Radial Schrödinger Equation Solution for Gaussian Potential

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Energy eigenvalues and eigenfunctions in quantum mechanics are studied through the exact solution of the radial Schrödinger equation for Gaussian potentials, using the Asymptotic Iteration Method. The method's efficiency in solving wave equations for different potentials is highlighted, with a focus on nuclear structure and exotic nuclei reactions.

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Solution of the Radial Schrdinger Solution of the Radial Schr dinger Equation for Gaussian Potential Equation for Gaussian Potential A. A. Hacisalihoglu* Hacisalihoglu*,a ,a, Y. , Y. Kucuk Kucukb b, , aDepartment of Physics, Recep Tayyip Erdogan University, 53100 Rize, Turkey bDepartment of Physics, Akdeniz University, 07100 Antalya, Turkey

CONTENTS I. Introduction II. Asymptotic Iteration Method III. Solution of the Radial Schr dinger Equation for Gaussian Potential IV. Conclusions

I. INTRODUCTION Over the last decades, the energy eigenvalues and corresponding eigenfunctions between interaction systems have raised a great deal of interest in the relativistic as well as in the non-relativistic quantum mechanics. The exact solution of the wave equations (relativistic or non- relativistic) are very important since the wave function contains all the necessary information regarding the quantum system under consideration.The Gaussian extensively used to describe the bound and the continuum states of the interactions systems potential has been

In general,the approximation methods have been used for the solution of the Gaussian potential such as the; Wentzel Kramers Brillouin Approximation [4], Perturbation and Variational Methods [5] , The Higher-Order Perturbation Theory combined with Hypervirial Pade Scheme [6], Large-N Expansion Method [7].

In this study, we apply the asymptotic iteration method (AIM) to solve the radial Schr dinger equation for the Gaussian potentail without any approximation to the potential. It has been shown that this method is very efficient to solve the Schr dinger equation for different potentials. This study, Investigation of the exotic nuclei reactions by using the Continuum Discretized Coupled Channels (CDCC) Model (project number: 110T388) from the scope of the project work was carried out to contribute to the structure of nuclear. At Nuclear structure process, Bohr Hamiltonian defined excitation energy levels of nuclei was aimed to be solved by using the method of asymptotic iteration potential for gauss potential.

II. THE ASYMPTOTIC ITERATION METHOD AIM is proposed to solve the second-order differential equations of the form. = + '' ' ( ) x y ( ) y s x y (2.1) 0 0 0( ) s x 0( ) x where The variables, differentiable. The differential Eq.(2.1) has a general solution. and are sufficiently 0( ) 0 x x ' x x ( ') a x dx ' ( ( '') x 0 = + '' '' ( ) y x 2 ( ) a x dx e c c e + (2.2) 2 1

If k > 0, for sufficiently large k, we obtain the values from ( ) kk s x = ( ) x ( ) ( ) x s x = ( ) x k (2.3) k ( ) kx ( ) ks x Where and = + + ' ( ) x ( ) x ( ) x ( ) x ( ) x s (2.4) (2.5) 1 0 ( ) ( ) ( ) k k k S x S x S x = + 1 1 0 1 k k k k ' k = ( ) x 1,2,3... 1 The energy eigenvalues are obtained from the quantization condition . The quantization condition of the method together with Eq. (2.5) can also be written as follows (2.6) 1 ( ) ( ) ( ) k k k x x S x = + = ( ) x S x ( ) 0 1 k k

the radial Schrdinger equation is converted to the form of Eq. (2.1). Then, and are determined and parameters are calculated. For a given potential such as the Gaussian potential one, 0( ) x 0( ) s x ( ) ( ) and kx ks x condition given by Eq. (2.6). In this equation, k shows the iteration number. The energy eigenvalues are determined by the quantization For the exactly solvable potentials, the energy eigenvalues are obtained from this equation if the problem is exactly solvable and the radial quantum number n is equal to the iteration number k for this case.

specific n radial quantum number, we choose a suitable point, determined generally as the maximum value of the asymptotic wave function or the minimum value of the potential. The approximate energy eigenvalues are obtained from the roots of this equation for sufficiently great values of k with iteration. For nontrivial potentials that have no exact solutions, for a 0 r

