Determination of energy levels, probabilities, and expectation values of particles in the three-dimensional box at quantum numbers ≤5

Abstract

Static potential boxes represent an area that is limited by an infinite potential wall, where there is no external potential in it. The solution of the Schrodinger equation gives a wave function that can explain the energy level and determine the probability of finding a particle at any point. The main focus of the research in this paperwas to solve the solution to the energy levels, probabilities, and expectation values of a particle in a three-dimensional box, where control limits are given for the potential box width, namely \(\frac{L}{4},\frac{L}{2},\frac{3L}{4}\) and L for each situation. In this research, the results showed that probability meetings depend on changes in box width and quantum number of particles, while particle energy levels depend on the length of the box and the quantum number of particles. Based on the relation between probabilities and expectation values, the most stable size of the box to find the particles in ground state to the fourth excitation state is at the width of \(\frac{L}{4}\) and L.