TITLE: Riemann Zeta Function connection to Quantum Mechanics.
QUESTION [31 upvotes]: I feel like this question is probably wrong for MO, (too low level, perhaps unclear) but my curiosity has got the better of me:
I hear that the Riemann Zeta Function and its zeros have applications to quantum mechanics, as well as other fields.  I do not understand these connections, and because of this the following question came up:
In theory, is it possible through physical experiments (particle experiments) to approximately calculate the first few zeros of the Riemann zeta function?  
In other words, (using the explicit formula) could we write down the $n^{th}$ prime number (up to a given margin of error/probability of correctness) only from doing quantum mechanical experiments?
(If there are conjectures/facts that we cannot prove, but would answer the question, I would be happy to hear those too)
Thanks!

REPLY [4 votes]: This may be buried in one of the references above, but for those don't wish to go through them all...
The zeta function can arise as the trace of Hamiltonians governing physical systems. For example in an experiment to measure the Casimir effect (two perfectly conducting plates placed very close to each other) the force they exert on each other has a formula that involved the the derivative of the Riemann zeta function evaluated at $-\frac{1}{2}$. This has been experimentally validated to a reasonable amount of precision. 
This may not get you zeros, but it gets you certain values.