Formula Method for Bound State Problems

Abstract

We present a simple formula for finding bound state solution of any quantum wave equation which can be simplified to the form of $\Psi "(s)+\frac{(k_1-k_{2}s)}{s(1-k_{3}s)}{\Psi }^{\prime }(s)+\frac{(A{s}^2+Bs+C)}{s^2(1-k_{3}s{)}^2}\Psi (s)=0$. The two cases where $k_3=0$ and $k_3\ne 0$ are studied. We derive an expression for the energy spectrum and the wave function in terms of generalized hypergeometric functions ${}_2{F}_1(\alpha ,\beta ;\gamma ;k_{3}s)$. 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 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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• Spectroscopic study of some diatomic molecules via the proper quantization rule
• The effect of tensor interaction in splitting the energy levels of relativistic systems
• Hydrogen atom in a Laser-Plasma
• Boosting energy levels in graphene magnetic quantum dots through magnetic flux and inhomogeneous gap
• Massive fermions interacting via a harmonic oscillator in the presence of a minimal length uncertainty relation
• An electron of helium atom under a high-intensity laser field