Analytical solution of the Bohr-Mottelson equation in minimal length effect for cotangent hyperbolic potential using the hypergeometric method

The rigid deformed nucleus of minimal length effect is investigated using the Bohr-Mottelson equation that influenced by cotangent hyperbolic potential. The Bohr-Mottelson equation in effect a minimum length resolved hypergeometric method for determining the energy spectrum and the wave functions. Energy spectrum was calculated using Matlab software and the wave function is displayed in the form of hypergeometric.

The concept of minimal length is connected by commutation relations between position and momentum operators in Heisenberg Uncertainty Principle.The minimal length occur at Heisenberg Uncertainty Principle is influenced by gravity quantum which is called General Uncertainty Principle (GUP) (Alimohammadi et al.: 2017;Chabab et al.: 2015;Chabab et al.: 2016;Hossenfelder.: 2004;Garay.: 1994).The General Uncertainty Principle modify Heisenberg Uncertainty Principle with additional a small constant (Chabab et al., 2015).
In this paper, equation Bohr-Mottelson in length at least to the potential effects of Hyperbolic cotangent solved using hypergeometric method.Energy spectrum and the wave functions obtained using methods hypergeometric.Hyperbolic cotangent potentially be used to describe the core excitation (Cari et al.: 2013;Suparmi et al.: 2017).This paper consists of four parts, the second part describes the Bohr-Mottelson equation in effect a minimum length and hypergeometric method.Furthermore, in Section 3 describes the results and discussion and final section 4 contains conclusions.

The Bohr-Mottelson equation in minimal length effect
The general canonical commutation between position and momentum is expressed (Hossenfelder.: 2004;Garay.: 1994)

 
, X P i  (1) where X is a position, P is a corresponding momentum.Then, the general canonical commutation between position and momentum is influenced by quantum gravity, it is becomes (Alimohammadi et al.: 2017;Chabab et al.: 2015;Chabab et al.: 2016;Hossenfelder.: 2004;Garay.: 1994), The equation ( 2) is called General Uncertainty Principle, where  was a minimal length parameter that has very small positive values.The uncertainty relation is caused by commutation relation.The equation ( 2) can be reduced becomes (Alimohammadi et al.: 2017;Chabab et al.: 2015;Chabab et al.: 2016;Hossenfelder.: 2004;Garay.: 1994 4) Then, equation (4) can be written (Alimohammadi et al.: 2017;Chabab et al.: (5) where  is Laplacian operator for nucleus that has three degrees of freedom :  , the Laplacian operator as follow (Alimohammadi et al.: 2017;Chabab et al.: 2016) 1 , with g and 1 ij g  are determinant and inverse of the matrix ij g , respectively.We get Laplacian operator, is given as (Alimohammadi et al.: 2017 where P is momentum operator,   V  is potential energy in  function and B m is a mass parameter.We obtain, The equation ( 9) is Bohr-Mottelson equation in minimal length effect.In the case of Bohr-Mottelson equation without the minimal length effect with 0 ML   (Elviyanti et al.: 2017) for equation ( 9), so yields square term is given as (Alimohammadi et al.: 2017), Equations ( 7) and ( 10) are inserted in equation ( 9) and multiplied by which is the separation variable method that used to solve equation ( 11), we have Euler angles part of Bohr-Mottelson Hamiltonian with minimal length,  in equation ( 13) so we have, (14) The Bohr-Mottelson equation for a   part in a minimal length effect for rigid deformed nucleus case is expressed by equation.

Hypergeometric method
The second-order differential equation of hypergeometric function as follow (Suparmi.: 2011;Elviyanti et al.: 2017), The energy eigenvalue is obtained from the condition in equation ( 15) , (Suparmi.: 2011;Elviyanti et al.: 2017) or a n b n     (16) where n=0,1,2,3….Equation ( 16) can be finite series of polynomials of rank n by equation ( 15) .The solution of a wave function is given as By applying the suitable variable change in equation and reduced to standard hypergeometri equation, we get energy eigenvalue and wave function (Suparmi.: 2011;Elviyanti et al.: 2017)

Result and discussion
Cotangent hyperbolic potential is expressed as follows: Equation ( 23) is a differential equation that has been simplified to the form using the hypergeometric differential equation by inserting the following new wave function (26) in equation ( 24) we get 27) is the hypergeometric differential equations are obtained following hypergeometric parameter2  20) -( 22), ( 25) and ( 28 29) is the equation of the energy spectrum of the Bohr-Mottelson in length at least to the potential effects of Hyperbolic cotangent.Then, to get the wave function using equation ( 17), ( 26) and Equation ( 31) and ( 32) is a function of the wave equation Bohr-Mottelson in effect a minimum length for n = 0 and n = 1.The value of the wave function depends on the value of the parameter hypergeomtric.

Conclusion
The Bohr-Mottelson equation in minimal length at least to the potential effects of cotangent hyperbolic can be solved by using hypergeometric.Hypergeometric method used to obtain the energy spectrum and the wave functions in the equation Bohr-Mottelson in effect a minimal length.Energy spectrum and the wave functions equation Bohr-Mottelson the minimal length effect can be shown by equation ( 29), (31) and equation ( 32).