system
Given the following post, respond as if you're commenting on Reddit:
human
Is the Hydrogen spectrum fully solved?

As an undergrad student, I used the Schrödinger equation to solve the Hydrogen atom, but the result would lead to missing transitions. The professor then said that the Dirac equation would give a better result, but it would still lack some corrections from the hyperfine structure and quantum field theory. 

So, my questions are:

1) Are all the known corrections enough? Or are there transitions that happen using a state that shouldn't exist according to the model? I.e., are all states from the Hydrogen atom in the model?

2) Do these corrections make a closed formula, or are they had to be done using numerical methods?
gpt
There is no analytic solution that takes into account everything, so we use numerical approximations. A particularly complicated aspect is the finite size of the nucleus. We use that dependency to measure the proton radius based on transition energies in hydrogen.

> Or are there transitions that happen, and we don't know why?

All transitions between different states can happen (if there is enough energy), some of them are just very unlikely.