In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle. The eigenvalue of the operator is the position vector of the particle.
Contents
Introduction
In one dimension, the square modulus of the wave function,
Accordingly, the quantum mechanical operator corresponding to position is
The circumflex over the x on the left side indicates an operator, so that this equation may be read The result of the operator x acting on any function ψ(x) equals x multiplied by ψ(x). Or more simply, the operator x multiplies any function ψ(x) by x.
Eigenstates
The eigenfunctions of the position operator, represented in position space, are Dirac delta functions. To show this, suppose that
recalling that
Although such a state is physically unrealizable and, strictly speaking, not a function, it can be thought of as an "ideal state" whose position is known exactly (any measurement of the position always returns the eigenvalue
Three dimensions
The generalisation to three dimensions is straightforward. The wavefunction is now
where the integral is taken over all space. The position operator is
Momentum space
In momentum space, the position operator in one dimension is
Formalism
Consider, for example, the case of a spinless particle moving in one spatial dimension (i.e. in 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:
with domain
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 discrete 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:
Since Q is just multiplication by x, its spectral resolution is simple. For a Borel subset B of the real line, let
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
where μ is the Lebesgue measure. After the measurement, the wave function collapses to either
or