Are coordinate operators Hermitian?

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The coordinate operator in quantum mechanics is considered Hermitian because it operates on real-valued coordinates, particularly in the context of a one-dimensional free particle. In the Schrödinger picture, the coordinate operator is directly linked to the canonical commutation relation [x,p] = ihbar, and it can be shown to be self-adjoint on its maximal domain. However, the coordinate operator is not a simple matrix; it necessitates the use of Rigged Hilbert Spaces and the Generalized Spectral Theorem for a comprehensive understanding. This rigorous treatment is complex and requires a solid mathematical foundation. Resources like Ballentine and Hall's "Quantum Theory For Mathematicians" are recommended for those looking to deepen their understanding.
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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?
 
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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.
 
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
 
Thanks for the feedback. I appreciate it.
 

Thread 'Two equivalent statements of time reversal symmetric Hamiltonian'

Time reversal invariant Hamiltonians must satisfy ##[H,\Theta]=0## where ##\Theta## is time reversal operator. However, in some texts (for example see Many-body Quantum Theory in Condensed Matter Physics an introduction, HENRIK BRUUS and KARSTEN FLENSBERG, Corrected version: 14 January 2016, section 7.1.4) the time reversal invariant condition is introduced as ##H=H^*##. How these two conditions are identical?
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