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Position operator

In quantum mechanics, the position operator, typically denoted as \hat{\mathbf{r}} or \hat{x} in one dimension, is a fundamental self-adjoint operator that represents the observable position of a particle within a Hilbert space, such as L^2(\mathbb{R}^3).^[1] In the position representation, it acts on a wave function \psi(\mathbf{r}) by simple multiplication with the position coordinate, yielding \hat{\mathbf{r}} \psi(\mathbf{r}) = \mathbf{r} \psi(\mathbf{r}), which encodes the probability distribution of finding the particle at a specific location.^[2] This operator has a continuous spectrum of real eigenvalues corresponding to all possible position values, with generalized eigenfunctions given by Dirac delta distributions \delta(\mathbf{r} - \mathbf{r}_0), though these are not normalizable in the strict Hilbert space sense and require the framework of rigged Hilbert spaces for rigorous treatment.^[3] The position operator plays a central role in formulating the dynamical laws of quantum systems, appearing prominently in the time-independent Schrödinger equation \hat{H} \psi = E \psi, where the Hamiltonian \hat{H} often includes potential terms dependent on \hat{\mathbf{r}}, such as \hat{V}(\hat{\mathbf{r}}).^[4] It does not commute with the momentum operator \hat{\mathbf{p}}, satisfying the canonical commutation relation [\hat{x}_i, \hat{p}_j] = i \hbar \delta_{ij}, which underpins the Heisenberg uncertainty principle, stating that the product of uncertainties in position and momentum must satisfy \Delta x \Delta p \geq \hbar/2.^[5] For multi-particle systems, position operators for each particle are defined similarly, enabling descriptions of relative positions and center-of-mass coordinates essential for molecular and solid-state physics.^[6] Beyond its basic properties, the position operator's unbounded nature and continuous eigenvalues distinguish it from discrete observables like spin, influencing measurement theory where position measurements collapse the wave function to a localized state.^[7] In relativistic quantum field theory, generalizations of the position operator arise, though they encounter challenges related to non-commutativity and causality, as explored in extensions like Newton-Wigner operators.^[8] These features make the position operator indispensable for bridging classical intuition with quantum probabilistic descriptions across applications in atomic physics, quantum optics, and beyond.

Historical context

The position operator emerged during the foundational development of quantum mechanics in the 1920s. In 1925, Werner Heisenberg introduced the operator formalism in matrix mechanics, representing observables such as position and momentum as non-commuting arrays. The next year, Erwin Schrödinger formulated wave mechanics, defining the position operator in the position representation as multiplication by the coordinate. These approaches were unified in subsequent work by Paul Dirac and John von Neumann, providing the modern Hilbert space framework.^[9]

Introduction

Definition and role in quantum mechanics

In quantum mechanics, the position operator \hat{\mathbf{r}} serves as the quantum analog of the classical position vector \mathbf{r}, representing the observable position of a particle and enabling the description of its spatial properties through operator actions on quantum states.^[10] This operator is Hermitian, ensuring that its eigenvalues are real numbers corresponding to possible outcomes of position measurements, as required by the foundational postulate that physical observables map to self-adjoint operators. The position operator plays a central role in quantifying particle localization, where the probability distribution for finding a particle at a specific position is given by the squared modulus of the wave function |\psi(\mathbf{r})|^2, and the expectation value \langle \hat{\mathbf{r}} \rangle = \int \psi^*(\mathbf{r}) \mathbf{r} \psi(\mathbf{r}) \, d^3\mathbf{r} yields the average position, akin to the center of mass in classical probability distributions.^[4] In the Heisenberg picture, where states are time-independent and operators evolve, the position operator's dynamics reflect the particle's motion under the Hamiltonian, providing insight into how quantum states translate through space over time.^[10] This formulation arises from Bohr's correspondence principle, which posits that quantum operators should reproduce classical observables in the limit of large quantum numbers, thereby quantizing classical variables like position to bridge the two theories.^[11] For instance, in the time-dependent Schrödinger equation i\hbar \frac{\partial \psi}{\partial t} = \hat{H} \psi, the position operator \hat{\mathbf{r}} enters the potential term V(\hat{\mathbf{r}}), allowing the incorporation of position-dependent interactions such as Coulomb potentials in atomic systems.^[10]

