Measurement, Commutators, and Uncertainty

If we measure the exact position of the electron in a hydrogen atom, we force it into a state having a Dirac delta function as its wavefunction. Since this function is also an eigenfunction for the dipole moment operator, it follows that we also know the (instantaneous) dipole moment for the atom at that instant. In effect, measuring position measures dipole moment too. But the delta function is not an eigenfunction for the hamiltonian operator of the atom, and so we have not simultaneously measured the electronic energy of the atom 

We have earlier seen that an eigenfunction for one operator can serve also as eigenfunction for another operator when the operators commute. In the above example, the 

The reader may suspect that there is some connection between commutators and the uncertainty principle, and this is indeed the case. It can be shown that the product of widths of simultaneous measurements (i.e., the uncertainty in their values) of two variables satisfies the relation 

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