Queries of proof of adiabatic theorem in QM

Question

I have a few questions regarding the proof of the adiabatic theorem in the book "Introduction to Quantum Mechanics" by Griffiths:

The assumptions are that if the Hamiltonian changes with time then the eigenfunctions and eigenvalues themselves are time-dependent: $$H(t)\psi_{n}(t) = E_{n} \psi_{n}(t).$$

1. Firstly, how would we know that the spectrum remains discrete so as to write it in this way?

He then states that the eigenfunctions form an orthonormal set which is complete, this I understand comes from the postulates of QM. But then he states that the general solution to the time-dependent Schrodinger equation can be expresses as a linear combination of them: $$\Psi(t) = \sum_{n} c_{n}(t) \psi_{n}(t)e^{i \theta_{n}(t)} ~~~\text{where }~~\theta_{n}(t) := - \frac{1}{\hbar}\int_{0}^{t}E_{n}(t')dt'$$

2. How does he know that this is the form of the phase factor (he uses this later in the proof)?

Answer 1

1. One doesn't need to assume that the spectrum is discrete. As the page "Adiabatic theorem" that you linked to rightfully says, what is needed is that there is an energy gap around the energy value $E_n$ that we want to keep track of, i.e. that there exists a positive $\epsilon\gt 0$ such that $E_n$ is the only eigenvalue (in the spectrum) among the numbers from the interval $(E_n-\epsilon,E_n+\epsilon)$. The remaining states away from the gap may belong to a discrete spectrum or a mixed spectrum (it cannot be a purely continuous spectrum because the $E_n$ makes it at least partly discrete). If the spectrum is mixed, the notation is consistent with that because $n$ may be assumed to be a continuous number/label, or a multi-index that stores more information that just one number.

2. Griffiths may make the statement about the phase because it is a correct statement. If I remember well, Griffiths even actually proves that this is the correct phase. The OP's opinion that Griffiths should calculate the phase before he proves it is a logically flawed opinion about derivations in mathematics and physics in general. That's simply not how it works. Mathematicians and physicists often have to "guess" some correct result by some ingenious or heuristic tricks or by trying many viable candidate answers. If they are able to prove that it is the correct (and in many cases, the only correct) answer, then they have everything that is actually needed.

The proof is easy. In Schrödinger's equation, the time derivative of $\Psi$ picks the usual terms from the time-independent Hamiltonian case but also a term where the phase $\theta$ is differentiated, and that differentiation turns the indefinite integral of $E_n(t')$ to $E_n$ itself, and that's what is needed for the Schrödinger equation to hold.

On top of the proof, one could want some insight into the thinking of the first people who guessed the right answer. Well, different people could have thought differently but there would always be some heuristic quasi-rational component to that. It may be insightful to explain "how I discovered something" but it is in no way a "necessary" part of a discussion in a textbook.

Answer 2

This is just a partial answer. He writes the time dependent coefficient as $c_n(t)e^{i\theta_n(t)}$ instead of just $c_n(t)$ because he already knows the result and knows writing it this way will make the proceeding calculations easier (albeit marginally). You can try the proof starting with $\Psi(t)=\sum_nc_n(t)\psi(t)$ and you should still get the same result as Griffiths.