Historical context

The evolution of the position operator concept began with foundational ideas in wave mechanics during the mid-1920s. In his 1924 doctoral thesis, Louis de Broglie proposed that particles possess wave properties, with position associated to the spatial characteristics of these matter waves, extending the wave-particle duality from light to matter and setting the stage for a position-dependent quantum description.^[12] This hypothesis influenced Erwin Schrödinger, who in 1926 derived the time-independent wave equation, incorporating position explicitly via the potential energy term V(\hat{x}) in the Hamiltonian operator, thereby treating position as a continuous variable governing the system's dynamics.^[13] Parallel to these developments, Werner Heisenberg introduced matrix mechanics in his 1925 paper, representing position and momentum as non-commuting infinite arrays, which highlighted the operator nature of observables and their failure to commute.^[14] The conceptual unity of Heisenberg's discrete matrix approach and Schrödinger's continuous wave mechanics was established through proofs of their mathematical equivalence in 1926 by Schrödinger and others, demonstrating that position functions equivalently in both frameworks as a fundamental quantum observable.^[15] In the late 1920s, Paul Dirac developed the transformation theory, contributing to the abstract operator framework, while John von Neumann in 1932 provided a rigorous axiomatic foundation for quantum theory in Hilbert space, defining unbounded operators like position and ensuring their self-adjointness for physical observables.^[16]^[17] The position operator's role solidified in the axiomatic quantum mechanics of the post-1950s, where it underpinned measurement postulates and operator algebra developments, drawing directly from von Neumann's framework to address foundational issues in quantum theory.^[12]

Mathematical Formulation

One-dimensional position operator

In the one-dimensional case, the position operator \hat{x} acts on the Hilbert space L^2(\mathbb{R}) of square-integrable wave functions \psi(x) by multiplication with the position variable: \hat{x} \psi(x) = x \psi(x).^[18] This form represents the observable position along the real line, where the operator is unbounded due to the unbounded nature of the multiplier x.^[19] To ensure the operator is densely defined and well-behaved, its domain is taken as the dense subspace consisting of smooth functions with compact support, C_c^\infty(\mathbb{R}), which is embedded in L^2(\mathbb{R}).^[20] On this domain, \hat{x} maps elements back into L^2(\mathbb{R}), and the subspace is dense because compactly supported smooth functions can approximate any square-integrable function arbitrarily well in the L^2 norm.^[18] The position operator is symmetric on this domain, as the inner product satisfies \langle \psi | \hat{x} \phi \rangle = \int_{-\infty}^\infty \psi^*(x) \, x \, \phi(x) \, dx = \int_{-\infty}^\infty [x \psi(x)]^* \phi(x) \, dx = \langle \hat{x} \psi | \phi \rangle for \psi, \phi \in C_c^\infty(\mathbb{R}), with no boundary terms arising since the functions vanish outside a finite interval.^[19] To establish self-adjointness, consider the maximal extension to the domain D(\hat{x}) = \{\psi \in L^2(\mathbb{R}) \mid x \psi(x) \in L^2(\mathbb{R})\}, where \int_{-\infty}^\infty |x \psi(x)|^2 \, dx < \infty. On this domain, symmetry extends directly without additional boundary conditions, as the real-valued multiplier x ensures the adjoint coincides with the operator itself, \hat{x}^\dagger = \hat{x}, and the domain is maximal.^[20] A sketch of self-adjointness involves verifying that for \psi, \phi \in D(\hat{x}), the difference \langle \psi | \hat{x} \phi \rangle - \langle \hat{x} \psi | \phi \rangle = 0 holds by direct computation of the integrals, with vanishing contributions at infinity due to square-integrability; integration by parts is not required here but confirms the absence of surface terms when considering related differential operators.^[18] The spectral decomposition of \hat{x} follows from the spectral theorem for self-adjoint operators with continuous spectrum, expressed informally in the position basis as

\hat{x} = \int_{-\infty}^{\infty} \lambda \, |\lambda\rangle \langle \lambda| \, d\lambda,

where the |\lambda\rangle denote the (generalized) position eigenstates, and the integral represents resolution over the real line spectrum \sigma(\hat{x}) = \mathbb{R}.^[19] This form highlights the continuous nature of position measurements in one dimension.

