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# 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 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

$Q \left(psi\left(x\right)\right) = x cdot psi \left(x\right)$

with domain

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

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:

$Q = int lambda d Omega_Q\left(lambda\right).$

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 ΩQ is given by

$Omega_Q\left(B\right) psi = chi _B cdot psi ,$

i.e. ΩQ 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

$|Omega_Q\left(B\right) psi |^2 = | Chi _B cdot psi |^2 = int _B |psi|^2 d mu ,$

where μ is the Lebesgue measure. After the measurement, the wave function collapses to $frac\left\{Omega_Q\left(B\right) psi\right\}\left\{ |Omega_Q\left(B\right) psi |\right\}$, 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

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

usually written in bra-ket notation as:

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

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).
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