Position operator

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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.

[edit] 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 χB denote the indicator function of B. We see that the projection-valued measure ΩQ is given by

\Omega_Q(B) \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(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).

[edit] 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

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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