Multi-dimensional extensions

In quantum mechanics, the position operator for a single particle in three-dimensional space is generalized to a vector operator \hat{\mathbf{r}} = (\hat{x}, \hat{y}, \hat{z}), where each component acts by multiplication in the position representation on wave functions \psi(\mathbf{r}) in the Hilbert space L^2(\mathbb{R}^3). Specifically, \hat{x}_i \psi(\mathbf{r}) = x_i \psi(\mathbf{r}) for i = x, y, z, with the components satisfying the canonical commutation relations [\hat{x}_i, \hat{p}_j] = i\hbar \delta_{ij} alongside the momentum operator components \hat{\mathbf{p}} = -i\hbar \nabla.^[21] This vector formulation extends naturally to other coordinate systems, such as spherical coordinates (r, \theta, \phi), where the position operator components become multiplication operators by the respective coordinates: \hat{r} \psi(r, \theta, \phi) = r \psi(r, \theta, \phi), \hat{\theta} \psi(r, \theta, \phi) = \theta \psi(r, \theta, \phi), and \hat{\phi} \psi(r, \theta, \phi) = \phi \psi(r, \theta, \phi). In curvilinear systems like spherical coordinates, these operators facilitate the separation of radial and angular dependencies in the Schrödinger equation, particularly for central potentials where the wave function decomposes into radial and spherical harmonic parts.^[22]^[23]/11%3A_Operators/11.03%3A_Operators_and_Quantum_Mechanics_-_an_Introduction) A key application of the multi-dimensional position operator arises in the definition of the orbital angular momentum operator, given by the vector cross product \hat{\mathbf{L}} = \hat{\mathbf{r}} \times \hat{\mathbf{p}}, which generates rotations in three-dimensional space. The components of \hat{\mathbf{L}} are \hat{L}_x = \hat{y} \hat{p}_z - \hat{z} \hat{p}_y, \hat{L}_y = \hat{z} \hat{p}_x - \hat{x} \hat{p}_z, and \hat{L}_z = \hat{x} \hat{p}_y - \hat{y} \hat{p}_x, with the z-component in spherical coordinates simplifying to \hat{L}_z = -i\hbar \frac{\partial}{\partial \phi} due to the azimuthal symmetry. This structure underpins the quantization of angular momentum in atomic and molecular systems, where eigenstates of \hat{\mathbf{L}}^2 and \hat{L}_z are the spherical harmonics Y_{lm}(\theta, \phi).^[22]^[23] For systems involving N particles, the position operator for the j-th particle is \hat{\mathbf{r}}_j = (\hat{x}_j, \hat{y}_j, \hat{z}_j), acting on the tensor product Hilbert space L^2(\mathbb{R}^{3N}) via multiplication by the coordinates of the j-th particle while leaving others unchanged. The total wave function \Psi(\mathbf{r}_1, \dots, \mathbf{r}_N) satisfies \hat{\mathbf{r}}_j \Psi = \mathbf{r}_j \Psi, enabling the description of relative positions and center-of-mass motion in multi-particle quantum mechanics, such as in molecular dynamics or many-body interactions. For indistinguishable particles, symmetrization or antisymmetrization of the wave function is imposed, but the operator structure remains a direct product over individual particle spaces./05%3A_Multi-Particle_Systems/5.01%3A_Fundamental_Concepts_of_Multi-Particle_Systems)^[24]^[25]

Operator Properties

Basic algebraic properties

The position operator \hat{x}, acting on the Hilbert space L^2(\mathbb{R}), is linear, satisfying \hat{x}(a\psi + b\phi) = a\hat{x}\psi + b\hat{x}\phi for any scalars a, b \in \mathbb{C} and wave functions \psi, \phi in its domain.^[26] This linearity follows directly from its definition as multiplication by the coordinate x in the position representation, \hat{x}\psi(x) = x \psi(x).^[26] As a linear operator, it maps superpositions to superpositions, preserving the vector space structure essential for quantum superpositions.^[26] The position operator is unbounded, meaning there exist sequences of normalized states for which \|\hat{x} \psi_n\| \to \infty as n \to \infty.^[26] Its domain is restricted to D(\hat{x}) = \{\psi \in L^2(\mathbb{R}) \mid \int_{-\infty}^{\infty} x^2 |\psi(x)|^2 \, dx < \infty \}, ensuring the operator is well-defined on a dense subspace.^[27] The spectrum of \hat{x} is the entire real line \mathbb{R}, reflecting the continuous range of possible position measurements without bounds.^[27] The position operator is self-adjoint, \hat{x}^\dagger = \hat{x}, on its domain, which guarantees real eigenvalues and orthogonal eigenprojections in the spectral decomposition.^[26] This property is verified by the inner product relation \langle \phi | \hat{x} \psi \rangle = \langle \hat{x} \phi | \psi \rangle^* for all \phi, \psi \in D(\hat{x}), confirming symmetry and essential self-adjointness.^[26] As an unbounded self-adjoint operator, \hat{x} admits a resolution of the identity via the spectral theorem:

\hat{1} = \int_{-\infty}^{\infty} |x\rangle \langle x | \, dx,

where |x\rangle denote the generalized position eigenstates, and the integral is understood as a projection-valued measure over Borel sets on \mathbb{R}.^[27] This resolution decomposes the identity operator into projections onto position subspaces, enabling the probabilistic interpretation of position measurements.^[27]

