Math 2.0

Quantum theory Definition

Bra–ket notation

Builds on Hilbert spaceContinuous dual spaceRiesz representation theoremOrthonormal basis

Definition

Let ℋ\mathcal{H} be a complex Hilbert space whose inner product ⟨⋅,⋅⟩\ip{\cdot}{\cdot} is conjugate-linear in the first argument and linear in the second. Bra–ket notation consists of the following conventions.

  1. A ket ∣ψ⟩\lvert \psi \rangle is a vector ψ∈ℋ\psi \in \mathcal{H}.
  2. A bra ⟨φ∣\langle \varphi \rvert, for φ∈ℋ\varphi \in \mathcal{H}, is the continuous linear functional ψ↦⟨φ,ψ⟩\psi \mapsto \ip{\varphi}{\psi}, an element of the continuous dual of ℋ\mathcal{H}.
  3. The bracket ⟨φ∣ψ⟩\langle \varphi \vert \psi \rangle is the value of ⟨φ∣\langle \varphi \rvert at ∣ψ⟩\lvert \psi \rangle, that is, ⟨φ,ψ⟩\ip{\varphi}{\psi}.
  4. For an operator AA on ℋ\mathcal{H} and ψ\psi in its domain, ⟨φ∣A∣ψ⟩=⟨φ,Aψ⟩\langle \varphi \vert A \vert \psi \rangle = \ip{\varphi}{A\psi}.
  5. The ket-bra ∣ψ⟩⟨φ∣\lvert \psi \rangle\langle \varphi \rvert is the operator χ↦⟨φ,χ⟩ ψ\chi \mapsto \ip{\varphi}{\chi}\, \psi on ℋ\mathcal{H}.

Remarks

By the Riesz representation theorem, every continuous linear functional on ℋ\mathcal{H} is a bra ⟨φ∣\langle \varphi \rvert for exactly one φ\varphi. The map ∣φ⟩↦⟨φ∣\lvert \varphi \rangle \mapsto \langle \varphi \rvert is conjugate-linear.

For an orthonormal basis (ek)(e_k), the expansion ψ=∑k⟨ek,ψ⟩ek\psi = \sum_k \ip{e_k}{\psi} e_k reads ∑k∣ek⟩⟨ek∣=I\sum_k \lvert e_k \rangle\langle e_k \rvert = I, with convergence of the sum applied to each vector.

Example: in ℂ2\C^2, let ∣0⟩=(1,0)\lvert 0 \rangle = (1, 0) and ∣+⟩=(1,1)/2\lvert + \rangle = (1, 1)/\sqrt{2}. Then ⟨0∣+⟩=1/2\langle 0 \vert + \rangle = 1/\sqrt{2}, and ∣0⟩⟨0∣=diag(1,0)\lvert 0 \rangle\langle 0 \rvert = \mathrm{diag}(1, 0) is the orthogonal projection onto the first axis.

Non-example: physicists write ∣x⟩\lvert x \rangle for a "position eigenstate" with ⟨x∣ψ⟩=ψ(x)\langle x \vert \psi \rangle = \psi(x). It is not a ket in the sense of clause 1: no vector of L2(ℝ)L^2(\R) gives the functional ψ↦ψ(x)\psi \mapsto \psi(x), which is not even defined on all of L2(ℝ)L^2(\R). Such symbols are made precise as distributions, for example in a rigged Hilbert space.

Reading path

To understand this page from the ground up, read these 29 pages in order. The foundations come first.

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Closed set, concept, no evidence levelClosed setconceptContinuous dual space, concept, no evidence levelContinuous dual spaceconceptConvex set, concept, no evidence levelConvex setconceptHilbert space, concept, no evidence levelHilbert spaceconceptLinear form, concept, no evidence levelLinear formconceptOrthogonal complement, concept, no evidence levelOrthogonal complementconceptOrthonormal basis, concept, no evidence levelOrthonormal basisconceptVector space, definition, no evidence levelVector spacedefinitionInner product, definition, no evidence levelInner productdefinitionLinear map, definition, no evidence levelLinear mapdefinitionLinear subspace, definition, no evidence levelLinear subspacedefinitionNorm, definition, no evidence levelNormdefinitionInner product space, definition, no evidence levelInner product spacedefinitionOrthogonal vectors, definition, no evidence levelOrthogonal vectorsdefinitionEndomorphism, definition, no evidence levelEndomorphismdefinitionLinear form, definition, no evidence levelLinear formdefinitionSum of subspaces, definition, no evidence levelSum of subspacesdefinitionNorm induced by an inner product, definition, no evidence levelNorm induced by an in…definitionOrthogonal complement, definition, no evidence levelOrthogonal complementdefinitionRiesz representation theorem in finite dimensions, statement, no evidence levelRiesz representation …statementDirect sum of subspaces, definition, no evidence levelDirect sum of subspac…definitionParallelogram law, statement, no evidence levelParallelogram lawstatementA space is the direct sum of a subspace and its orthogonal complement, statement, no evidence levelA space is the direct…statementProjection, definition, no evidence levelProjectiondefinitionOrthogonal projection, definition, no evidence levelOrthogonal projectiondefinitionOrthogonal projection is the closest point, statement, no evidence levelOrthogonal projection…statementHilbert projection theorem, statement, no evidence levelHilbert projection th…statementRiesz representation theorem, statement, no evidence levelRiesz representation …statementBra–ket notation, definition, no evidence levelBra–ket notationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Closed set concept
  2. Continuous dual space concept
  3. Convex set concept
  4. Hilbert space concept
  5. Linear form concept
  6. Orthogonal complement concept
  7. Orthonormal basis concept
  8. Vector space definition
  9. Inner product definition
  10. Linear map definition
  11. Linear subspace definition
  12. Norm definition
  13. Inner product space definition
  14. Orthogonal vectors definition
  15. Endomorphism definition
  16. Linear form definition
  17. Sum of subspaces definition
  18. Norm induced by an inner product definition
  19. Orthogonal complement definition
  20. Riesz representation theorem in finite dimensions statement
  21. Direct sum of subspaces definition
  22. Parallelogram law statement
  23. A space is the direct sum of a subspace and its orthogonal complement statement
  24. Projection definition
  25. Orthogonal projection definition
  26. Orthogonal projection is the closest point statement
  27. Hilbert projection theorem statement
  28. Riesz representation theorem statement
  29. Bra–ket notation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Bra%E2%80%93ket_notation
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TagPY1C87VH5Y
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Kinddefinition
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MSC81Q10, 46C05
DefinesBra–ket notation
UsesHilbert space, Continuous dual space, Riesz representation theorem, Orthonormal basis
Changed3 November 2026

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