## Small Group Activity: $|\pm\rangle$ Forms an Orthonormal Basis

Quantum Fundamentals 2021
Student explore the properties of an orthonormal basis using the Cartesian and $S_z$ bases as examples.
• This ingredient is used in the following sequences
What students learn
• The properties of an orthonormal basis
• How to write these properties using Dirac notation

The state vectors $\left|{+}\right\rangle$ and $\left|{-}\right\rangle$ form an orthormal basis. Orthonormal bases have properties listed below.

For each property, write an equation that illustrates the property for both:

• The Cartesian basis vectors $\hat{x}$, $\hat{y}$, and $\hat{z}$.
• The spin state vectors $\left|{+}\right\rangle$ and $\left|{-}\right\rangle$.
1. The elements are norm 1.

2. The elements are orthogonal.

3. The elements form a complete set. Any vector in the space can be written as a linear combination of the two.

Keywords
Cartesian Basis $S_z$ basis completeness normalization orthogonality basis Quantum Mechanics
Learning Outcomes