The Exact Solution to the Schr\"{o}dinger Equation with the Octic Potential

Abstract

The Schr\"{o}dinger equation with the central potential is first studied in the arbitrary dimensional spaces and obtained an analogy of the two-dimensional Schr\"{o}dinger equation for the radial wave function through a simple transformation. As an example, applying an ${\it ansatz}$ to the eigenfunctions, we then arrive at an exact closed form solution to the modified two-dimensional Schr\"{o}dinger equation with the octic potential, $V(r)=ar^2-br^4+cr^6-dr^4+er^{10}$.