# Proving the existence of bound states for attractive potentials in 1-d and 2-d without calculus

## TL;DR

This paper demonstrates an algebraic method, based on Schrödinger's operator approach, to prove the existence of bound states in 1D and 2D attractive potentials, making quantum mechanics more accessible without calculus.

## Contribution

It simplifies Schrödinger's operator method for pedagogical use and applies it to prove bound states in low-dimensional attractive potentials algebraically.

## Key findings

- Proved bound states exist in 1D and 2D attractive potentials algebraically.
- Streamlined Schrödinger's operator method for educational purposes.
- Accessible approach requiring minimal mathematical background.

## Abstract

Schroedinger developed an operator method for solving quantum mechanics. While this technique is overshadowed by his more familiar differential equation approach, it has found wide application as an illustration of supersymmetric quantum mechanics. One reason for the reticence in its usage for conventional quantum instruction is that the approach for simple problems like the particle-in-a-box is much more complicated than the differential equation approach, making it appear to be less useful for pedagogy. We argue that the approach is still quite attractive because it employs only algebraic methods, and thereby has a much lower level of math background needed to use it. We show how Schroedinger's operator method can be streamlined for these particle-in-a-box problems greatly reducing the complexity of the solution and making it much more accessible. As an application, we illustrate how this approach can be used to prove an important result, the existence of bound states for one- and two-dimensional attractive potentials, using only algebraic methods. The approach developed here can be employed in undergraduate classes and possibly even high school classes because it employs only algebra and requires essentially no calculus.