Position Operator

The symbols $\hat x$ and $\hat{\mathbf r}$ denote position observables. In one dimension, $\hat x$ is a single self-adjoint operator. In three dimensions,

$$\hat{\mathbf r} = (\hat x,\hat y,\hat z)$$

is an ordered triple of commuting component operators. A hat distinguishes the operator from a coordinate value, although many advanced texts omit hats when context is unambiguous.

Default Meaning

In the position representation on the line, the operator acts by multiplication:

$$(\hat x\psi)(x) =x\psi(x).$$

In three Cartesian dimensions,

$$(\hat{\mathbf r}\psi)(\mathbf r) = \mathbf r\,\psi(\mathbf r),$$

or componentwise,

$$(\hat r_i\psi)(\mathbf r) = r_i\psi(\mathbf r).$$

The symbol on the left is an operator. The $x$ or $\mathbf r$ on the right is the numerical coordinate labeling the wavefunction argument.

Mathematical Type

On $L^2(\mathbb R)$, the position operator is an unbounded self-adjoint multiplication operator with natural domain

$$\mathcal D(\hat x) = \left\lbrace \psi\in L^2(\mathbb R) : \int_{-\infty}^{\infty} x^2\lvert\psi(x)\rvert^2\,dx <\infty \right\rbrace.$$

The condition says that $\hat x\psi$ must itself be square-integrable. A normalized state can belong to $L^2(\mathbb R)$ while lying outside $\mathcal D(\hat x)$, so its mean position need not exist.

On the full line, the spectrum of $\hat x$ is continuous and equals $\mathbb R$. The formal eigenvalue equation is

$$\hat x\lvert x\rangle = x\lvert x\rangle.$$

The generalized eigenkets $\lvert x\rangle$ are not normalizable Hilbert-space vectors. They are distributional objects satisfying

$$\langle x\rvert x'\rangle = \delta(x-x'), \qquad \int_{-\infty}^{\infty} \lvert x\rangle \langle x\rvert\,dx =I.$$

The same glyph $x$ labels the generalized eigenket and denotes its eigenvalue. Context, especially the ket delimiters, separates the two roles.

Units and Dimensions

Every Cartesian component of position has dimensions of length:

$$[\hat x]=[\hat r_i]=L.$$

Accordingly,

$$\langle\hat x\rangle = \langle\psi\rvert \hat x \lvert\psi\rangle$$

has length units, while

$$(\Delta x)^2 = \langle\hat x^2\rangle -\langle\hat x\rangle^2$$

has units of length squared. Dimensionless coordinates such as $\xi=x/a$ must be distinguished from the physical operator $\hat x=a\hat\xi$.

Position Measurement

For a normalized wavefunction,

$$\psi(x) = \langle x\rvert\psi\rangle,$$

and the probability of finding the particle in a measurable region $\Delta$ is

$$\Pr(x\in\Delta) = \int_{\Delta} \lvert\psi(x)\rvert^2\,dx.$$

In spectral notation, the position projector for that region is

$$P_x(\Delta) = \int_{\Delta} \lvert x\rangle \langle x\rvert\,dx,$$

so that

$$\Pr(x\in\Delta) = \langle\psi\rvert P_x(\Delta) \lvert\psi\rangle.$$

The spectral projector is a bounded operator even though $\hat x$ itself is unbounded.

The expectation value and variance are

$$\langle\hat x\rangle = \int_{-\infty}^{\infty} x\lvert\psi(x)\rvert^2\,dx,$$ $$(\Delta x)^2 = \int_{-\infty}^{\infty} \left( x-\langle\hat x\rangle \right)^2 \lvert\psi(x)\rvert^2\,dx,$$

provided the required moments exist.

Momentum Representation

Adopt the Fourier convention

$$\langle x\rvert p\rangle = \frac{1}{\sqrt{2\pi\hbar}} \exp\left( \frac{ipx}{\hbar} \right).$$

Then the position operator in momentum representation is

$$(\hat x\widetilde\psi)(p) = i\hbar \frac{d}{dp} \widetilde\psi(p).$$

In several dimensions,

$$\hat{\mathbf r} = i\hbar\nabla_{\mathbf p}.$$

Changing the sign in the Fourier kernel changes the sign of the derivative representation. The Fourier convention should therefore be checked before importing either formula.

Canonical Commutators

The Cartesian position components commute:

$$[\hat r_i,\hat r_j]=0.$$

Together with canonical momenta,

$$[\hat r_i,\hat p_j] = i\hbar\delta_{ij}I.$$

In one dimension this becomes

$$[\hat x,\hat p] =i\hbar I.$$

The identity operator is often suppressed. The relation holds on a suitable common domain; for unbounded operators it is not an unrestricted algebraic identity on every Hilbert-space vector.

Position and Translations

With the active translation operator

$$T(a) = \exp\left( -\frac{ia\hat p}{\hbar} \right),$$

the translated wavefunction is

$$(T(a)\psi)(x) = \psi(x-a).$$

The position operator transforms as

$$T^\dagger(a)\hat xT(a) = \hat x+aI.$$

The sign depends on whether a source uses active state translations or passive coordinate changes. The displayed pair fixes the convention internally: a state translated by positive $a$ has its mean position increased by $a$.

