Product Wavefunction and the Nikiforov-Uvarov Method for the Schrodinger Equation

 In a series of notes (1) , we suggested one may compare the Schrodinger equation, with the ground state solution removed, to a general hypergeometric equation:

p(x) d/dx d/dx W1(x) + q(x) d/dx W1(x) + C1 W(1x) = 0 ((1)) where p(x) is at most quadratic in x and q(x) is at most linear. Such an equation has known Rodrigues polynomial solutions (2).To remove the ground state, we write W(x)=Wo(x)W1(x) where W(x) is the full wavefunction of a bound state and Wo(x), the ground state (or another state). Inserting into the time-independent Schrodinger equation leads to the coupled equation:

   -1/2m d/dx d/dx W1(x) - 1/m d/dx W1(x) [d/dx Wo(x) / Wo(x)] = (E-Eo) W1(x)  ((2))

Recently in 2006, a more general method was presented in the literature (3). In this method, W(x)=f(x)F(x), E=Ef + EF and V=Vf + VF. These values arre inserted into the time-independent Schrodinger equation and  compared with the Nikiforov-Uvarov method equation. This general method was applied to a specific case of Jacobi polynomials and VF found to be 0. 

   In the case that VF=0, f(x) is a solution of the time-independent Schrodinger equation and this general method is the same as the one presented in (1). In fact, it seems in the example in (3), VF is forced to equal 0 which may imply the more general method is very similar to the method of (1) The objective of this note is to compare the two methods.