + ( ) 0, F r nκ

2 2 b r r b r r h e h e + − + +

κ κ ( 1) 2 1 1

( )

0 1 2 2 b r r e b r r h e h e − −

(17) where                      

2 µ ε + − =

( ) c E C s νκ

2 2

1 c η

2 2 4 2 µ µ ε − + − =

E c c E C s νκ νκ

2 2 2 c η (18) To solve Equation (17), we use the transformation ) ( er r h b e s − − = , which gives

2

2 ( ) (1 ) ( ) 0 (1 ) (1 ) ds dF Ps n n n ds d F s qs s Qs R F s qs s qs κ κ κ − + + + + = − − , (19) where,

 + − − − + − + + − =

)1 ( 2 )1 ( )1 ( 1

r D q V C q r D r D q q b P

κ κ ε ε κ κ κ κ ε  

2 2 2 1 2 2

s    e e h

e     + − + + − + =

)1 ( )1 ( 2 2 2 1

r D r D q q V C q b Q

κ κ κ κ ε ε ε  

.

2 1 2 0 2 0 1 1

 e e s h    

    + − − − − =

)1 ( 2 1 r D V C b R

κ κ ε ε ε  

2 0 0 1 1 2

s  h e (20) The constants are calculated from Equation (5) as 

, , ,1 R q − = = = =

α α α α

4 3 2 1  

  Using Eqs. (6) and (4), the relativistic energy and unnormalized eigen function are derived as                                       + −      + +                            − + −   + −     +    = + +         + − +   + − + + +       + −     + +        −

2 2 4 2 2 µ µ µ κ κ E c c E C c E s C s D C s V r c c V c E D q s q qn n b r c c E C s n c e ( ) ( 1) 0 2 2 ( ) ( 1) 1 ( 1) ( 1) 2 ( ) 1 2 2 0 2 2 2 ( 1) 4 ( 1) νκ νκ νκ

2 b h µ νκ κ κ + − + + η

2 ( ) 2 2 2 0 2 ( 1) 4 2 ( 1) c E C s q qn c h D V q q r b e

2 2 2 (22) − R n n = −

α κ κ α ( ) (1 )

F r N s qs

5

2 1, 1     − +        

− + − + , 2 ;2

F n n R

R qs

5 ,(23) κ n N is a normalization condition.

Conclusion

In this work, we present solutions of Dirac equations with Hua potential energy model using the Formula method. Using the spin symmetry and the Pekeris form of approximation, we evaluated the relativistic rotation- vibrational energy equation for diatomic molecules under the Hua potential. In the nonrelativistic limit, the relativistic energy expression becomes the nonrelativistic rotation- vibrational energy equation. Numerical results are also computed for the selected molecules. The results show considerable agreement with reports in literature. This study can be applied in molecular physics, spectroscopy and other fields of science.

Declaration of Interest

The authors declare that they have no known conflict of interest.

Funding statement

This work did not receive any grant or support from funding agencies.