Levene's test

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.

Introduction

In one dimension, the wave function $\psi$ represents the probability density of finding the particle at position $x$. Hence the expected value of a measurement of the position of the particle is

$⟨x⟩=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}x|\psi {|}^{2}dx=\underset{-\mathrm{\infty }}{\overset{+\mathrm{\infty }}{\int }}{\psi }^{*}x\psi dx$

Accordingly, the quantum mechanical operator corresponding to position is $\hat{x}$, where

$(\hat{x}\psi )(x)=x\psi (x)$

Eigenstates

The eigenfunctions 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,

$\hat{x}\psi (x)=x\psi (x)=x_0\psi (x)$

recalling that $\hat{x}$ simply multiplies the function by $x$ in position representation. Since $x$ is a variable while $x_0$ is a constant, $\psi$ must be zero everywhere except at $x=x_0$. The normalized solution to this is

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

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 $x_0$). Hence, by the uncertainty principle, nothing is known about the momentum of such a state.

Three dimensions

The generalisation to three dimensions is straightforward. The wavefunction is now $\psi (\mathbf{r},t)$ and the expectation value of the position is

$⟨\mathbf{r}⟩=\int \mathbf{r}|\psi {|}^2{d}^3\mathbf{r}$

where the integral is taken over all space. The position operator is

$\hat{\mathbf{r}}\psi =\mathbf{r}\psi$

Momentum space

In momentum space, the position operator in one dimension is

$\hat{x}=i\hslash \frac{d}{dp}$

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:

$Q(\psi )(x)=x\psi (x)$

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.

Measurement

As with any quantum mechanical observable, in order to discuss measurement, we need to calculate the spectral resolution of Q:

$Q=\int \lambda d{\mathrm{\Omega }}_Q(\lambda ).$

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

${\mathrm{\Omega }}_Q(B)\psi ={\chi }_B\psi ,$

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

$|{\mathrm{\Omega }}_Q(B)\psi {|}^2=|{\chi }_B\psi {|}^2=\underset{B}{\int }|\psi {|}^{2}d\mu ,$

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

$\frac{{\mathrm{\Omega }}_Q(B)\psi }{\Vert {\mathrm{\Omega }}_Q(B)\psi \Vert }$

or

$\frac{(1-{\chi }_B)\psi }{\Vert (1-{\chi }_B)\psi \Vert }$, where $\Vert \cdots \Vert$ is the Hilbert space norm on L²(R).

See also

• Position and momentum space
• Momentum operator

References

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