General Solution of the Schr\"odinger Equation with Potential Field Quantization

TL;DR

This paper introduces a simple, approximation-free method for solving the time-independent Schrödinger equation in one dimension using potential quantization, providing explicit solutions for bound states, scattering, and decay scenarios.

Contribution

It presents a novel approach to solve the Schrödinger equation via potential quantization without approximations, applicable to various potential problems and validated against experimental data.

Findings

Derived explicit symmetric and antisymmetric wave functions.

Validated solutions against experimental data for scattering and decay.

Provided a general procedure for calculating energy levels and wave functions.

Abstract

It has been found a simple procedure for the general solution of the time-independent Schr\"odinger equation (SE) with the help of quantization of potential area in one dimension without making any approximation. Energy values are not dependent on wave functions, so to find the energy values; it is enough to find the classic turning points of the potential function. Two different solutions were obtained, namely, symmetric and antisymmetric at bound states. These normalized wave functions are always periodic. It is enough to take the integral of the square root of the potential energy function to find the normalized wave functions. If these calculations cannot be made analytically, it should then be performed by numerical methods. SE has been solved for a particle in many one-dimension and the spherical symmetric central potential well as examples. It has been found their energies and normalized wave functions as examples. These solutions were also applied to the theories of scattering and alpha decay. The results obtained with the experimental values were compared with the calculated values. One has been seen to be very fit.