Position operator

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 "L"²(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 "L"²(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:

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

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).