III. THE ENERGY EIGENVALUES FOR THE GAUSSIAN POTENTIAL The radial Schr dinger equation is given by (3.1) 2 ( ) 2 d R r dr m + E V = ( ) r ( ) 0 nl R r eff nl 2 2 where m is the mass and Veff(r) is effective potential which is the sum of centrifugal attractive radial Gaussian potentials. + 2 ( 1) l l and the = ( ) r centrifugal V 2 2 mr 2 = ( ) r (3.2) ( ) gauss V r Ae

Veff(r) is effective potential which is the sum of centrifugal and the attractive radial Gaussian potential: + 2 ( 1) l l 2 (3.3) ( ) effective V r Ae = + ( ) r 2 2 mr where A is the depth of the Gaussian potential. Inserting Eq. (2.7) into Eq. (2.6) and using the following ansatzs 2 mE 2 mA A= = n 2 2

We can easily obtain + 2 ( ) ( 1) d R r l l 2 (3.4) 2 Ae dr + + = r ( ) 0 nl R r nl 2 r In order to solve this equation with AIM, we should transform this equation into the form of Eq. (2.1). It is clear that when r goes to zero, and at infinity, therefore we can write the physical wavefunction as follows (3.5) nl nl 2 ( ) exp( ) R r r + 1 l ( ) R r r nl nl 2 + = 1 l r ( ) ( ) r R r r e f

If we insert this wave function into Eq. (3.4), we have the second-order homogeneous linear differential equations in the following form: ( ) ( ) ( 1) 2 2 2 2 r A r dr + 2 4 d f r df r l Ar = + + + + + 2 2 2 (3.6) 6 4 4 ( ) r nl nl Ar r l r f nl 2 4 dr By comparing this equation with Eq. (2.1), we can easily write the and values: (3.7) (3.8) 4 0( ) r 0( ) s r + ( 1) l = + ( ) r 2 2 r 0 r Ar 4 A Ar = + + + 2 2 2 ( ) 2 2 6 4 4 s r r l r 0

( ) ks r 0( ) r 0( ) s r Using and values we calculate and values as follows: 1 ( ) r r ( ) kr 2 = + 4 r (3.9) 4 Ar = A Ar + + ( 3 2 + (3.10) )) 2 2 1( ) s r 2 2( r 4

It is important to point out that if this equation can be solved at every point, then the problem is called as the exactly solvable. r In our potential, since the problem is not exactly solvable, we have to choose a suitable the equation to find 0 ( , ) 0 0 r point and solve values. = kr 0 r In this study, we determine the maximum point of the asymptotic wavefunction, which is the same as the root of , namely, value from the 0( ) r = + 0 1 l = r 0 2

Here, is convergence constant which affects the speed of the convergence [19 21]. We have noticed that the optimum value of is problem dependent. Therefore, we have conducted a series of calculations for and observed that the best convergence is obtained when the constant value, which gives the energy eigenvalues around k = 130 iterations. For other values of and high number of quantum states, the convergence needs more iteration to obtain the energy eigenvalues. from 1 to 40 = 10

ekil ekil- -1: 1: ,m=0.5, A=400, , = 10 and for l=0,1,2,3,4,5,6,7 values changes r- dependent of potential. ( ) eff V r = = 1 1

V. CONCLUSIONS In this study, we have presented the iterative solutions of the radial Schr dinger equation for the attractive radial Gaussian potentials for any n and l quantum states by using the asymptotic iteration method. We have obtained the bound state energy eigenvalues for any n and l states using AIM iteration procedure without any approximation. The obtained results of AIM calculations have been compared the results of other studies. As shown in Table I the very agreement between AIM calculations and other calculations has been obtained.

The advantage of the asymptotic iteration method is that it gives the eigenvalues directly by transforming the second-order differential equation into a form of = + '' ' ( ) ( ) ( ) r f r ( ) ( ) s r f r f r 0 0 n n n analytical solutions if there is and provides the closed-forms for the energy eigenvalues. Where there is no such a solution, The asymptotic iteration method results in exact iterative approach [12, 14, 19 21]. As it is presented, AIM puts no constraint on the potential parameter values involved and it is easy to implement. The results are sufficiently accurate for the practical purposes. It is worth extending this method to examine other interacting systems. The energy eigenvalues are obtained by using an

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