Position in the Heisenberg Equation

For a Schrödinger-picture operator with no explicit time dependence,

$$\frac{d\hat x_H}{dt} = \frac{i}{\hbar} [H_H,\hat x_H].$$

For

$$H = \frac{\hat p^2}{2m} +V(\hat x),$$

this gives

$$\frac{d\hat x_H}{dt} = \frac{\hat p_H}{m}.$$

This relation motivates identifying $\hat p/m$ with velocity for the simple nonrelativistic Hamiltonian. With electromagnetic coupling, spin-orbit terms, lattices, or relativistic dispersion, the velocity operator must instead be computed from the commutator.

Multiparticle Coordinates

For particles labeled by $a=1,\ldots,N$, write

$$\hat{\mathbf r}_a$$

for the position of particle $a$. The operator acts nontrivially on that particle’s factor of the tensor-product Hilbert space, with identity factors usually suppressed.

For two particles, common collective coordinates are

$$\hat{\mathbf R} = \frac{ m_1\hat{\mathbf r}_1 +m_2\hat{\mathbf r}_2 }{ m_1+m_2 },$$ $$\hat{\mathbf r} = \hat{\mathbf r}_1 -\hat{\mathbf r}_2.$$

Capital $\mathbf R$ conventionally denotes center-of-mass position, while lowercase $\mathbf r$ denotes a relative coordinate. A source may choose the opposite sign for the relative coordinate, so the definition should be recorded before conjugate momenta are introduced.

Curvilinear, Compact, and Lattice Coordinates

The Cartesian position vector remains a multiplication operator after a change of coordinates, but the inner-product measure changes. In spherical coordinates,

$$d^3r = r^2\sin\theta \,dr\,d\theta\,d\phi.$$

The radial coordinate $r$ is not a Cartesian component, and its conjugate momentum is not obtained safely by replacing $x$ with $r$ in $-i\hbar\,d/dx$. The Jacobian and operator domain matter.

On a circle, a globally single-valued angle operator has subtleties that the Cartesian line does not. Periodic observables such as $e^{i\hat\phi}$ can be better behaved than a branch-dependent angle $\hat\phi$.

On a discrete lattice, position is often written

$$\hat X = \sum_n x_n \lvert n\rangle \langle n\rvert,$$

where $\lvert n\rangle$ labels a site and $x_n$ is its assigned coordinate. Translation symmetry, periodic boundaries, and polarization observables can make this finite-system position operator more subtle than the formula suggests.

Typography and Symbol Collisions

Form	Usual meaning
$x$	Coordinate value, eigenvalue, or classical variable
$\hat x$	One-dimensional position operator
$\lvert x\rangle$	Generalized position eigenket
$\mathbf r$	Three-dimensional coordinate vector
$\hat{\mathbf r}$	Vector of position operators
$\hat{\mathbf e}_x$	Unit vector in the $x$ direction
$X$ in a qubit circuit	Pauli-$X$ gate, not a spatial position
$X$ in lattice theory	Often a discrete position operator

A roman unit vector, an operator hat, and boldface can all appear together. For example, $\hat{\mathbf e}_x$ is a dimensionless basis vector, whereas $\hat x$ is a length-valued observable.

Hat Policy

This reference uses $\hat x$ when the distinction from a coordinate is important. In an explicitly operator-valued equation, authors often write

$$[x,p]=i\hbar$$

without hats. In a position-space differential equation, the same author may use $x$ as the coordinate argument and $\hat x$ for the multiplication operator. Neither convention is wrong; mixing them without an announced rule is.

Canonical Home

Coordinate Representation owns the position basis and multiplication representation. The longer Position operator card collects operational properties.

Canonical Commutation Relations develops the relation to momentum. Expectation Values owns moment calculations, and Fourier-Transform Conventions fixes the position-momentum transform.

Convention Warnings

• $x$ can denote a coordinate, an eigenvalue, or an operator with its hat suppressed.
• Generalized position eigenkets are not normalizable Hilbert-space vectors.
• A normalized state need not have a finite mean or variance of position.
• The derivative representation in momentum space depends on the Fourier-sign convention.
• Canonical commutators of unbounded operators require a common domain.
• $\hat{\mathbf r}$ is a vector of operators, not a single scalar operator.
• Radial position and Cartesian position have different measures and conjugate-momentum subtleties.
• Position on a compact periodic space is not identical to position on the real line.
• $X$ in quantum-information notation usually means the Pauli-$X$ gate.
• Identity operators on untouched tensor factors are often omitted.

References

• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994, chs. 1, 4, and 7.
• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020, chs. 1 and 2.
• C. Cohen-Tannoudji, B. Diu, and F. Laloë, Quantum Mechanics, vol. I, Wiley, 1977, complements A and B.
• M. Reed and B. Simon, Methods of Modern Mathematical Physics I: Functional Analysis, rev. ed., Academic Press, 1980, ch. VIII.