I am a undergrad student interested in math taking quantum mechanics.
Yesterday I was introduced to what physicists call perturbation theory, non-degenerate case. According to authors Griffiths, Shankar and Sakuraii, they all state the procedure approximately in the same way. In the following, I will quote Griffiths over the others, for no particular reason. He states that this is the procedure:
The hamiltonian $H = H^0 + \lambda H'$, where $H'$ is the perturbation and the superscript $0$ always identifies the unperturbed quantity. For the moment we'll take $\lambda$ to be a small number, later we will crank it up to $1$ to get the true Hamiltonian. Now, we write $\psi_n$ and $E_n$ as power series in $\lambda$: $$ \psi_n = \psi_n^0 + \lambda \psi_n^1 + \lambda^2\psi_n^2 + \cdots $$ $$ E_n = E_n^0 + \lambda E_n^1 + \lambda^2E_n^2 + \cdots $$
The term $E_n^k$ is called the $k$-th order correction to the $n$-th energy eigenvalue and the term $\psi_n^k$ is called the $k$-th order correction to the $n$-th energy eigenfunction. I dont get any further, because at this point, and don't understand what is going on. So here are some questions to begin with:
Regarding the two power series in $\lambda$, how do we know that the solution to $H$ has this form? It may well have any other form. Why this?
How do we know that the assumptions on which the method is derived still hold when we increase $\lambda$ to $1$? In particular, how do we know that the two power series still converge for $\lambda = 1$? Why did we even have to to let $\lambda$ be small, when we might as well have let it be $1$ anyway?
For some operators $H^0, H'$ it can be shown that the eigenvalues and eigenvectors are analytic functions of $\lambda$ so we can express them in a Taylor series. In other cases, the series may not converge, but for small $ \lambda $ the first few terms may still be a good approximation for physics purposes. This is a whole branch of mathematical physics. The classic text is Kato's Perturbation Theory for Linear Operators (This link downloads the entire book from University of Edinburgh).