Commutation relations

The canonical commutation relation between the position operator \hat{x} and the momentum operator \hat{p} in one dimension is given by [\hat{x}, \hat{p}] = i\hbar \hat{1}, where \hbar is the reduced Planck's constant and \hat{1} is the identity operator.^[28] This relation arises from the quantization procedure that replaces the classical Poisson bracket \{x, p\} = 1 with the quantum commutator, scaled by i\hbar.^[28] It encodes the fundamental non-commutativity of position and momentum in quantum mechanics, distinguishing it from classical mechanics where these variables commute.^[28] In three dimensions, the position operator \hat{\mathbf{x}} = (\hat{x}_1, \hat{x}_2, \hat{x}_3) and momentum operator \hat{\mathbf{p}} = (\hat{p}_1, \hat{p}_2, \hat{p}_3) satisfy the generalized canonical commutation relations [\hat{x}_j, \hat{p}_k] = i\hbar \delta_{jk} \hat{1} for j, k = 1, 2, 3, where \delta_{jk} is the Kronecker delta.^[28] Components of position commute among themselves, as do components of momentum: [\hat{x}_j, \hat{x}_k] = [\hat{p}_j, \hat{p}_k] = 0.^[28] These relations extend the one-dimensional case and form the basis for the algebraic structure of quantum mechanics in multi-particle and field theories. A key consequence of the commutation relations is their role in the time evolution of expectation values, as described by the Ehrenfest theorem. For a particle of mass m under a potential V(\hat{\mathbf{x}}), the time derivative of the position expectation value is \frac{d}{dt} \langle \hat{x}_j \rangle = \frac{1}{m} \langle \hat{p}_j \rangle, derived by inserting the Heisenberg equation of motion \frac{d\hat{A}}{dt} = \frac{i}{\hbar} [\hat{H}, \hat{A}] + \frac{\partial \hat{A}}{\partial t} (with Hamiltonian \hat{H} = \frac{\hat{\mathbf{p}}^2}{2m} + V(\hat{\mathbf{x}})) and using the canonical commutators.^[29] Similarly, \frac{d}{dt} \langle \hat{p}_j \rangle = -\left\langle \frac{\partial V}{\partial x_j} \right\rangle.^[29] This demonstrates how quantum expectation values mimic classical equations of motion on average, bridging quantum and classical dynamics.^[29] More generally, the position operator commutes with arbitrary functions of momentum according to [\hat{x}, f(\hat{p})] = i\hbar \frac{\partial f}{\partial p}, where the partial derivative treats f as a function of the classical variable p.^[30] To derive this, assume f(\hat{p}) admits a power series expansion f(\hat{p}) = \sum_{n=0}^\infty a_n \hat{p}^n. Then, the commutator is

[\hat{x}, f(\hat{p})] = \sum_{n=0}^\infty a_n [\hat{x}, \hat{p}^n].

Using the identity [\hat{x}, \hat{p}^n] = i\hbar n \hat{p}^{n-1} (proved inductively from the canonical relation, with base case [\hat{x}, \hat{p}] = i\hbar and recursion [\hat{x}, \hat{p}^n] = [\hat{x}, \hat{p}] \hat{p}^{n-1} + \hat{p} [\hat{x}, \hat{p}^{n-1}]), the series becomes i\hbar \sum_{n=1}^\infty a_n n \hat{p}^{n-1} = i\hbar \frac{\partial f}{\partial p}.^[30] This result holds for analytic functions and extends to other canonical pairs.^[30] These commutation relations imply that position and momentum cannot be simultaneously measured with arbitrary precision, a principle central to quantum uncertainty.^[28]

Representations

Position representation

In the position representation of quantum mechanics, the position basis is formed by the eigenstates |x\rangle of the position operator \hat{x}, which satisfy the eigenvalue equation \hat{x} |x\rangle = x |x\rangle, where x labels the continuous set of position eigenvalues spanning the real line.^[5] These states provide a complete basis for the Hilbert space of square-integrable wave functions, allowing any physical state to be expanded in terms of position eigenstates.^[1] The wave function \psi(x) associated with a quantum state |\psi\rangle is defined as the projection \psi(x) = \langle x | \psi \rangle, representing the amplitude for the system to be found at position x.^[2] In this representation, the position operator acts as a simple multiplication operator: \langle x | \hat{x} | \psi \rangle = x \psi(x), which underscores its diagonal form in the position basis and facilitates the computation of expectation values and uncertainties for position-dependent observables.^[5] The inner product between two states |\psi\rangle and |\phi\rangle in the position representation is given by the integral \langle \psi | \phi \rangle = \int_{-\infty}^{\infty} \psi^*(x) \phi(x) \, dx, ensuring the Hilbert space structure with orthonormality and completeness relations among the basis states.^[31] According to the Born rule, the probability density for measuring the position of the system in the state |\psi\rangle at x is |\psi(x)|^2, so the probability of finding the particle in an infinitesimal interval dx is |\psi(x)|^2 \, dx, normalized such that \int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = 1.^[32] This probabilistic interpretation directly ties the wave function's modulus squared to measurable position outcomes in quantum experiments.^[33] In contrast to the momentum representation, where \hat{x} assumes a differential form, the position basis diagonalizes \hat{x} for straightforward multiplication-based calculations.^[1]

