# 5: Particle in Boxes

The complexities of quantum mechanics are demonstrated mostly in trapped particles.

- Free-Electron Model
- The particle in a box quantum-mechanical problem can provide an instructive application to chemistry: the free-electron model (FEM) for delocalized π -electrons.

- Particle in a 1-Dimensional box
- A particle in a 1-dimensional box is a fundamental quantum mechanical approximation describing the translational motion of a single particle confined inside an infinitely deep well from which it cannot escape.

- Particle in a 2-Dimensional Box
- A particle in a 2-dimensional box is a fundamental quantum mechanical approximation describing the translational motion of a single particle confined inside an infinitely deep well from which it cannot escape.

- Particle in a 3-Dimensional box
- A particle in a 3-dimensional box is a fundamental quantum mechanical approximation describing the translational motion of a single particle confined inside an infinitely deep well from which it cannot escape.

- Particle in a Sphere
- Particle on a sphere is one out of the two models that describe rotational motion. A single particle travels on the surface of the sphere. Unlike particle in a box, the particle on a sphere requires angular momentum, JJ .

- Particle on a Ring
- The case of a particle in a one-dimensional ring is similar to the particle in a box.