The D-dimensional non-relativistic particle in the Scarf Trigonometry plus Non-Central Rosen-Morse Potentials

The D-Dimensional Non-Relativistic Particle Properties in the Scarf Trigonometry plus Non-Central Rosen-Morse Potentials was investigated using an analytical method. The bound state energy is given approximately in the closed form. The approximate wave function for arbitrary l-state in D-dimensions are expressed in the form of generalised Jacobi Polynomials. The energy spectra of the particle are increased when the dimensions are higher. The relationship between the orbital number in each dimension is recursive. The special case in 3 dimensions is given to the ground state.

The Scarf potential describes the periodically arranged particles like a crystal [5,6,14,15]. The application of this potential is a crystal model in the solid state physics [5,6]. On the other hand, the Rosen-Morse potential describes the quark-gluon dynamics of Quantum Chromodynamics [16]. The solution of this potential remains in the radial part [16][17][18]. The combination of these potential in D-Dimensions with the centrifugal term is separable potential. Hence, this system can be solved by the separation of variable method. The solution of radial part of Schrodinger equation is one solution. Nevertheless, the solution of angular part of Schrodinger equation is D-1 solutions.

The Nikiforov-Uvarov Method
The Schrodinger equation in D-dimensions of any shape invariant potential can be reduced into hypergeometric-type of a differential equation using a suitable variable transformation [19][20][21]. The hypergeometric-type of differential equation in Nikiforov-Uvarov method is:  (2) into equation (1), we get the hypergeometric type equation, that is: Where ϕ (s) is a logarithmic derivative. The solution of ϕ (s) is obtained from condition: ' while the function π (s) and the parameter λ are defined as: The value of k in equation (5) can be found from the condition that the expression under the square root of equation (5) must be a square of a polynomial. This polynomial is mostly first-degree polynomial, therefore the discriminate of the quadratic expression is zero. A new eigenvalue of equation (3) is: where 2      (8) The new bound state energy obtained using the equation (6) and (7). To generate the bound state energy and the corresponding eigenfunction, the condition that τ' < 0 is required. The solution of the second part of the wave function, yn (s), is connected to Rodrigues relation [20]: where Cn is normalization constant, and the weight function ρ(s) must satisfies the condition: The wave function of the system obtained from equation (4), (9) and (10).

Scarf Trigonometry plus Non-Central Rosen-Morse Potentials
The Scarf Trigonometry potential can be expressed as: The Non Central Rosen-Morse potential can be expressed as: The Schrödinger Equation in D-Dimensions for Scarf Hyperbolic plus Non-Central Pocshl-Teller Potential can be expressed as:  is hypersperical harmonics, that is, D-Dimensional angular momentum operator. For 2 ≤ k ≤ D -1, we have: and for k = 1, we have: Subtitute (14), (15), (16), using variable substitution, and separation variable, that is,           for k = 2, 3, 4, …, D -1. im Ae   (22) where A is normalization constant.

Solution of radial part of Schrodinger equation in D-dimensions
The radial part of Schrodinger equation in D-Dimensions (Eq. 19) is hypergeometric differential equation. Equation (19) can be solved using some approximation and variable substitution, that is, and AD-1 is centrifugal term which depends on the eigenvalue of polar part of Schrodinger equation in D-Dimensions.
By using eigenfunction of NU method, we obtain: From Figure 1, we have the effect of extra dimensions in radial wave function is increase the amplitude of radial wave function. It was found to agree with previous work [6]. Based on superstring theory [22], the number of dimensions in the universe restricted to 10-spatial dimensions and 1-time dimension. If the amount of spatial dimension more than 10, the universe unstable and collapse. Thus, the maximum value of D is limited to 10 spatial dimensions.

Solution of polar part of Schrodinger equation in D-dimensions
According to the value of k, there are two equations at polar part of Schrodinger equation in D-Dimensions. First, for k = 2, 3, 4, …, D -1, we solve equation (20) using variable substitution, that is,