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I have been following a series of on-line lectures by Dr Physics A. He clearly describes what Hermitian operators for polarization and spin are and what they do. But when he gets to the position and momentum operators I am rather lost. They are no longer represented by square matrices. The position operator, Xhat in one dimension seems to be only defined by the fact that it delivers the eignenvalue, position. What is the mathematical representation for the operator itself, analogous to the Pauli matrices for spin? As for the momentum operator, he simply asserts it is -i d/dx and proves that it is Hermitian. Where does it come from.