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?
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.
Frequency of all 'things' is deturmined by purity. Hence there are variations in base frequencies according to the quantity and quality of whatever is added. Alloys, semiconductor doping, and close proximity of 'pairing' affects 'true' frequency. Just because man does not have the equipement or technical ability to 'read' extremely small variations, does not mean they do not exist.