Formula Method for Bound State Problems

We present a simple formula for finding bound state solution of any quantum wave equation which can be simplified to the form of $Ψ"(s)+\frac{(k_1-k_2s)}{s(1-k_3s)}Ψ'(s)+\frac{(As^2+Bs+C)}{s^2(1-k_3s)^2}Ψ(s)=0$. The two cases where $k_3=0$ and $k_3\neq 0$ are studied. We derive an expression for the energy spectrum and the wave function in terms of generalized hypergeometric functions $_2F_1(α, β; γ; k_3s)$. In order to show the accuracy of this proposed formula, we resort to obtaining bound state solutions for some existing eigenvalue problems in a rather more simplified way. This method has been shown to be accurate, efficient, reliable and very easy to use particularly when applied to vast number of quantum potential models.

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