INVOLVING NIKIFOROV-UVAROV METHOD IN SCHRODINGER EQUATION OBTAINING HARTMANN POTENTIAL †

The total wave function and the bound state energy are investigated by involving Nikiforov-Uvarov method to Schrodinger equation in spherical coordinates employing Hartmann Potential (HP). The HP is considered as non-central potential that is mostly recognized in nuclear field potentials. Every wave function is specified by principal quantum number n , angular momentum number 𝑙 , and magnetic quantum number m . The radial part of the wave function is obtained in terms of the associated Laguerre polynomial, using the coordinate transformation 𝑥 = cos 𝜃 to obtain the angular wave function that depends on inverse associated Legendre polynomials.


INTRODUCTION
The Hartmann Potential is a kind of non-central potentials that have been studied in nuclear physics field, which consider as the coulomb potential surrounded by the ring-shaped inverse square potential.Organic molecules such as cyclic polyenes and benzene, handling this potential since 1972 [1][2].In spherical coordinate the Hartmann potential is coulomb potential adding a one potential proportional to  tan  .So, it defined by [3][4]: Where α, β are constants that consider as positive real numbers.Obtaining this HP in Schrodinger equation, assuming ( = ℏ = 1) [5]: , ,  + 2 , ,  = 0. ( The equation (2) depends on the total coordinates in spherical coordinates (, , ), to find the total wave function we need apply separation of variables method on equation (2).The main object in this work is to determine the bound state energy and the wave function.

SEPARATION OF VARIABLES METHOD
Obviously, in spherical potential, we let [6]: Now, by Separating variables in equation ( 2), we hold that: The Radial Part where  is separation constant.And, to solve the equation, we let Where ` and ` are positive integers or zero.
Where the angular part can displayed as: The Angular part Insert a new variable  = cos  and using Chain rules technique, to get the angular Schrodinger equation.
After applying all assumes into equation ( 6), we get The Azimuthal Part ( ) Where  is the magnetic quantum number.

NIKIFOROV-UVAROV METHOD
The Nikiforov-Uvarov method is a one of the methods used to predict the solutions of generalized second order liner differential equation like Schrodinger equation with particular orthogonal function, we could be obtaining the solution by NU-method when make some transforming to Schrodinger equation to be the same of the below equation [7] ``() + ( ) Where equation ( 9) is considering the standard form of NU-method.Where, () and  () are polynomials with a maximum degree of 2; ̌ ()is polynomial with a maximum degree of 1; () is a hypergeometric function type, and the primes intending the derivatives respect to z. by supposing that: The equation ( 9) become as hypergeometric form: Where Where π(z) is a parameter of 1 st polynomial degree and introduces by equation (13): While  is introduced by equation ( 14) Since () is 1 st degree polynomial, this implies that second order function under square root must be equal to zero, then the quadratic equation can determine k.
To obtain () we can use the integral below equation: And the parameter  in equation ( 14) defined by; The weight function () is obtained in (Eq.17).

DEVELOPING HARTMANN POTENTIAL IN SCHRODINGER EQUATION
The Radial Schrodinger Equation From equation ( 5) we can write the radial part by the below form By comparing equation ( 9) by equation ( 19) to obtain the NU-Coefficients, we get Now, taking this quadratic equation's discriminant equal zero, then the value of constant  could be determined.
The quadratic equation ( 22 Taking  () where τ(r) is negative in equation ( 12) to hold the well value by NU method; so: Returning to the equations ( 14) and ( 16) respectively, and developing equation (25) we get: Comparing equations ( 26) and ( 27) one can predict bound state energy.
Where  is given by equation ( 4).Depending on previous result especially equation 25 we can hold the function () and the weight function () that in equations ( 15) and (17) in a new form: By obtaining equation ( 18) and (30); one can establish the polynomial  (): Involving associated Laguerre polynomial in equation [4] then  () can be defined as: By substituting √−2 as in equation ( 28) where  =  , then the radial wave function (), which defined as () = ( ) is obtained as: Where,  is the normalized constant for orthogonally associated Laguerre polynomial.So, the normalized constant equal Substituting equation (34) to write the final form of radial Schrodinger equation The Angular Schrodinger Equation Now, to determine the angular wave function (); compare equation (7) with equation ( 9) to obtain We have obtained the constants  and  from the equations ( 14) and ( 16) respectively.
Comparing equation (41) with equation (42), we get: Now, depending on the upon result we return to use equations ( 15) and ( 17 Now we can determine the polynomial  () by equation ( 18) and (45).
And by using () =  () () that are defined by equations ( 46) and ( 44) and where By use some relations in associated Legendre polynomials Now apply the equation (48) into equation (47); Where: Where associated Legendre polynomials is giving by equation [13]; So, equation (49) become: Where the normalization constant is So  ()become  () =  `  ` `().
To find the normalized constant, use the normalized condition  () = 1.By use associated Legendre polynomials orthogonally [13], we get So, after replacing  = cos  the angular wave function equal;