In quantum mechanics, the position operator corresponds to the position observable of a particle. Consider, for example, the case of a spinless particle moving on a line. The state space for such a particle is L²(R), the Hilbert space of complex-valued and square-integrable (with respect to the Lebesgue measure) functions on the real line. The position operator, Q, is then defined by

$Q\; (psi(x))\; =\; x\; cdot\; psi\; (x)$

with domain

$D(Q)\; =\; \{\; psi\; in\; L^2(\{mathbf\; R\})\; ,|,\; Q\; psi\; in\; L^2(\{mathbf\; R\})\; \}.$

Since all continuous functions with compact support lie in D(Q), Q is densely defined. Q, being simply multiplication by x, is a self adjoint operator, thus satisfying the requirement of a quantum mechanical observable. Immediately from the definition we can deduce that the spectrum consists of the entire real line and that Q has purely continuous spectrum, therefore no eigenvalues. The three dimensional case is defined analogously. We shall keep the one-dimensional assumption in the following discussion.

Measurement

As with any quantum mechanical observable, in order to discuss measurement, we need to calculate the spectral resolution of Q:

$Q\; =\; int\; lambda\; d\; Omega\_Q(lambda).$

Since Q is just multiplication by x, its spectral resolution is simple. For a Borel subset B of the real line, let $chi\; \_B$ denote the indicator function of B. We see that the projection-valued measure Ω_Q is given by

$Omega\_Q(B)\; psi\; =\; chi\; \_B\; cdot\; psi\; ,$

i.e. Ω_Q is multiplication by the indicator function of B. Therefore, if the system is prepared in state ψ, then the probability of the measured position of the particle being in a Borel set B is

$|Omega\_Q(B)\; psi\; |^2\; =\; |\; Chi\; \_B\; cdot\; psi\; |^2\; =\; int\; \_B\; |psi|^2\; d\; mu\; ,$

where μ is the Lebesgue measure. After the measurement, the wave function collapses to $frac\{Omega\_Q(B)\; psi\}\{\; |Omega\_Q(B)\; psi\; |\}$, where $|\; cdot\; |$ is the Hilbert space norm on L²(R).

Unitary equivalence with momentum operator

For a particle on a line, the momentum operator P is defined by

$P\; psi\; =\; -i\; hbar\; frac\{partial\}\{partial\; x\}\; psi$

usually written in bra-ket notation as:

$langle\; x\; |\; hat\{p\}\; |\; psi\; rangle\; =\; -\; i\; hbar\; \{partial\; over\; partial\; x\}\; psi\; (x\; )$

with appropriate domain. P and Q are unitarily equivalent, with the unitary operator being given explicitly by the Fourier transform. Thus they have the same spectrum. In physical language, P acting on momentum space wave functions is the same as Q acting on position space wave functions (under the image of Fourier transform).