Student explore the properties of an orthonormal basis using the Cartesian and \(S_z\) bases as examples.
1. << Dimensional Analysis of Kets | Completeness Relations |
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 \(\left|{-}\right\rangle \).
The elements are norm 1.
The elements are orthogonal.
- The elements form a complete set. Any vector in the space can be written as a linear combination of the two.