Complete set of operators ↩

Metadata

shorthands: {}
aliases: [Complete set of observables]
created: 2022-01-06 02:52:27
modified: 2022-01-10 04:13:04

The set of Hermitian operators

is called a complete set of operators or observables if these operators all commute with each other and if the common set of eigenvectors is no longer degenerate.

These eigenvectors then can be characterized by the corresponding eigenvalues :

Comments

If for a given set of operators the eigenvectors are still degenerate, there exists an additional symmetry of these operators and the generator of this symmetry operation also commutes with this set of operators
Commuting operators have a common set of eigenvectors
Operators with a common set of eigenvectors commute

Examples

For one-dimensional potentials (Schrödinger equation in 1D): or
For three-dimensional potentials (For the general form see here): or
For three-dimensional spherically symmetric potentials: or or