Momentum representation

In the momentum representation, the state of a quantum system is described by the momentum-space wave function \tilde{\psi}(p) = \langle p | \psi \rangle, which is the Fourier transform of the position-space wave function \psi(x) = \langle x | \psi \rangle:

\tilde{\psi}(p) = \frac{1}{\sqrt{2\pi \hbar}} \int_{-\infty}^{\infty} \psi(x) \, e^{-i p x / \hbar} \, dx.

This transformation relates the two representations through the overlap between position and momentum eigenstates.^[34] The position operator \hat{x} acts on the momentum-space wave function as a differential operator:

\hat{x} \tilde{\psi}(p) = i \hbar \frac{\partial}{\partial p} \tilde{\psi}(p).

This form arises from the canonical commutation relation [\hat{x}, \hat{p}] = i \hbar. To derive it, consider the action in position space where \hat{x} \psi(x) = x \psi(x) and \hat{p} = -i \hbar \frac{\partial}{\partial x}. Transforming to momentum space via the Fourier relation and applying the commutator yields the differential form after integration by parts, ensuring consistency with the algebra of operators.^[34]^[5] The matrix elements of the position operator in the momentum basis are given by

\langle p | \hat{x} | p' \rangle = i \hbar \frac{\partial}{\partial p} \delta(p - p'),

reflecting the non-local nature of \hat{x} in momentum space, where it couples states differing infinitesimally in momentum. This expression follows directly from the operator action on the completeness relation \int dp' |p'\rangle \langle p' | = \hat{1}.^[34] As an illustrative example, consider the ground state of the quantum harmonic oscillator, where the position-space wave function is \psi_0(x) = \left( \frac{\alpha}{\pi} \right)^{1/4} e^{-\alpha x^2 / 2} with \alpha = m \omega / \hbar. The corresponding momentum-space wave function is \tilde{\psi}_0(p) = \left( \frac{1}{\pi \alpha \hbar^2} \right)^{1/4} e^{-p^2 / (2 \alpha \hbar^2)}, also a Gaussian. Applying \hat{x} gives \hat{x} \tilde{\psi}_0(p) = -i \frac{p}{\alpha \hbar} \tilde{\psi}_0(p), which is proportional to p \tilde{\psi}_0(p). This demonstrates that the position operator mixes momentum states around p = 0, as the result is not an eigenstate of momentum but a linear combination weighted by the derivative, highlighting the operator's role in connecting nearby momenta.^[34]

Eigenstates and Spectrum

Position eigenstates

In quantum mechanics, the position eigenstates, denoted as |x\rangle, are the eigenvectors of the position operator \hat{x}, satisfying the eigenvalue equation \hat{x} |x\rangle = x |x\rangle, where x is a real number representing the position eigenvalue.^[35] These states provide an idealized description of a particle localized precisely at position x. Any state |\psi\rangle in the Hilbert space can be expanded in terms of these eigenstates as |\psi\rangle = \int_{-\infty}^{\infty} \psi(x) |x\rangle \, dx, where \psi(x) = \langle x | \psi \rangle is the wave function in the position representation.^[36] The position eigenstates form a complete basis for the space of square-integrable functions L^2(\mathbb{R}), enabling the resolution of the identity operator via \hat{I} = \int_{-\infty}^{\infty} |x\rangle \langle x | \, dx.^[35] Their orthogonality is expressed using the Dirac delta function as the kernel: \langle x | x' \rangle = \delta(x - x'), which ensures that distinct position eigenstates are orthogonal in the continuous spectrum sense.^[36] Physically, the state |x\rangle corresponds to a particle with infinitely precise position knowledge, represented by a Dirac delta function in position space, but this idealization is unphysical because it implies a completely delocalized momentum distribution, leading to infinite expectation value of kinetic energy.^[37] Such states cannot be realized in practice due to this divergence, though they serve as useful mathematical limits; their normalization challenges are addressed in the context of continuous spectra.^[35]

