General solution of the Schrödinger equation with potential field quantization

Abstract

A simple procedure has been found for the general solution of the time-independent Schrödinger equation(SE) with the help of quantization of potential area in one dimension without making any approximation. Energyvalues are not dependent on wave functions. So, to find the energy values, it is enough to find the classic turningpoints of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric in boundstates. These normalized wave functions are always periodic. It is enough to take the integral of the square root of thepotential energy function to find the normalized wave functions. If these calculations cannot be made analytically, theyshould then be performed by numerical methods. The SE has been solved for a particle in many one-dimension and thespherical symmetric central potential wells as examples. Their energies and normalized wave functions were found asexamples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with theexperimental values were compared with the calculated values. One was seen to be very fit.