# On the boundary conditions for eliminating negative energy states in the infinite square well model

## TL;DR

This paper explores various boundary conditions in the infinite square well model that prevent negative energy states, highlighting multiple physically valid configurations beyond the standard continuity condition.

## Contribution

It identifies alternative boundary conditions that eliminate negative energy states in the infinite square well, expanding the understanding of possible physical systems within this model.

## Key findings

- Standard boundary condition ensures no negative energy states.
- Other boundary conditions can also prevent negative energy states.
- Multiple physical configurations are possible in the infinite well model.

## Abstract

For a quantum confinement model, the wave function of a particle is zero outside the confined region. Due to this, the negative energy states are, in fact, square integrable. As negative energy states are not physical, we need to impose some boundary conditions in order to avoid these states. For the case of the infinite square well model, we show the possible boundary conditions that avoid the existence of the negative energy states. The well-known boundary condition requiring the wave function be continuous on both sides of the well is one of the boundary conditions that avoids negative energy states. However, there are other types of boundary condition that can also avoid the negative energy states. This shows that there are the different branch of physical systems that can be established on the infinite square well model.