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I am new to QM and was wondering how it is deduced that the position operator has a continuous spectrum of eigenvalues and eigenvectors. In particular, is the equation $\hat{x}|x\rangle=x|x\rangle$ taken to be a principle / postulate of QM or does this equation pop out as a consequence of something?

Also, does writing the eigenvalue equation for the position operator as above mean that the eigenvectors of the position operator are non-degenerate, say in 1D?

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