- #1

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- To describe conjugate variable pairs in order to derive uncertainty principles (not counting the Energy-time one), there appear to be two descriptions: one by their relationship by a Fourier transform (modulo a constant), and another by the lack of commutativity of the corresponding operators. What is the link between these two descriptions?

If I understand correctly (a big caveat), one shows that if one can get from one function to the other via a Fourier transform and multiplication by a constant, then the width of the corresponding Gaussian wave of one gets larger as that of the other gets smaller, and vice-versa, and by a bit of manipulation you have a pair of variables to which an uncertainty principle applies (and which are non-commuting).

Alternatively, if the variables correspond to two non-commuting Hermitian operators, then by direct algebraic manipulation (e.g., the Cauchy-Schwarz inequality) and some clever substitutions, the uncertainty principle results in a straightforward manner.

I am sure that the connection between the two methods is straightforward, but it eludes me. I will be grateful for any indications to set me on the right path.

Alternatively, if the variables correspond to two non-commuting Hermitian operators, then by direct algebraic manipulation (e.g., the Cauchy-Schwarz inequality) and some clever substitutions, the uncertainty principle results in a straightforward manner.

I am sure that the connection between the two methods is straightforward, but it eludes me. I will be grateful for any indications to set me on the right path.