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Suppose there is a charged particle moving in an Electromagnetic field for which the vector potential is A.

Now we change A by gradient of some scalar(which is time independent) Hence, the scalar potential remains unchanged.

Now, I want to investigate the effects of such a gauge transformation in Quantum Mechanics.

  1. The expectation value of $\hat{x}$ (position operator) and Mechanical Momenta would remain unchanged under gauge transformation, right? ( 1st question)

But the expectation value of Canonical Momenta is not gauge invariant despite the physical situation being the same.

  1. Does that imply that we cannot associate an observable with the Canonical Momenta i.e we cannot directly measure it by equipments. Is it not an observable? (2nd question)
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You are correct on both counts. Mechanical momentum is a gauge invariant physical quantity, but canonical momentum isn't. That means that canonical momentum is as measurable a quantity as the vector potential - which is to say, not at all unless you further specify what gauge you want it in.

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