we denote the unitary eigenvector of the position operator corresponding to the eigenvalue
is the Dirac delta (function) distribution centered at the position
It is fundamental to observe that there exists only one linear continuous endomorphism
It's possible to prove that the unique above endomorphism is necessarily defined by
Consider representing the quantum state of a particle at a certain instant of time by a square integrable wave function
For now, assume one space dimension (i.e. the particle "confined to" a straight line).
represents the probability density of finding the particle at some position
then the probability to find the particle in the position range
which is simply the canonical embedding of the position-line into the complex plane.
In the above definition, which regards the case of a particle confined upon a line, the careful reader may remark that there does not exist any clear specification of the domain and the co-domain for the position operator.
In literature, more or less explicitly, we find essentially three main directions to address this issue.
The last case is, in practice, the most widely adopted choice in Quantum Mechanics literature, although never explicitly underlined.
thereby providing a mathematically rigorous notion of eigenvectors and eigenvalues.
[3] The eigenfunctions of the position operator (on the space of tempered distributions), represented in position space, are Dirac delta functions.
To show that possible eigenvectors of the position operator should necessarily be Dirac delta distributions, suppose that
We write the eigenvalue equation in position coordinates,
Clearly, no continuous function satisfies such properties, and we cannot simply define the wave-function to be a complex number at that point because its
This suggest the need of a "functional object" concentrated at the point
and with integral different from 0: any multiple of the Dirac delta centered at
Indeed, recalling that the product of any function by the Dirac distribution centered at a point is the value of the function at that point times the Dirac distribution itself, we obtain immediately
Usually, in quantum mechanics, by representation in the momentum space we intend the representation of states and observables with respect to the canonical unitary momentum basis
where: Consider the case of a spinless particle moving in one spatial dimension.
; the Hilbert space of complex-valued, square-integrable functions on the real line.
Immediately from the definition we can deduce that the spectrum consists of the entire real line and that
has a strictly continuous spectrum, i.e., no discrete set of eigenvalues.
As with any quantum mechanical observable, in order to discuss position measurement, we need to calculate the spectral resolution of the position operator
is the so-called spectral measure of the position operator.
denote the indicator function for a Borel subset
, then the probability of the measured position of the particle belonging to a Borel set
After any measurement aiming to detect the particle within the subset B, the wave function collapses to either