# Solution of the 1d Schr\"odinger Equation for a Symmetric Well

## TL;DR

This paper explores analytical and numerical solutions to the 1D Schrödinger equation for a symmetric potential well, comparing methods like perturbation theory, WKB, and Numerov to enhance understanding of quantum mechanics techniques.

## Contribution

It introduces a comprehensive comparison of analytical and numerical methods for solving the 1D Schrödinger equation in a symmetric well, serving as an educational resource.

## Key findings

- Analytical solutions using perturbation theory and WKB provide insights into the potential well.
- Numerov method offers accurate numerical solutions for the Schrödinger equation.
- Comparison highlights the strengths and limitations of each approach.

## Abstract

We suggest a mathematical potential well with spherical symmetry and apply to the 1d Schr\"odinger equation. We use some well-known techniques as Stationary Perturbation Theory and WKB to gain insight into the solutions and compare them to each other. Finally, we solve the 1d Schr\"odinger equation using a numerical approach with the so-called Numerov technique for comparison. It can be a good exercise for undergrad students to grasp the above-cited techniques in a quantum mechanics course.