Continuous spectrum and normalization

The position operator \hat{x} in one dimension possesses a purely continuous spectrum \sigma(\hat{x}) = \mathbb{R}, spanning the entire real line with no discrete eigenvalues, reflecting the unbounded nature of position measurements in quantum mechanics.^[38] This continuous spectrum arises because the operator, defined on the Hilbert space L^2(\mathbb{R}) as multiplication by x, admits generalized eigenstates that are not square-integrable, necessitating advanced mathematical frameworks to handle them rigorously.^[38] To accommodate these non-normalizable eigenstates |x\rangle, the rigged Hilbert space (RHS) formulation is employed, structuring the space as a Gel'fand triple \Phi \subset L^2(\mathbb{R}) \subset \Phi', where \Phi is the Schwartz space of smooth, rapidly decaying test functions serving as the domain for operator actions.^[38] In this setup, the position eigenstates reside in the dual space \Phi', the antidual of \Phi, allowing the operator to extend continuously while preserving key quantum properties like self-adjointness.^[38] The RHS resolves the limitations of the standard Hilbert space by embedding distributions, such as Dirac deltas, which formalize the eigenstates without violating Hilbert space norms. The normalization of position eigenstates follows the Dirac delta convention, \langle x | x' \rangle = \delta(x - x'), ensuring orthogonality and completeness via the resolution of the identity \int_{-\infty}^{\infty} |x\rangle \langle x| \, dx = \hat{1}.^[38] However, the inner product \langle x | x \rangle = \delta(0) diverges, indicating that these states are not proper vectors in L^2(\mathbb{R}) and cannot be normalized to unity in the conventional sense.^[38] This divergence is regularized through limiting procedures, where sequences of normalizable states approximate the eigenstates; for instance, Gaussian wave packets centered at x with variance \sigma^2 \to 0 yield wave functions \psi_\sigma(y) = (2\pi \sigma^2)^{-1/4} \exp\left[ -(y - x)^2 / (4\sigma^2) \right], whose probability densities approach \delta(y - x) while maintaining finite norms.^[39] As \sigma \to 0, the position uncertainty \Delta x = \sigma \to 0, effectively mimicking the idealized eigenstate in the RHS framework.^[39]

Applications

Position measurements

In quantum mechanics, measuring the position of a particle involves the position operator \hat{x}, which is a self-adjoint operator on the Hilbert space of the system, corresponding to the observable for position. According to the von Neumann measurement formalism, the measurement outcomes are determined by a projection-valued measure (PVM) associated with \hat{x}, where for any Borel set \Delta \subseteq \mathbb{R}, the projector is given by

E(\Delta) = \int_{\Delta} |x\rangle \langle x| \, dx,

with |x\rangle denoting the (improperly normalized) position eigenstates. This setup ensures that the projectors satisfy the resolution of the identity \int_{-\infty}^{\infty} E(\{x\}) \, dx = \mathbb{I} and orthogonality for disjoint sets, allowing the observable to be expressed as \hat{x} = \int_{-\infty}^{\infty} x \, E(\{x\}) \, dx. The probability of obtaining a position measurement outcome x \in \Delta for a system in state |\psi\rangle is provided by the Born rule, which states P(x \in \Delta) = \langle \psi | E(\Delta) | \psi \rangle = \int_{\Delta} |\psi(x)|^2 \, dx, where \psi(x) = \langle x | \psi \rangle is the position-space wave function normalized such that \int_{-\infty}^{\infty} |\psi(x)|^2 \, dx = 1. This probabilistic interpretation connects the squared modulus of the wave function to the likelihood of measurement results, forming a cornerstone of quantum prediction. Upon measurement yielding an outcome in \Delta, the collapse postulate dictates that the system's state updates instantaneously to the normalized projection onto the corresponding subspace: |\psi'\rangle = \frac{E(\Delta) |\psi\rangle}{\sqrt{\langle \psi | E(\Delta) | \psi \rangle}}. In the position representation, this corresponds to \psi'(x) \propto \chi_{\Delta}(x) \psi(x) / \sqrt{P(\Delta)}, where \chi_{\Delta}(x) is the indicator function that is 1 if x \in \Delta and 0 otherwise, effectively localizing the wave function within \Delta. This projection ensures the post-measurement state is an eigenstate (or superposition within the degenerate subspace) of the measured observable. A key feature of this measurement process is repeatability: an immediate re-measurement of position on the collapsed state |\psi'\rangle will yield an outcome in \Delta with probability 1, as E(\Delta) |\psi'\rangle = |\psi'\rangle, satisfying the von Neumann repeatability hypothesis for ideal measurements. This property underscores the non-demolition nature of the projection for compatible observables and distinguishes quantum measurements from classical ones.

