It's action on the position operator leaves it invariant:
2.
However, unlike the Hamiltonian, the position operator lacks proper eigenfunctions.
3.
For the expectation value of the position operator, one then has the formula
4.
This is because the position operator is unbounded, and \ psi has to be chosen from its domain of definition.
5.
Each component of the position operator is a one-electron operator, they can be represented in second quantization as follows:
6.
A single eigenstate of the position operator when represented in the momentum picture again requires a linear combination of the momemtum operator eigenstates.
7.
The LT is defined as a second order moment cumulant of the position operator divided by the number of electrons in the system.
8.
Informally stated, with certain technical assumptions, every representation of the Heisenberg group is equivalent to the position operators and momentum operators on.
9.
An example of a quantum mechanical observable which has purely continuous spectrum is the position operator of a free particle moving on a line.
10.
Also, since the momentum operator is unitarily equivalent to the position operator, via the Fourier transform, they have the same spectrum.
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