In quantum theory, also operators with non-discrete spectrum are in use, such as the position operator Q in quantum mechanics.
12.
Using the spectral representation of the position operator x = \ int dx'| x'\ rangle \ langle x'| we can write
13.
That means that the less the uncertainity that we measure the particles position, the closer it is to being in a single eigenstate of the position operator.
14.
In this context, the position operator has a complete set of eigen-distributions, labeled by the points " y " of the real line, given by
15.
Finally, an analogous statement is true for the position operator : When the system is rotated, the three different components of the position operator are effectively interchanged or mixed.
16.
Finally, an analogous statement is true for the position operator : When the system is rotated, the three different components of the position operator are effectively interchanged or mixed.
17.
The reason for this is that the eigenstates ( or state of the system corresponding to an observed value ) of the position operator are not eigenstates of the momentum operator.
18.
Taking the position operator on the wave function yields a position with an imaginary part, and taking the magnitude of the time derivative of a complex position should be able to yield a real solution.
19.
Classically, the Fourier coefficients give the intensity of the emitted radiation, so in quantum mechanics the magnitude of the matrix elements of the position operator were the intensity of radiation in the bright-line spectrum.
20.
Using the fact that the position operator \ mathbf { \ hat r } is Hermitian and | \ mathbf r \ rangle is the eigenvector of \ mathbf { \ hat r } with eigenvalue \ mathbf r,
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