Relation to uncertainty principle

The non-commutativity of the position operator \hat{x} and the momentum operator \hat{p}, satisfying [\hat{x}, \hat{p}] = i\hbar, implies fundamental limits on the simultaneous precision of position and momentum measurements for any quantum state, as established in the commutation relations section. This leads to the Heisenberg-Robertson uncertainty principle, which quantifies these limits through the standard deviations (or uncertainties) \Delta x = \sqrt{\langle (\hat{x} - \langle \hat{x} \rangle)^2 \rangle} and \Delta p = \sqrt{\langle (\hat{p} - \langle \hat{p} \rangle)^2 \rangle}, yielding the inequality \Delta x \, \Delta p \geq \frac{\hbar}{2}. More generally, for any pair of non-commuting Hermitian operators \hat{A} and \hat{B}, the variance-based form is \mathrm{Var}(\hat{A}) \, \mathrm{Var}(\hat{B}) \geq \frac{|\langle [\hat{A}, \hat{B}] \rangle|^2}{4}; for position and momentum, this specializes to \mathrm{Var}(\hat{x}) \, \mathrm{Var}(\hat{p}) \geq \left( \frac{\hbar}{2} \right)^2, since \langle [\hat{x}, \hat{p}] \rangle = i\hbar. A sketch of the proof relies on the Cauchy-Schwarz inequality applied to the deviation operators \delta \hat{x} = \hat{x} - \langle \hat{x} \rangle and \delta \hat{p} = \hat{p} - \langle \hat{p} \rangle. Consider the expectation value \langle (\delta \hat{x} + i \lambda \delta \hat{p})^\dagger (\delta \hat{x} + i \lambda \delta \hat{p}) \rangle \geq 0 for real \lambda > 0, which expands to \langle (\delta \hat{x})^2 \rangle + \lambda^2 \langle (\delta \hat{p})^2 \rangle - 2\lambda \mathrm{Im} \langle [\hat{x}, \hat{p}] \rangle / 2 \geq 0. Minimizing over \lambda gives the bound \Delta x \, \Delta p \geq |\langle [\hat{x}, \hat{p}] \rangle| / 2 = \hbar / 2. Equality holds for states that saturate the inequality, known as minimum-uncertainty states, which include Gaussian wave packets where the position-space wave function is \psi(x) \propto \exp\left( -\frac{(x - x_0)^2}{4\sigma_x^2} + i p_0 x / \hbar \right) with \Delta x \, \Delta p = \hbar / 2. A prominent example is the ground state of the quantum harmonic oscillator, |0\rangle, which is Gaussian in position space with \Delta x = \sqrt{\frac{\hbar}{2 m \omega}} and \Delta p = \sqrt{\frac{m \omega \hbar}{2}}, achieving the minimum product \Delta x \, \Delta p = \frac{\hbar}{2}. This principle imposes physical consequences by restricting the simultaneous knowledge of position and momentum, preventing arbitrary precision in both for any particle. For instance, in electron microscopy, achieving high spatial resolution requires short-wavelength electrons (small \Delta x), but the corresponding large momentum transfer (\Delta p \approx h / \lambda) disturbs the sample's position, limiting ultimate resolution to scales set by \hbar.^[40]

