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The basic commutator relation $$[J_1,J_2]=i \hbar J_3$$ of quantum mechanics yields the uncertainty relation $$\Delta(J_1)\Delta(J_2)\ge \frac{\hbar}{2}|\langle J_3\rangle|.$$ However, unlike the position-momentum or energy-time uncertainty relations, I'm not familiar with any intuitive explanation of this uncertainty.

I would like to see a heuristic derivation for this uncertainty relation which can be understood by a mathematically untrained layman (i. e. something you could tell your non-physicist friends or print in a book on popular physics). If possible, the explanation should be based on a real-life experiment in which this uncertainty has immediately visible consequences.

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What intuitive explanation do you have in mind for position/momentum and time/energy? –  WillO Feb 3 at 15:08

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