I have been wondering why anyone ever thought that we could find an expansion for eigenstates of some perturbed Hamiltonian in terms fo those for the basis Hamiltonian. My lecturer insisted that this was because the eigenstates of the Hamiltonian are complete', however from my (limited) understanding of the mathematical basis of QM:
When you are given a Hamiltonian, it comes with the Hilbert space of functions of which it acts (subject to some conditions we impose, such as the Lebesgue square-integrability of functions)
When you are given another Hamiltonian, there is no guarantee that it will act on the same Hilbert space
Therefore, that the zeroth order Hamiltonian has eigenstates forming a complete basis for the space on which ti acts does not guarantee that these same eignestates form a complete basis for the Hilbert space on which the purturbed Hamiltonian acts.
As a simple example. I can imagine adding a small perturbing magnetic field to a Hamiltonian in which there initially is none, such that we lose the degeneracy in the eigenstates accounting for different spins of a particle. One could say that we have the same Hilbert space before and after but we lose degeneracy; or I would have been tempted to say that our initial Hilbert space did not have these extra dimensions corresponding to the spin states. So rather than the eigenstates of the basis Hamiltonian also forming a complete basis for the purtubed Hamiltonian, but having eigenvalue degeneracy which is lifted on the purtubration, I would have said that our Hilbert space has changed. After all, if we take the former view (i.e. a basis accouting for spin but with degeneracy) then it seems like for any problem we would need to know and account for EVERY possible system property that can be discerned. This doesn't seem reasonable to me....
NOTE:
1) My question was asked here previously: Perturbation theory in quantum mechanics: assumptions on eigenvectors
although there is only a discussion and no conclusive answer given. I see that the distinction is important in relativistic QM, and the case of adding a potential onto a free particle Hamiltonian seems to be a special one. Although I am still not sure what the answer is for general purturbations.
2) Similar question here: Nature of perturbed state in perturbation theory?
but it seems to completely avoid answering the question.
Answer
Some potential pitfalls of perturbation theory have been well observed in references, such as problems with the fundamental claim that a 'small' perturbation corresponds to a 'small' deviation from the original eigenvectors/eigenvalues. For instance, take a look of chapter XII in Reed and Simon's Methods of Mathematical Physics, Vol 4 or chapter 2 of Kato's Perturbation Theory for Linear Opperators, 2ed. Nevertheless, as shown in those references, under some strong assumptions one can find a regular behavior.
In the cases that one usually explores in perturbation theory, the original and perturbed Hamiltonian are as self-adjoint as one usually claims in physics contexts (of course, the problem of true self-adjointness of operators is a constant concern in physics and surely has been explored in many other questions in this site). This grants you completeness under the assumptions of the spectral theorem.
In that sense, I would not be as worried as you are about operators acting on different spaces, particularly in the example you suggest about the Zeeman effect. The fact that you were ignoring spin before the perturbation simply shows that your were skipping the study of that degeneracy. This degenerate space is not particularly sick, as you can find an orthonormal basis and do physics without issues. If you think about it, we could be equally concerned about other degeneracies coming, for instance, from the nucleus spin or even the electromagnetic field excitations. Formalisms have been developed to include them, we simply start by focusing on a simpler subspace. Ultimately, once we write $H = H_0 + V$, we are implicitly assuming that they all act on the same Hilbert space. Experimentally, the perturbation is just opening up a window to a subspace we had previously ignored.
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