Position operator

Basic properties

In the above definition, which regards the case of a particle confined upon a line, the careful reader may remark that there does not exist any clear specification of the domain and the co-domain for the position operator. In literature, more or less explicitly, we find essentially three main directions to address this issue.

1. The position operator is defined on the subspace ${\displaystyle D_X}$ of ${\displaystyle L^2}$ formed by those equivalence classes ${\displaystyle \psi }$ whose product by the embedding ${\displaystyle \mathrm{x}}$ lives in the space ${\displaystyle L^2}$. In this case the position operator $${\displaystyle X:D_X\subset L^2\to L^2:\psi \mapsto \mathrm{x}\psi }$$ reveals not continuous (unbounded with respect to the topology induced by the canonical scalar product of ${\displaystyle L^2}$), with no eigenvectors, no eigenvalues and consequently with empty point spectrum.
2. The position operator is defined on the Schwartz space ${\displaystyle \mathcal{S}}$ (i.e. the nuclear space of all smooth complex functions defined upon the real-line whose derivatives are rapidly decreasing). In this case the position operator $${\displaystyle X:\mathcal{S}\subset L^2\to \mathcal{S}\subset L^2:\psi \mapsto \mathrm{x}\psi }$$ reveals continuous (with respect to the canonical topology of ${\displaystyle \mathcal{S}}$), injective, with no eigenvectors, no eigenvalues and consequently with empty point spectrum. It is (fully) self-adjoint with respect to the scalar product of ${\displaystyle L^2}$ in the sense that $${\displaystyle ⟨X(\psi )|\varphi ⟩=⟨\psi |X(\varphi )⟩,\mathrm{\forall }\psi ,\varphi \in \mathcal{S}.}$$
3. The position operator is defined on the dual space ${\displaystyle {\mathcal{S}}^{\times }}$ of ${\displaystyle \mathcal{S}}$ (i.e. the nuclear space of tempered distributions). As ${\displaystyle L^2}$ is a subspace of ${\displaystyle {\mathcal{S}}^{\times }}$, the product of a tempered distribution by the embedding ${\displaystyle \mathrm{x}}$ always lives ${\displaystyle {\mathcal{S}}^{\times }}$. In this case the position operator $${\displaystyle X:{\mathcal{S}}^{\times }\to {\mathcal{S}}^{\times }:\psi \mapsto \mathrm{x}\psi }$$ reveals continuous (with respect to the canonical topology of ${\displaystyle {\mathcal{S}}^{\times }}$), surjective, endowed with complete families of generalized eigenvectors and real generalized eigenvalues. It is self-adjoint with respect to the scalar product of ${\displaystyle L^2}$ in the sense that its transpose operator $${\displaystyle {}^{t}X:\mathcal{S}\to \mathcal{S}:\varphi \mapsto \mathrm{x}\varphi ,}$$ is self-adjoint, that is $${\displaystyle ⟨{}^{t}X(\varphi )|\psi ⟩=⟨\varphi |{}^{t}X(\psi )⟩,\mathrm{\forall }\psi ,\varphi \in \mathcal{S}.}$$

The last case is, in practice, the most widely adopted choice in Quantum Mechanics literature, although never explicitly underlined. It addresses the possible absence of eigenvectors by extending the Hilbert space to a rigged Hilbert space: $${\displaystyle \mathcal{S}\subset L^2\subset {\mathcal{S}}^{\times },}$$ thereby providing a mathematically rigorous notion of eigenvectors and eigenvalues.

Eigenstates

The eigenfunctions of the position operator (on the space of tempered distributions), represented in position space, are Dirac delta functions.

Informal proof. To show that possible eigenvectors of the position operator should necessarily be Dirac delta distributions, suppose that ${\displaystyle \psi }$ is an eigenstate of the position operator with eigenvalue ${\displaystyle x_0}$. We write the eigenvalue equation in position coordinates, $${\displaystyle \hat{\mathrm{x}}\psi (x)=\mathrm{x}\psi (x)=x_0\psi (x)}$$ recalling that ${\displaystyle \hat{\mathrm{x}}}$ simply multiplies the wave-functions by the function ${\displaystyle \mathrm{x}}$, in the position representation. Since the function ${\displaystyle \mathrm{x}}$ is variable while ${\displaystyle x_0}$ is a constant, ${\displaystyle \psi }$ must be zero everywhere except at the point ${\displaystyle x_0}$. Clearly, no continuous function satisfies such properties, and we cannot simply define the wave-function to be a complex number at that point because its ${\displaystyle L^2}$-norm would be 0 and not 1. This suggest the need of a "functional object" concentrated at the point ${\displaystyle x_0}$ and with integral different from 0: any multiple of the Dirac delta centered at ${\displaystyle x_0}$. The normalized solution to the equation $${\displaystyle \mathrm{x}\psi =x_0\psi }$$ is $${\displaystyle \psi (x)=\delta (x-x_0),}$$ or better $${\displaystyle \psi ={\delta }_{x_0},}$$ such that $${\displaystyle \mathrm{x}{\delta }_{x_0}=x_0{\delta }_{x_0}.}$$ Indeed, recalling that the product of any function by the Dirac distribution centered at a point is the value of the function at that point times the Dirac distribution itself, we obtain immediately $${\displaystyle \mathrm{x}{\delta }_{x_0}=\mathrm{x}(x_0){\delta }_{x_0}=x_0{\delta }_{x_0}.}$$ Although such Dirac states are physically unrealizable and, strictly speaking, are not functions, the Dirac distribution centered at ${\displaystyle x_0}$ can be thought of as an "ideal state" whose position is known exactly (any measurement of the position always returns the eigenvalue ${\displaystyle x_0}$). Hence, by the uncertainty principle, nothing is known about the momentum of such a state.

