Determine the Eigen Function of Schrodinger Equation with Non-Central Potential by Using NU Method

So far, Schrodinger equation with central potential has been solved in different methods but solving this equation with non-central potentials is less dealt with. Solving such equations are way more difficult and complicated and a certain and limited number of non-central potentials can be solved. In this paper, we introduce one of the solvable kinds of such potentials and we will use NU method for solving Schrodinger equation and then by using this method we have calculated particular figures of its energy and function.


Introduction
One of the important tasks of quantum mechanic is finding accurate answers of Schrodinger equation with a certain potential.It is obvious that finding exact answers of SE by the usual and traditional methods is impossible, except certain cases such as a system with Qualeny potential or a coordinating oscillator.Thus, it is inevitable to use methods to help us solve this problem.Among the cases where we have to refuse ordinary methods and seek new methods is solving SE with non-central potentials.Such potentials are of high importance in quantum chemistry and nuclear physics.Recently, a lot of studies are being done about such potentials.Accordingly, different methods are used to solve SE with non-central potentials among which we can name symmetrical cloud, SUSY, SIP idea [1,2], route integral [3] and Factorial method [4].
There is also another method known as NU (Nikiforov-Uvarov) which gives a clear instruction for obtaining exact answers of certain states ,Eigen value of energy and the related functions based on Orthogonal polynomials [5].NU method is based on reducing a second degree differential equation of SE into an equation of hyper geometric type [6][7][8].Based on this, in this paper we will try to solve SE with a suitable potential without any limits.Therefore, we consider a potential as follows to meet all our needs in order to solve SE by NU method: After choosing a suitable change of variable   s s r  , the transformed equation will be as follows: In which  and  are polynomials of maximum second degree and  is a polynomial of maximum first degree.By considering wave function Equation ( 1) will be reduced to the following equation which is of hyper geometric type [5]; That in equations:  is a parameter which is defined as follows [5]: The polynomial ( ) s

 
shows that its first derivative must be negative.We should notice that  and n  are obtained from a particular answer of     n y s y  s which is a polynomial of n degree.Furthermore, statement   n y s ,wave function of Equation (2), is a function of hyper geometric type which is obtained from the following Rodriguez equation [5].
In which n is the normalization constant and B   s  is a weight function which has to meet the following condition [5,6].
has to be a polynomial of maximum first degree, the statements under the radical in Equation (9) have to be sorted in the form of a first degree polynomial and this is possible when its determiner, , is zero.In this case, an equation is obtained for K and after solving the equation, the obtained figures for K are placed in Equation (9) and by comparing with Equations ( 6) and (10) we will calculate Eigen value of energy.

Schrodinger Equation with Central and Non-Central Potentials
We consider time-independent SE as follows: Wave function elaborates certain states and their related energy levels, n , for a particle in a potential field.First, central and non-central potentials are considered in spiracle coordinates and put them in the above equation.( ) By considering the whole wave function as: And replacing in Equation ( 12), we can write the equation separately as follows: In which m 2 and   are the separation fix amounts.The answer for Equation ( 16) is as follows [9].
Equations ( 14) and (15) are radial and angular equations respectively which we are going to solve by NU method.

Solving Radial Equation and Calculating Eigen Values of Energy by Using NU Method
For solving radial part of SE, by considering in Equation ( 14) we will have: Now if we compare the above equation with Equation (1), the general form of equation in NU method, we will have: Therefore, according to the definition of   r  in Equation ( 9), we can write   r  as follows: From the above equation, considering that under the radical there should be a of a first degree polynomial.
We will determine K and we will have: To continue, we choose the suitable amount of   r  which can meet the condition 0 Now, for (i), we can write   r  2 4 r r from Equation (5) as follows: And from Equations ( 6) and (10) we will calculate n  and  respectively So: And eventually, we can obtain Eigen value of energy for (i) from the above equation: In the same way, for (ii), Equation (22), by repeating the above process we can obtain Eigen value of energy.Therefore, from Equation (5), we write as: And from Equations ( 6) and ( 10) we calculate n  and  : From the above equations, Eigen value of energy is obtained: We considered 2 E    and from ( 27) and (32), we have Eigen value of energy as:

Calculating Eigen Functions Related to Radial Share of Wave Function
In order to obtain Eigen radial functions, by using Equation (4) we have: Thus, for (i) and (ii) in Equation ( 22) we have: We will have: On the other hand, from Equation (8) we have: From Equation (37), as we have   2 r r

 
   , we will have: From Rodriguez sequence polynomial, we will have Laguerres dependent functions as [9]: By comparing the two functions, (39) and ( 40) and by considering k    and 2 r x   we will have: In the same way for (ii) in which from Equation (37) we will have: And finally, from Equation ( 7 Therefore, from Equations (2), ( 35), (36), (42), (46) the whole radial wave function can be written as: 1 by taking orthogonal condition of associated Laguerre polynomials and we will have:

Solving Angular Equation and Calculating Eigen Values
With a variable change as cos x   we will tion (15) as the following transformation: have Equa- By comparing Equations ( 49) with (1) we will have: By using ( 9), we write   x  as: We determine K on condition that there gree polynomial under the radical: And for we have the right choice as: m Eq Also, fro uation (5), we can obtain   x  e will hav Then w e: According to Equations ( 6) and ( 10), we will calculate n  and  respectively: Finally we will obtain  from the above equation:

Calculation Eigen Function of A Equation
To fi by using Equation (4) we will have:

s ngular
nd Eigen function related to angular equation, Thus, for   8) we have: On the oth m Equation Therefore, we can write the weight function And eventually, from Equation (7), we wr e ) (x acobi's From Rodriguez sequence, we will have J polynomials as [10,11]: ) and (64) and c sidering This way, we calculated Eigen of functions and Eigen value of energy.According to this wave function, we can determine static characteristics of this system.We can also use this method for solving SE with other non-central potentials.

7.
rtain non-central potential by using the rse, we should note that this method, ethods for solving SE, does not funcon with any non-central potential and we can get a 0375-9601(00)00252-8 x x x  (66)

Conclusions
In this paper we showed that Eigen value of energy and its Eigen dependent functions can be obtained for a system under a ce NU method.Of cou too, like previous m ti suitable result out of this method only with certain types of potentials which meet the requirements of the method.And in this paper, by considering all the requirements, we have introduced the potential in mind and gotten the results mentioned in the article.