Contents
What does it mean for 2 operators to commute?
If two operators commute, then they can have the same set of eigenfunctions. If two operators commute and consequently have the same set of eigenfunctions, then the corresponding physical quantities can be evaluated or measured exactly simultaneously with no limit on the uncertainty.
Does Hamiltonian commute?
The Hamiltonian operator for a quantum mechanical system is represented by the imaginary unit times the partial time derivative. Hence the time derivative and the gradient commute.
Does energy commute with Hamiltonian?
When a particle is under the influence of a central (symmetrical) potential, then L commutes with potential energy V(r). If L commutes with Hamiltonian operator (kinetic energy plus potential energy) then the angular momentum and energy can be known simultaneously.
Do commuting operators share eigenvalues?
Commuting matrices do not necessarily share all eigenvector, but generally do share a common eigenvector.
Does L 2 commute with R?
L2 is independent of r; therefore it commutes with any function of r. This seems related to a problem in Schaum’s Quantum Mechanics (Amazon link) dealing with a particle in a spherical potential well.
Do Hermitian operators commute with Hamiltonian?
When we have a QM system in an energy eigenstate (say after a measurement of energy) then we can measure any time another quantity that is described by an hermitian operator that commutes with the Hamiltonian and expect to get a precisely predictable result, namely an eigenvalue.
Does an operator commute with itself?
The super-commutator of D operator (1) with itself is not zero: [D,D]SC = 2D2 = 2ddt≠0. II) More generally, the fact that a Grassmann-odd operator (super)commute with itself is a non-trivial condition, which encodes non-trivial information about the theory. This is e.g. used in supersymmetry and in BRST formulations.