Is (energetic) degeneracy a physical property?

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 (vector) 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 on a side: I can imagine that the case is for "symmetry imposed degeneracy" where the state corresponds to higher-dimensional irreducible representations is different than for the general thing which shall include what is sometimes called "random degeneracy".

• en.wikipedia.org/wiki/Stern%E2%80%93Gerlach_experiment Jul 28 '20 at 7:37
• @CharlesFrancis: Thank you, I know that of course. But my questions was about the degeneracy ITSELF such that the dimension of the space would be the directly measured quantity. Jul 28 '20 at 7:41
• The degeneracy of a state enters into the partition function, which influences the thermodynamic properties of your system. So I think it should be possible to measure the degree of a degeneracy just by looking at the temperature dependence of your state populations. Jul 28 '20 at 9:08
• OK, good point! But this exceeds Quantum mechanics. Even within QM I see immediately possibilites to somehow find out the dimension with a sequence of measurements. The real question is: Can a QM operator like $\mathcal{D}_\mathcal{H}$ exist, that does the job in one measurement? Jul 28 '20 at 18:51