12.1 The position operator

12.1 The position operator

Concept links · terms present in this machine draft; source roles are unverified: adjoint operator · eigenvalue · spectral theorem

On a state space $\mathcal{H}$ of functions (or distributions) of a position variable $q,$ one can define:

Definition (Position operator)

The position operator $Q$ is given by

$$Q\psi (q)=q\psi (q)$$

Note that this operator has similar problems of definition to those of the momentum operator $P;$ it can take a function in $L^2(\mathbf{R})$ to one that is no longer square-integrable. Like $P,$ it is well-defined on the Schwartz space $\mathcal{S}(\mathbf{R})$ , as well as on the distributions $S^{\prime }(\mathbf{R})$ . Also like $P,$ it has no eigenfunctions in $\mathcal{S}(\mathbf{R})$ or $L^2(\mathbf{R})$ , but it does have eigenfunctions in $S^{\prime }(\mathbf{R})$ . Since

$${\int }_{-\infty }^{+\infty }q\delta (q-q^{\prime })f(q)dq=q^{\prime }f(q^{\prime })={\int }_{-\infty }^{+\infty }{q}^{\prime }\delta (q-q^{\prime })f(q)dq$$

one has the equality of distributions

$$q\delta (q-q^{\prime })=q^{\prime }\delta (q-q^{\prime })$$

so $\delta (q-q^{\prime })$ is an eigenfunction of $Q$ with eigenvalue $q^{\prime }$ .

The operators $Q$ and P do not commute, since

$$[Q,P]f=-iq\frac{d}{dq}f+i\frac{d}{dq}(qf)=if$$

and we get (reintroducing ℏ for a moment) the fundamental operator commutation relation

$$[Q,P]=i\hslash 1$$

the Heisenberg commutation relation. This implies that $Q$ and the free particle Hamiltonian $\begin{array}{r}H=\frac{1}{2m}{P}^2\end{array}$ also do not commute, so the position, unlike the momentum, is not a conserved quantity.

For a finite dimensional state space, recall that the spectral theorem (4.1) for a self-adjoint operator implied that any state could be written as a linear combination of eigenvectors of the operator. In this infinite dimensional case, the formula

$$\psi (q)={\int }_{-\infty }^{+\infty }\delta (q-q^{\prime })\psi (q^{\prime })d{q}^{\prime }\text{\hspace{1em}(12.1)}$$

can be interpreted as an expansion of an arbitrary state in terms of a continuous linear combination of eigenvectors of $Q$ with eigenvalue $q^{\prime }$ , the $\delta$-functions $\delta (q-$ $q^{\prime })$ . The Fourier inversion formula (11.4)

$$\psi (q)=\frac{1}{\sqrt{2\pi }}{\int }_{-\infty }^{+\infty }{e}^{ikq}\tilde{\psi }(k)dk$$

similarly gives an expansion in terms of eigenvectors $\frac{1}{\sqrt{2\pi }}{e}^{ikq}$ of $P.$ , with eigenvalue $k.$ .

Chapter contents · Next section: 12.2 Momentum space representation · 中文版本

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