A Quantum Mechanics question that no one can answer One of the professors in our department asked the following question on a Quantum Mechanics exam:

Write me a Quantum Mechanics question below. If I can answer it, you
  will get 6 points and if I cannot you will get 12!

Now, I wonder whether there is such a question or not. What I mean is: Are there any open questions in the formalism of Quantum Mechanics? Is QM a complete theory as it is?
 A: Interesting question, but there are many possible answers. In the area of molecular physics one might ask...
"Calculate the probability of an electron with 100eV ionizing a C$_{60}$ molecule"
this is a QM question that (s)he would not have been able to answer, but it is not really about the formalism of QM. The question could probably be described in QM, but it would be way too complicated to solve without a powerful computer and various approximations. 
A: I initially said that quantum mechanics is generally not considered a complete theory, but several people in the comments disagree, so I guess I'll have to make a more explicit case for this. In any case, it seems this matter is not as "settled" as I thought it was. If you just want my examples of questions, just skip to the bulleted points. 
Specifically, the following unanswered question is a central one of modern day theoretical physics:


*

*What is the best way to unify quantum mechanics and gravity?


Notice that I said gravity, not special relativity. Quantum field theories are effective in unifying quantum mechanics with special relativity, but they don't touch gravity. Indeed, when you apply quantum mechanics to regimes where both quantum effects and relativistic effects are expected to be important, e.g. the area around black holes, quantum mechanics gives us answers that are wrong. Perhaps we have different definitions of complete, but I'm not sure how you can say that a theory that makes wrong predictions is complete.
Some people were also arguing that this example lies outside the domain of quantum mechanics. I'd argue against this notion. Quantum mechanics, while only noticeable at very small scales, is expected to (and largely does) reduce to the classical results at larger scales. Quantum mechanics does not e.g. predict that people will randomly pass through walls (with any reasonable change), it only predicts that particles will. Similarly, Special relativity and General relativity correctly reduce to our observations of slow-moving objects and less-extreme gravity. So, I think it is imprecise to argue that certain regimes lie outside quantum mechanics in any sense that allows quantum mechanics to be straight-up wrong in those domains. I think that a complete theory would not be so clearly inconsistent with reality. Again, perhaps we are thinking of different notions of complete.
Another example of a question that has not been answered:


*

*What is the correct interpretation (e.g. Copenhagen, Many-Worlds, etc...) of quantum mechanics? This question might never be answered, and indeed might not be a meaningful question, but it is also possible that it is a meaningful question, and might be answered in the future by new data that takes these ideas farther from meta-physics. In any case, while we certainly don't have an answer to this question, I did not mean to imply that the status of this question implies the incompleteness of quantum mechanics.


These are very "high level" questions, and perhaps you were looking for something more concrete. Some examples of more concrete questions:


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*Find an analytic 3-body (i.e. wavefunctions of the two electrons and the nucleus) solution to the helium atom including the coulomb potential and and electron-electron repulsion.

*Develop an accurate theory of high-temperature superconductivity.


I might be mistaken, but I'm pretty sure none of the questions above have accepted answers. A couple examples of computationally difficult problems that do have solutions:


*

*Solve the hydrogen atom potential for an electron in parabolic-cylindrical coordinates.

*Find the entropy of a system of neutral atoms trapped in an optical lattice.


Hope this helps
A: Open up the chapter on perturbation theory in any quantum mechanics textbook and pick any Hamiltonian you see. Chances are an exact solution to that quantum system isn't known. Ask your Professor to find the exact solution. For example one question could be:
Find the exact energy spectrum and eigenfunctions of the following Hamiltonian:
$$H = \frac{1}{2}p^2 + \frac{1}{2}x^2 + \frac{\lambda}{4!}x^4.$$
A: 
Calculate the fine structure constant from first principles (i.e. without recourse to the empirical values of $\epsilon_0$, $e$, $\hbar$, and $c$)
Comment: The fine structure constant is undoubtedly the most fundamental pure (dimensionless) number in all of physics. It relates the basic constants of electromagnetism (the change of the electron), relativity (the speed of light), and quantum mechanics (Plank's constant). If you can solve [this], you will have the most certain Nobel Prize in history waiting for you. But I wouldn't recommend spending a lot of time on it right now; many smart people have tried, and all (so far) have failed.

Problem 6.11 b "Introduction to Quantum Mechanics" David J. Griffiths
A: Ask him why you treat the molecules in a gas as particles if according to QM their wave function will quickly spread out and will become   planar unlocalized waves (it has been asked in this forum a few times in different formats but never well answered)
