Are coordinate operators Hermitian?

[Heisenberg]

I can't figure this one out given that the coordinate operator is continuous, it's hard to imagine "matrix elements". But presumably since the coordinates of the system (1d free particle) are always real valued, would this make the coordinate operator Hermitian?

 

[dextercioby]

Coordinate operators make most sense and are to be directly interpreted in the Schrödinger picture, where one builds a representation of the canonical commutation relation [x,p] =ihbar on a Hilbert space. One can show that on the maximal domain of definition, these operators are self-adjoint.

 

[bhobba]

Sure - its the position operator.

But its not a matrix - it requires Rigged Hilbert Spaces and the Generalized Spectral Theorem to fully flesh out what's going on.

Be warned however the rigorous treatment of such is HARD.

Work your way up to it from Ballentine then a mathematically more orientated treatment like Hall - Quantum Theory For Mathematicians.

Thanks
Bill

 

[Heisenberg]

Thanks for the feedback. I appreciate it.