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A prime example is the. position operator. Let's first consider a particle in d = 1. We define x as the position operator, with corresponding eigenstates |x0> ...
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A new look at the position operator in quantum theory - NASA ADS
Different components of the new position operator do not commute with each other and, as a consequence, there is no wave function in coordinate representation.Missing: mechanics | Show results with:mechanics
[9]
[PDF] Quantum Physics I, Lecture Note 5 - MIT OpenCourseWare
Feb 21, 2016 · introducing a position operator x that acting on functions of x gives another function of x as ... One can in fact formulate Quantum Mechanics ...
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A history of Quantum Mechanics - MacTutor - University of St Andrews
In 1932 von Neumann put quantum theory on a firm theoretical basis. Some of the earlier work had lacked mathematical rigour, but von Neumann put the whole ...
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[PDF] An undulatory theory of the mechanics of atoms and molecules - ISY
The paper gives an account of the author's work on a new form of quantum theory. §1. The Hamiltonian analogy between mechanics and optics. §2. The.
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Understanding Heisenberg's 'Magical' Paper of July 1925 - arXiv
Apr 1, 2004 · In July 1925 Heisenberg published a paper [Z. Phys. 33, 879-893 (1925)] which ended the period of `the Old Quantum Theory' and ushered in the new era of ...
[13]
Mathematical foundations of quantum mechanics : Von Neumann ...
Jul 2, 2019 · Mathematical foundations of quantum mechanics. by: Von Neumann, John, 1903-1957. Publication date: 1955. Topics: Matrix mechanics. Publisher ...
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[PDF] Functional Analysis Princeton University MAT520 Lecture Notes
Aug 18, 2023 · However, applying the position operator on this function yields x 7→ x. √. 1 + x2 which is not in L2 (R). So it can't be that X : L2 (R) ...
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None
### Summary of Position Operator from https://home.ba.infn.it/~facchi/lectures/operatori%20autoaggiunti.pdf
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[PDF] Self-adjoint operators and solving the Schrödinger equation
Jun 13, 2014 · In this tutorial we collect facts from the theory of self-adjoint operators, mostly with a view of what is relevant for applications in ...<|control11|><|separator|>
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3 The Formalism of Quantum Mechanics - DAMTP
For a particle moving in 3d there are three position operators, one for each direction. ... With these results in mind, let's now return to Hermitian operators ...
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Spherical Coordinates and the Angular Momentum Operators
We now proceed to calculate the angular momentum operators in spherical coordinates. The first step is to write the ...
[19]
[PDF] Orbital Angular Momentum and Spherical Harmonics
Wave functions in quantum mechanics are single-valued functions of position because ψ(x) is just the expansion of the state |ψi with respect to the position ...
[20]
[PDF] MULTIPARTICLE STATES AND TENSOR PRODUCTS
Dec 14, 2013 · Below is a description of the quantum mechanics and family of operators associated with each particle: • Particle 1: its quantum mechanics is ...
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Fundamental Concepts - Richard Fitzpatrick
By analogy, in a multi-particle system, the position of the $i$ th particle is represented by the algebraic operator $x_i$ , whereas the corresponding ...<|control11|><|separator|>
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[PDF] arXiv:1508.02682v2 [cond-mat.mes-hall] 29 Nov 2016
Nov 29, 2016 · 2 Specifically, for an unbounded operator to be self-adjoint, it is necessary but not sufficient for it to be Hermitian (symmetric). There- fore ...
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[PDF] arXiv:1809.03356v2 [math.FA] 9 Nov 2020
Nov 9, 2020 · It includes the case of the position operator in Quantum Mechanics. This example is prototypical since the spectral theorem guarantees that any ...
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[PDF] Principles Of Quantum Mechanics Dirac
Sep 28, 2020 · Dirac emphasized commutation relations between operators corresponding to observables, which determine the fundamental uncertainty relations and ...
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https://farside.ph.utexas.edu/teaching/qmech/Quantum/node57.html
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Commutation relations for functions of operators - AIP Publishing
Jun 2, 2005 · For the canonical position and momentum in quantum mechanics, the constant c takes the value i ℏ ⁠. III. GENERAL RESULT. The most general result ...
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[PDF] 1 Introduction 2 Wave functions - University of Oregon
Apr 11, 2006 · . The space of states for quantum mechanics comes with an inner product that associates a complex number (φ, ψ) to two vectors φ and ψ. The ...
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Wave Functions – University Physics Volume 3 - UCF Pressbooks
. It gives the probability that a particle will be found at a particular position and time per unit length, also called the probability density. The probability ...
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[PDF] Quantum Mechanics - Lehman College
Feb 26, 2019 · Traveling wave solutions + time evolution of probability density. P(x, t) = ψ. ∗. (x, t)ψ(x, t) = ψ. ∗. (x)eiωt ψ(x)e. −iωt = ψ. ∗. (x)ψ(x) (33).
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[PDF] Chapter 5 The Dirac Formalism and Hilbert Spaces
Note, since the position operator has a continuous spectrum of eigenvalues the orthogonality relation is a dirac delta function rather than a delta-symbol.
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[PDF] CHM 502 - Module 1 - The Postulates of Quantum Mechanics
• The position eigenstates form an orthonormal set so <x0|x> = δ(x - x0) ... complete basis sets, we can express any one of these wavefunctions from ...
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[PDF] Vacuum Energy Density near Fluctuating Boundaries - arXiv
density at that point for essentially the same reason that a position eigenstate in single particle quantum mechanics has infinite energy. In the case of ...
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[PDF] The role of the rigged Hilbert space in Quantum Mechanics - arXiv
Feb 9, 2005 · There is compelling evidence that, when continuous spectrum is present, the natural mathematical setting for Quantum Mechanics is the rigged ...
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[PDF] Position eigenstates via application of an operator on the vacuum
Aug 31, 2013 · that would give an approximation for how to obtain a position eigenstate from the vacuum. ... Gaussian wave packet. To confirm this, we use ...
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[PDF] Photon Momentum and Uncertainty Principle - MIT OpenCourseWare
The electron microscope is a device which uses the wave behavior of electrons to make images which are otherwise too small for visible light! IMPORTANT POINT:.