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I've just worked through a derivation of the H.U.P. that uses the Cauchy Schwarz inequality to come up with the expression [tex] (\Delta A)^2(\Delta B)^2 \geq \frac{1}{4}|<[A,B]>|^2 [/tex]. This much I am happy with, but then it seems that when dealing with two "canonically conjugate observables" you set [tex] [A,B] = i\hbar [/tex] to find the uncertainty principle [tex] (\Delta A)(\Delta B) \geq \frac{\hbar}{2} [/tex].

It clearly gives the result I was expecting, but I cannot seem to find out where this [tex][A,B] = AB - BA = i\hbar [/tex] comes from.

Is this something that can be figured out? Or, is it just something that some quantum mechanic somewhere has found out by working out the commutators of loads of operators and discovered that the commutators of conjugate observables just happen to be equal to [tex] i\hbar [/tex]?

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# Heisenberg uncertainty principle derivation and canonically conjugate vairables?

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