# Chapter 13: Multi-dimensional Wells

Thus far we have concerned ourselves with one-dimensional (non-relativistic) problems in quantum mechanics. We now consider the extension to systems with more than one degree of freedom in more than one dimension. Doing so extends our discussion of quantum-mechanical systems to include more real-world-like situations. We finish with the Coulomb potential which is the potential energy function responsible for basic atomic structure.

# Table of Contents

## Sections

- Section 13.1: The Two-dimensional Infinite Square Well.
- Section 13.2: Two Particles in a One-dimensional Infinite Well.
- Section 13.3: Exploring Superpositions in the Two-dimensional Infinite Well.
- Section 13.4: Exploring the Two-dimensional Harmonic Oscillator.
- Section 13.5: Particle on a Ring.
- Section 13.6: Angular Solutions of the Schrödinger Equation.
- Section 13.7: The Coulomb Potential for the Idealized Hydrogen Atom.
- Section 13.8: Radial Representations of the Coulomb Solutions.
- Section 13.9: Exploring Solutions to the Coulomb Problem.

## Problems

- Problem 13.1: Shown are the solutions to an unknown potential energy function in two dimensions (rectangular).
- Problem 13.2: Shown are the solutions to a two-dimensional infinite square well with an added, unknown potential energy function.
- Problem 13.3: Degeneracy in a rectangular infinite well.
- Problem 13.4: The relationship between n and l and the number of zero crossings.
- Problem 13.5: Determine most-probable values for Coulomb wave functions.
- Problem 13.6: The probability density for an electron in an idealized Hydrogen atom.
- Problem 13.7: Calculating <r> for the Coulomb wave functions.

## Alternate Visualizations

- Section 13.5: Particle on a Ring.
- Section 13.6: Angular Solutions of the Schrödinger Equation.
- Section 13.9: Exploring Solutions to the Coulomb Problem.