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 ) + ( k 1 - k 2 s ) s ( 1 - k 3 s ) Ψ ′ ( s ) + ( A s 2 + B s + C ) s 2 ( 1 - k 3 s ) 2 Ψ ( s ) = 0 . The two cases where k 3 = 0 and k 3 ≠ 0 are studied. We derive an expression for the energy spectrum and the wave function in terms of generalized hypergeometric functions 2 F 1 ( α , β ; γ ; k 3 s ) . 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 shown to be accurate, efficient, reliable and very easy to use particularly when applied to vast number of quantum potential models.