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.
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