In quantum mechanics, observable properties correspond to expectation- or eigenvalues of (hermitian) operators.

After measurement (of an eigenvalue) the system is in an eigenstate that corresponds to an eigenfunction of the operator. Sometimes, however an eigenvalue (\(\lambda\)) can not only arise from one certain well-defined state-function but from a whole (Hilbert sub) space that is spanned by *n* eigenfunctions with the same eigenvalue \(\lambda\) for n>1.

What I am interested in, is if the degeneracy itself is something that can be measured? For example can there exist an operator ,say \(\mathcal{D}_\mathcal{H}\)that measures the degeneracy of a state function\(\Psi_\lambda\) (for a certain operator, for example the Hamiltonian $\mathcal{H})$:

$$ \mathcal{D}_\mathcal{H} \Psi_\lambda = n \Psi_\lambda $$

This question arises from some considerations of special symmetry properties of degenerate states I am investigating. In the course of these works the question arose, if the degeneracy of a state is some physical property or rather only something like an "mathematical artifact" that cannot be directly probed experimentally.

**Note:** I would expect a possible answer to consider both cases, for the one a "symmetry imposed" degeneracy where the state corresponds to higher-dimensional irreducible representation of the symmetry group of the Hamiltonian and for the other the *general* thing that is sometimes called *accidental* degeneracy.