Momentum space

Usually, in quantum mechanics, by representation in the momentum space we intend the representation of states and observables with respect to the canonical unitary momentum basis $${\displaystyle \eta ={\left(\left[(2\pi \hslash {)}^{-\frac{1}{2}}{e}^{(\iota /\hslash )(\mathrm{x}|p)}\right]\right)}_{p\in \mathbb{R}}.}$$

In momentum space, the position operator in one dimension is represented by the following differential operator $${\displaystyle {\left(\hat{\mathrm{x}}\right)}_P=i\hslash \frac{d}{d\mathrm{p}}=i\frac{d}{d\mathrm{k}},}$$

where:

• the representation of the position operator in the momentum basis is naturally defined by ${\displaystyle {\left(\hat{\mathrm{x}}\right)}_P(\psi {)}_P={\left(\hat{\mathrm{x}}\psi \right)}_P}$, for every wave function (tempered distribution) ${\displaystyle \psi }$;
• ${\displaystyle \mathrm{p}}$ represents the coordinate function on the momentum line and the wave-vector function ${\displaystyle \mathrm{k}}$ is defined by ${\displaystyle \mathrm{k}=\mathrm{p}/\hslash }$.

Formalism in L2(R, C)

Consider the case of a spinless particle moving in one spatial dimension. The state space for such a particle contains ${\displaystyle L^2(\mathbb{R},\mathbb{C})}$; the Hilbert space of complex-valued, square-integrable functions on the real line.

The position operator is defined as the self-adjoint operator $${\displaystyle Q:D_Q\to L^2(\mathbb{R},\mathbb{C}):\psi \mapsto \mathrm{q}\psi ,}$$ with domain of definition $${\displaystyle D_Q=\left\{\psi \in L^2(\mathbb{R})\mid \mathrm{q}\psi \in L^2(\mathbb{R})\right\},}$$ and coordinate function ${\displaystyle \mathrm{q}:\mathbb{R}\to \mathbb{C}}$ sending each point ${\displaystyle x\in \mathbb{R}}$ to itself, such that $${\displaystyle Q(\psi )(x)=x\psi (x)=\mathrm{q}(x)\psi (x),}$$ for each pointwisely defined ${\displaystyle \psi \in D_Q}$ and ${\displaystyle x\in \mathbb{R}}$.

Immediately from the definition we can deduce that the spectrum consists of the entire real line and that ${\displaystyle Q}$ has a strictly continuous spectrum, i.e., no discrete set of eigenvalues.

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

Measurement theory in L2(R, C)

As with any quantum mechanical observable, in order to discuss position measurement, we need to calculate the spectral resolution of the position operator $${\displaystyle X:D_X\to L^2(\mathbb{R},\mathbb{C}):\psi \mapsto \mathrm{x}\psi }$$ which is $${\displaystyle X={\int }_{\mathbb{R}}\lambda d{\mu }_X(\lambda )={\int }_{\mathbb{R}}\mathrm{x}{\mu }_X={\mu }_X(\mathrm{x}),}$$ where ${\displaystyle {\mu }_X}$ is the so-called spectral measure of the position operator.

Let ${\displaystyle {\chi }_B}$ denote the indicator function for a Borel subset ${\displaystyle B}$ of ${\displaystyle \mathbb{R}}$. Then the spectral measure is given by $${\displaystyle \psi \mapsto {\mu }_X(B)(\psi )={\chi }_B\psi ,}$$ i.e., as multiplication by the indicator function of ${\displaystyle B}$.

Therefore, if the system is prepared in a state ${\displaystyle \psi }$, then the probability of the measured position of the particle belonging to a Borel set ${\displaystyle B}$ is $${\displaystyle \Vert {\mu }_X(B)(\psi ){\Vert }^2=\Vert {\chi }_B\psi {\Vert }^2={\int }_B|\psi {|}^2\mu ={\pi }_X(\psi )(B),}$$ where ${\displaystyle \mu }$ is the Lebesgue measure on the real line.

After any measurement aiming to detect the particle within the subset B, the wave function collapses to either $${\displaystyle \frac{{\mu }_X(B)\psi }{\Vert {\mu }_X(B)\psi \Vert }=\frac{{\chi }_B\psi }{\Vert {\chi }_B\psi \Vert }}$$ or $${\displaystyle \frac{(1-{\chi }_B)\psi }{\Vert (1-{\chi }_B)\psi \Vert },}$$ where ${\displaystyle \Vert \cdot \Vert }$ is the Hilbert space norm on ${\displaystyle L^2(\mathbb{R},\mathbb{C})}$.