Position operator

In quantum mechanics, the position operator is the operator that corresponds to the position observable of a particle.

Contents

Statement

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 \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 discrete eigenvalues. The three dimensional case is defined analogously. We shall keep the one-dimensional assumption in the following discussion.

Eigenstates

The eigensfunction of the position operator, represented in position basis, are dirac delta functions.

To show this, suppose \psi is an eigenstate of the position operator with eigenvalue x_0 . We write the eigenvalue equation in position coordinates,

Q\psi(x) = x \psi(x) = x_0 \psi(x)

recalling that Q simply multiplies the function by x in position representation. Clearly, \psi must be zero everywhere except at x = x_0 . Since we want a normalized solution,

\psi(x) = \delta(x - x_0)

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 ΩQ is given by

\Omega_Q(B) \psi = \chi _B \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 \psi |^2 = \int _B |\psi|^2 d \mu ,

where μ is the Lebesgue measure. After the measurement, the wave function collapses to either

\frac{\Omega_Q(B) \psi}{ \|\Omega_Q(B) \psi \|}

or

\frac{(1-\chi _B) \psi}{ \|(1-\chi _B) \psi \|} , where \| \cdots \| 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).

Quantum
\hat{E} = i\hbar\frac{\partial}{\partial t} \,\!