Do commuting operators have common eigenvectors?

are also degenerate eigenstates of ˆ A. In other words, commuting operators have common eigenvectors, but we need to do some extra work to find them if degeneracy is present.

What are commuting operators?

In quantum mechanics, a complete set of commuting observables (CSCO) is a set of commuting operators whose eigenvalues completely specify the state of a system. It is therefore not necessary to specify the order in which the different observables are measured.

Which pair of operators do not commute?

Thus we have shown that the operator product of x and p is non-commuting. and this is called the commutator of ˆA and ˆB (in that order!). If [ ˆA, ˆ B] = 0, then one says that ˆA and ˆB do not commute, if [ ˆA, ˆ B] = 0, then ˆA and ˆB are said to commute with each other.

Which of the following pair of operator will commute?

Commutator. The commutator is a mathematical operation which tells about operators involved that they commute with each other or not. It can be represented as[A,B] or (A, B)

What does it mean if 2 operators 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.

Do commuting operators share eigenvalues?

Commuting matrices do not necessarily share all eigenvector, but generally do share a common eigenvector.

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.

Do all Hermitian operators commute?

If two operators have a complete set of simultaneous eigenfunctions, they must commute. That is, in this case the operators A and B must commute.

Do Hermitian operators always commute?

Can Hermitian operators be degenerate?

Hence, we conclude that the eigenstates of an Hermitian operator are, or can be chosen to be, mutually orthogonal. If the eigenvalues of two eigenfunctions are the same, then the functions are said to be degenerate, and linear combinations of the degenerate functions can be formed that will be orthogonal to each other.

Are there any operators that have common eigenfunctions?

According to the proof that malawi_glenn posted, if two operators commute they SHOULD have common eigenfunctions regardless of anything else. Which means that H,lx,ly,lz should all have common eigenfunctions. The fact that they dont (only H and one of the l i does) shows the proof to have a flaw?

Is the random eigenvalue of a commuting operator infinite?

So you`ve shown that if two opperators commute then they can be both represented by diagonal matrices in the same set of base kets. You`ve also shown that the elements on the diagonal are the operator`s eigenvalues. That means that the random eigenvalue is infinite?

Which is an example of a degenerate eigenvalue?

Degenerate eigenvalues/eigenvectors are those which don’t satisfy this uniqueness property. Note that some references use the expression \\degenerate state” with respect to Equation (1) but referring only to the energy operator, see for example [2].

Which is a member of the eigenspace A?

Since the commutation relation AB=BA implies that A (B|a>)=a (B|a>), B|a> is a obviously a member of the eigenspace associated with the eigenvalue a.