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**What's Really so "Quantum" About Heisenberg's Uncertainty?**

I've never really understood what was so interesting and strange about Heisenberg's Uncertainty (or Robertson's Inequalities). If we take as axiom that particles exist as wave functions that satisfy schrodinger's equation then what is Heisenberg's Uncertainty other than a quip about the nature of a Fourier Transform? To make a Fourier Series of a wave that is very localized in position then one must have a broad range of k values with large coefficients in their series. Why is this so mind blowing?