Solutions to the Schrödinger Equation with Inversely Quadratic Yukawa Plus Inversely Quadratic Hellmann Potential Using Nikiforov-Uvarov Method

The purpose of the present work is to present the solution to the Schrödinger equation with the inversely quadratic Yukawa potential [11] plus inversely quadratic Hellmann potential [12] of the forms The sum of these potentials can be written as where r represents the internuclear distance, a and b are the strengths of the Coulomb and Yukawa potentials, respectively, δ is the screening parameter, and V₀ is the dissociation energy. Equation (2) is then amenable to Nikiforov-Uvarov method. Ita [13] has solved the Schrödinger equation for the Hellman potential and obtained the energy eigenvalues and their corresponding wave functions using expansion method and Nikiforov-Uvarov method. Also, Hamzavi and Rajabi [14] have used the parametric Nikiforov-Uvarov method to obtain tensor coupling and relativistic spin and pseudospin symmetries of the Dirac equation with the Hellmann potential. Kocak et al. [15] solved the Schrödinger equation with the Hellmann potential using asymptotic iteration method and obtained energy eigenvalues and the wave functions. However, not much has been achieved in the area of solving the radial Schrödinger equation for any angular momentum quantum number, l, with IQYIQH potential using Nikiforov-Uvarov method in the literature.

The Nikiforov-Uvarov (NU) method is based on the solutions to a generalized second-order linear differential equation with special orthogonal functions [16]. The Schrödinger equation of the type: could be solved by this method. This can be done by transforming (3) into an equation of hypergeometric type with appropriate coordinate transformation z = z(r) to get To find the exact solution to (4), we write ψ(z) as Substitution of (5) into (4) yields (6) of hypergeometric type In (5), the wave function ϕ(z) is defined as the logarithmic derivative [17] with π(z) being at most first-order polynomials. Also, the hypergeometric-type functions in (6) for a fixed integer n are given by the Rodrigue relation as where B_n is the normalization constant and the weight function ρ(z) must satisfy the condition with In order to accomplish the condition imposed on the weight function ρ(z) it is necessary that the polynomial τ(z) be equal to zero at some points of an interval (a, b) and its derivative at this interval at σ(z) > 0 will be negative [17]. That is, The function π(z) and the parameter λ required for the NU method are then defined as [17] The z-values in (12) are possible to evaluate if the expression under the square root is square of polynomials. This is possible if and only if its discriminant is zero. Therefore, the new eigenvalue equation becomes [18] A comparison between (13) and (14) yields the energy eigenvalues.

In spherical coordinate, the Schrödinger equation with the potential V(r) is given as [19] Using the common ansatz for the wave function in (9) we get the following set of equations: where λ = l(l + 1) and m² are the separation constants. Y_lm(θ, ϕ) = Θ_ml(θ)Φ_m(ϕ) is the solution to (18) and (19) and their solutions are well known as spherical harmonic functions [19].

Equation (17) is the radial part of the Schrödinger equation which we are interested in solving. Equation (17) together with the potential in (2) and with the transformation z = r² yields the following equation: where the radial wave function is R(z) and Equation (14) is then compared with (4) and the following expressions are obtained: We then obtain the function π by substituting (22) into (12): According to the NU method, the quadratic form under the square-root sign of (23) must be solved by setting the discriminant of this quadratic equation equal to zero; that is, D = b² − 4ac = 0. This discriminant gives a new equation which can be solved for the constant k to get the two roots as Thus, we have When the two values of k given in (25) are substituted into (23), the four possible forms of π(z) are obtained as One of the four values of the polynomial π(z) is just proper to obtain the bound state solution since τ given in (4) must have negative derivative. Therefore, the most suitable expression of π(z) is chosen as for $$. We obtain $$ from (10) and the derivative of this expression would be negative; that is, $$. From (19) and (20) we obtain When we compare these expressions, λ = λ_n, we obtain the energy of the IQYIQH potential as Let us now calculate the radial wave function, R(z). Using σ and π (7) and (9), the following expressions are obtained: Then from (8) one has B_n is a normalization constant. The wave function R(z) can be obtained in terms of the generalized Laguerre polynomials as N_n is the normalization constant.

In summary, we have obtained the energy eigenvalues and the corresponding unnormalized wave function using the NU method for the Schrödinger equation with the inversely quadratic Yukawa plus inversely quadratic Hellmann potential. If we set the parameters V_o = b = 0 and a = ze², it is easy to show that (29) reduces to the bound state energy spectrum of a particle in the Coulomb potential; that is, $$, where n_p = n + l + 1 is the principal quantum number. Similarly, if we set a = b = 0, (29) results in the bound state energy spectrum of a vibrating-rotating diatomic molecule subject to the inversely quadratic Yukawa potential as follows: This result is similar to that obtained in [11]. Also, if we set V_o = 0, a ≠ 0, and b ≠ 0, equation reduces to the bound state energy spectrum of a vibrating-rotating diatomic molecule subject to inversely quadratic Hellmann potential as follows: Equation (34) is also similar to the one obtained in [12]. These show the accuracy of our calculations.