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I have a mathematics and computer science background with very little physics. I have read that the resonating frequency of an atom of some element is always exactly the same as the resonating frequency of any other atom of that element. This doesn't sit too well with me, I would have thought the resonating frequency could be viewed as a continuous random variable with an arbitrarily small standard deviation, but to say the resonating frequency is exactly an extremely strong statement to make. Is this actually the case?

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All atoms of the same kind are indistinguishable, see en.wikipedia.org/wiki/Identical_particles for details. –  Guo Qianyi Apr 30 at 15:49
    
@Guo Qianyi that is not always true. The atom of hydrogen in my eye and your mouth are distinguishable - and they are very far away from each other. The above article is about kets symmetric and antisymmetric with respect to exchange of labels. True, these kets do not provide way to distinguish which particle is which. But this is by design, to describe situations where particles are not distinguishable, like electrons in an atom. For situations where they are distinguishable, this description is obviously not exhausting. –  Ján Lalinský Apr 30 at 17:25

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I don't think it is true that resonating freqeuency is some exact frequency for every atom of an element. I'm not sure what type of resonating you have in mind. Nuclear Magnetic Resonance (NMR) spectroscopy is the first thing that comes to mind for me. The proton (or other nucleus) resonates at a different frequency depending upon what molecule, if any, it is in, magnetic field strength, temperature, and other factors. The line shape in any type of spectroscopy is not a Dirac delta function. There is always a range of frequencies due at least to the uncertainty princple.

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