Question on perturbation theory

[Couchyam]

Hey everyone,

I'm studying quantum mechanics from Griffiths (Introduction to Quantum Mechanics, 2nd edition), and I'm puzzling over his derivation of the nth order corrections to the energies and corresponding eigenstates for a perturbed Hamiltonian. The steps that are outlined in Griffiths are all very clear, but the derivation seems to dodge some of the deeper mathematical ideas, which could really give more insight into the physics of perturbation theory.

Firstly, if I am interpreting the text correctly, Griffiths starts off the non-degenerate case by asserting that for each n, there exists fixed ψ_n^j and E_n^j 0≤j≤∞, such that for any λ in [0,1], (H°+λH')ψ_n,λ = E_n,λ, where
ψ_n,λ = Ʃ λ^jψ_n^j, and
E_n,λ = Ʃ λ^jE_n^j

Although I can intuitively see how this could be true, I am not sure how to make the justification precise.
Also, I noticed that the technique for moving eigenstates from the non-perturbed Hamiltonian to the full Hamiltonian looks a lot like the continuity argument when proving Rouche's theorem (when you "perturb" a holomorphic function f by some holomorphic function g [with norm(g) less than norm(f) on a circle whose interior is contained in the open set on which f is holomorphic], the number of zeros of f inside the circle is equal to the number of zeros of f+g inside the same circle). Is the justification for perturbation theory related to this?

Thanks!

 

[eljose79]

Thank you for your post and for sharing your thoughts on Griffiths' derivation of nth order corrections to the energies and eigenstates in perturbation theory. As a scientist studying quantum mechanics, I can understand your interest in gaining a deeper understanding of the mathematical ideas behind this theory.

To answer your first question, the assertion that for each n there exists fixed ψnj and Enj 0≤j≤∞ such that (H°+λH')ψn,λ = En,λ is a direct result of the perturbation theory. This statement is based on the assumption that the perturbed Hamiltonian can be written as a sum of the unperturbed Hamiltonian and a perturbation term, and that the eigenstates and eigenvalues of the unperturbed Hamiltonian are known. The justification for this can be found in the mathematical proofs of perturbation theory, which involve series expansions and the use of the Dyson series.

Regarding your observation about the similarity between the technique for moving eigenstates in perturbation theory and the continuity argument in Rouche's theorem, there is indeed a connection between the two. Both involve finding solutions to equations by perturbing the system in a controlled manner. However, the justification for perturbation theory goes beyond this and involves a more rigorous mathematical analysis.

I hope this helps to clarify your doubts and provides some insight into the deeper mathematical ideas behind perturbation theory. Keep up the good work in your studies, and don't hesitate to reach out if you have any further questions.