How does the commutator of position and momentum operators produce an imaginary component?

[Bob3141592]

A reply in a different thread got me thinking. The Heisenberg Uncertainty principly is really a mathematical expression about the noncomutative behavior of operators, that is (using the standard position and momentum) p q - q p >= i h / 2 pi.

But aren't both position and momentum strictly defined in the reals? How does a difference in operations between two reals produce an imaginary component?

 

[selfAdjoint]

No, in quantum mechanics p and q are not real numbers. Thery are operators that act on the Hilbert space of quantum states. Check out the elementary derivation of the Schroedinger equation.

 

[Bob3141592]

No, in quantum mechanics p and q are not real numbers. Thery are operators that act on the Hilbert space of quantum states. Check out the elementary derivation of the Schroedinger equation.

Thanks. I only took one course in quantum physics, and I'd always had problems with the way imaginary numbers were used in it. Maybe it was a hang up because of the name. But after cranking through the equations and applying the results to an actual measurement, it always seemed there was an imaginary component left over that was just thrown away, and I was uncomfortable with that. Maybe after these three decades if I get the chance to study it again I'd do better.

 

[Janitor]

Operators of observables such as position and momentum need to be hermitian to ensure that the eigenvalues are real numbers, since the eigenvalues are the numbers that are supposed to correspond to the various possible results that an experiment can yield for anyone particular measurement.