| Hilbert space | |
|---|---|
| Name | Hilbert space |
| Type | Vector space |
| Field | Functional analysis |
| Introduced by | David Hilbert |
| First proposed | 20th century |
| Related | Banach space, Inner product space, Rigged Hilbert space |
Hilbert space
A Hilbert space is a complete inner product space that generalizes the notion of Euclidean space to possibly infinite dimensions. In Quantum Physics it provides the mathematical arena for states, observables, and unitary evolution, enabling a rigorous formulation of wavefunctions, operators, and the spectral decomposition that underpins measurement theory. Hilbert spaces connect functional analysis with physical structures developed by figures such as David Hilbert and John von Neumann.
A Hilbert space is a vector space H over the field of complex numbers (or occasionally the real numbers) furnished with an Inner product ⟨·,·⟩ that induces a norm ||ψ|| = √⟨ψ,ψ⟩ and a metric; H is complete with respect to this metric. Standard examples include finite-dimensional Euclidean spaces ℂ^n and infinite-dimensional spaces such as L^2(ℝ^n), the space of square-integrable functions used in quantum mechanics. Fundamental properties include linearity, completeness, the parallelogram law, and existence of orthogonal projections onto closed subspaces. Hilbert spaces are a central object of Functional analysis and are closely related to Banach space theory.
The inner product defines geometry on H: lengths, angles, and orthogonality. The induced norm and metric generate a topology making H a complete metric space. In quantum applications one typically uses the L^2 inner product ⟨φ,ψ⟩ = ∫ φ*(x) ψ(x) dx on L^2 spaces. The Riesz representation theorem identifies continuous linear functionals with elements of H and underlies bra–ket notation developed by Paul Dirac. Topological concepts such as convergence in norm (strong topology) and weak convergence are essential for limits of sequences of states and for operator convergence studied in Operator theory.
Orthogonality and orthonormal sets structure Hilbert spaces. An orthonormal basis is a maximal orthonormal set whose linear span is dense; every separable Hilbert space admits a countable orthonormal basis and is isomorphic to ℓ^2(ℕ), the space of square-summable sequences. Separability is a key physical assumption: Hilbert spaces used in quantum theory (e.g., for a single particle in a box or the harmonic oscillator) are typically separable, facilitating spectral expansions into basis vectors like eigenfunctions of the Hamiltonian. Gram–Schmidt orthonormalization constructs bases in finite or countable settings. Concepts of Schauder basis and Hamel basis differ in infinite dimensions and are relevant in mathematical subtleties.
Bounded and unbounded linear operators on Hilbert spaces represent physical transformations and observables. Self-adjoint operators correspond to measurable quantities via the spectral theorem, which generalizes diagonalization: a self-adjoint operator admits a spectral measure and can be represented through functional calculus. Key classes include bounded operators, compact operators, and densely defined unbounded operators such as differential operators arising from the Schrödinger equation. The classification of spectra (point, continuous, residual) underlies scattering theory and the study of resonances in quantum systems. Foundational contributions were made by John von Neumann and Marshall H. Stone.
The tensor product of Hilbert spaces provides the structure for composite quantum systems: H_total = H_A ⊗ H_B models systems A and B combined, with product states and entangled states emerging naturally. Tensor products require completion with respect to the Hilbert space norm; commonly used constructions include the algebraic tensor product and the Hilbert space completion yielding spaces isomorphic to ℓ^2 of product indices. Tensorial structure enables discussion of quantum entanglement, Bell's theorem, and operations in quantum information theory such as quantum teleportation and density matrix formalism employed by Nielsen and Chuang.
In the standard formulation of Quantum mechanics, pure states are unit vectors in a Hilbert space modulo global phase, or equivalently rays in projective Hilbert space. Mixed states are trace-class positive operators (density operators) on H. Observables are self-adjoint operators whose spectral measures determine probability distributions for measurement outcomes via the Born rule. Time evolution is given by unitary one-parameter groups generated by the Hamiltonian through Stone's theorem. Practically important models live in L^2(ℝ^n), Fock space for varying particle number, and spin spaces ℂ^2 for qubits used in Quantum computing research at institutions like IBM and Google.
Rigged Hilbert spaces (Gelfand triples) extend Hilbert spaces to include generalized eigenvectors and distributions, resolving formal uses of Dirac delta "kets" and continuous spectrum eigenfunctions. A rigged Hilbert space consists of a nested triple Φ ⊂ H ⊂ Φ', where Φ is a nuclear space of test vectors and Φ' its dual. This formalism, developed following work by Israel Gelfand and applied by Paul Dirac and John von Neumann, underpins scattering theory, resonance phenomena, and the rigorous treatment of unbounded operators and generalized eigenfunction expansions. Rigged spaces are widely used in mathematical physics, quantum field theory, and the theory of S-matrix and asymptotic states.