| Dirac notation | |
|---|---|
| Name | Dirac notation |
| Field | Quantum mechanics |
| Introduced | 1930s |
| Introduced by | Paul Dirac |
| Related | Bra–ket notation, Hilbert space |
Dirac notation
Dirac notation, commonly called bra–ket notation, is a symbolic notation for vectors and linear functionals in Hilbert space that simplifies the formalism of Quantum mechanics. Introduced by Paul Dirac in the early 20th century, it unifies state vectors, linear operators, and inner products into a compact algebraic language widely used in theoretical physics, quantum information, and mathematical formulations of quantum theory.
Dirac notation originated in the work of Paul Dirac and contemporaries such as Werner Heisenberg and Erwin Schrödinger during the development of matrix mechanics and wave mechanics in the 1920s–1930s. It provided a concise way to connect matrix mechanics with wave function formulations and to express observables and amplitudes used in experiments like the Stern–Gerlach experiment. Influential texts that adopted and popularized the notation include works by John von Neumann and later textbooks by Dirac himself and by J. J. Sakurai. The notation became central to formal developments such as the spectral theorem and to institutional research at places like Cavendish Laboratory, Institute for Advanced Study, and Bell Labs where quantum theory was applied to atomic physics and early quantum electronics.
In Dirac notation a state vector is denoted by a "ket" |ψ› (written |ψ⟩), while the dual vector is written as a "bra" ‹φ| (written ⟨φ|). The pairing of a bra and a ket, the inner product, is written ⟨φ|ψ⟩ and yields a complex amplitude related to transition probabilities via the Born rule. Kets correspond to elements of a complex Hilbert space H; bras correspond to elements of the dual space H*. The notation emphasizes linearity and conjugate-linearity properties and is compatible with mathematical treatments by authors such as John von Neumann and within frameworks used at institutions like CERN and Los Alamos National Laboratory.
Linear operators acting on kets are written as Â|ψ⟩ or with operator symbols such as Ĥ for the Hamiltonian operator. The outer product |φ⟩⟨ψ| denotes a rank-one operator; sums of such outer products form projections and density operators used in statistical descriptions. Representing operators by matrices arises by choosing an orthonormal basis and mapping bras and kets to column and row vectors respectively, a procedure connected to the matrix formalism of Heisenberg and practical computations in quantum chemistry and solid-state physics. Important named operators include the Pauli matrices, the position operator, and the momentum operator which obey commutation relations exemplified by the canonical commutation relation [x,p]=iħ.
A complete orthonormal basis {|n⟩} satisfies ⟨m|n⟩ = δ_{mn} and the completeness relation ∑_n |n⟩⟨n| = I, where I is the identity operator. Continuous bases, such as the position |x⟩ and momentum |p⟩ eigenkets, satisfy orthogonality expressed with Dirac delta distributions and completeness via integrals, e.g., ∫ |x⟩⟨x| dx = I. These relations underpin expansions of wavefunctions ψ(x)=⟨x|ψ⟩ and spectral decompositions used in proofs of the spectral theorem and in scattering theory developed by researchers at Harvard University and Princeton University.
Composite systems are described by tensor products of Hilbert spaces, with product kets written |ψ⟩⊗|φ⟩ or more compactly |ψ,φ⟩ or |ψφ⟩. Dirac notation handles entanglement, partial traces, and reduced density matrices fundamental to quantum information theory and to experiments in quantum optics and ion trap platforms. Concepts such as Bell states, Schmidt decomposition, and quantum teleportation protocols are naturally expressed in bra–ket form; laboratories like MIT's Research Laboratory of Electronics and corporate research groups at IBM and Google exploit these tools in quantum computing research.
Dirac notation succinctly expresses time evolution, measurement, and state preparation. The Schrödinger equation iħ∂_t|ψ(t)⟩ = Ĥ|ψ(t)⟩ uses kets for time-dependent states. Projective measurements are represented by projection operators P_n = |n⟩⟨n| and generalised measurements by POVM elements E_i. Density operators ρ = ∑_i p_i |ψ_i⟩⟨ψ_i| describe mixed states; expectation values take the form ⟨A⟩ = Tr(ρÂ). Practical applications include computations in atomic physics, condensed matter physics, and protocols in quantum information science such as quantum error correction developed by groups at Caltech and Microsoft Research.
Mathematically rigorous treatments of bra–ket notation leverage functional analysis, the theory of distributions, and the concept of a rigged Hilbert space (Gelfand triplet) to justify bras and continuous-spectrum kets like |x⟩. The rigged Hilbert space framework, developed following ideas by I. M. Gelfand and applied in quantum theory, places H between a space of test functions Φ and its dual Φ*, giving meaning to generalized eigenvectors. This approach connects to the spectral decomposition of unbounded self-adjoint operators and underpins scattering theory and resonances studied in mathematical physics departments at institutions such as University of Cambridge and University of California, Berkeley.
Category:Quantum mechanics Category:Mathematical notation