--- title: "Bra–ket notation" chunk: 7/8 source: "https://en.wikipedia.org/wiki/Bra–ket_notation" category: "reference" tags: "science, encyclopedia" date_saved: "2026-05-05T14:40:03.882193+00:00" instance: "kb-cron" --- | ψ ⟩ | ϕ ⟩ , | ψ ⟩ ⊗ | ϕ ⟩ , | ψ ϕ ⟩ , | ψ , ϕ ⟩ . {\displaystyle |\psi \rangle |\phi \rangle \,,\quad |\psi \rangle \otimes |\phi \rangle \,,\quad |\psi \phi \rangle \,,\quad |\psi ,\phi \rangle \,.} See quantum entanglement and the EPR paradox for applications of this product. == The unit operator == Consider a complete orthonormal system (basis), { e i | i ∈ N } , {\displaystyle \{e_{i}\ |\ i\in \mathbb {N} \}\,,} for a Hilbert space H, with respect to the norm from an inner product ⟨·,·⟩. From basic functional analysis, it is known that any ket | ψ ⟩ {\displaystyle |\psi \rangle } can also be written as | ψ ⟩ = ∑ i ∈ N ⟨ e i | ψ ⟩ | e i ⟩ , {\displaystyle |\psi \rangle =\sum _{i\in \mathbb {N} }\langle e_{i}|\psi \rangle |e_{i}\rangle ,} with ⟨·|·⟩ the inner product on the Hilbert space. From the commutativity of kets with (complex) scalars, it follows that ∑ i ∈ N | e i ⟩ ⟨ e i | = I {\displaystyle \sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|=\mathbb {I} } must be the identity operator, which sends each vector to itself. This, then, can be inserted in any expression without affecting its value; for example ⟨ v | w ⟩ = ⟨ v | ( ∑ i ∈ N | e i ⟩ ⟨ e i | ) | w ⟩ = ⟨ v | ( ∑ i ∈ N | e i ⟩ ⟨ e i | ) ( ∑ j ∈ N | e j ⟩ ⟨ e j | ) | w ⟩ = ⟨ v | e i ⟩ ⟨ e i | e j ⟩ ⟨ e j | w ⟩ , {\displaystyle {\begin{aligned}\langle v|w\rangle &=\langle v|\left(\sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|\right)|w\rangle \\&=\langle v|\left(\sum _{i\in \mathbb {N} }|e_{i}\rangle \langle e_{i}|\right)\left(\sum _{j\in \mathbb {N} }|e_{j}\rangle \langle e_{j}|\right)|w\rangle \\&=\langle v|e_{i}\rangle \langle e_{i}|e_{j}\rangle \langle e_{j}|w\rangle \,,\end{aligned}}} where, in the last line, the Einstein summation convention has been used to avoid clutter. In quantum mechanics, it often occurs that little or no information about the inner product ⟨ψ|φ⟩ of two arbitrary (state) kets is present, while it is still possible to say something about the expansion coefficients ⟨ψ|ei⟩ = ⟨ei|ψ⟩* and ⟨ei|φ⟩ of those vectors with respect to a specific (orthonormalized) basis. In this case, it is particularly useful to insert the unit operator into the bracket one time or more. For more information, see Resolution of the identity, I = ∫ d x | x ⟩ ⟨ x | = ∫ d p | p ⟩ ⟨ p | , {\displaystyle {\mathbb {I} }=\int \!dx~|x\rangle \langle x|=\int \!dp~|p\rangle \langle p|,} where | p ⟩ = ∫ d x e i x p / ℏ | x ⟩ 2 π ℏ . {\displaystyle |p\rangle =\int dx{\frac {e^{ixp/\hbar }|x\rangle }{\sqrt {2\pi \hbar }}}.} Since ⟨x′|x⟩ = δ(x − x′), plane waves follow, ⟨ x | p ⟩ = e i x p / ℏ 2 π ℏ . {\displaystyle \langle x|p\rangle ={\frac {e^{ixp/\hbar }}{\sqrt {2\pi \hbar }}}.} In his book (1958), Ch. III.20, Dirac defines the standard ket which, up to a normalization, is the translationally invariant momentum eigenstate | ϖ ⟩ = lim p → 0 | p ⟩ {\textstyle |\varpi \rangle =\lim _{p\to 0}|p\rangle } in the momentum representation, i.e., p ^ | ϖ ⟩ = 0 {\displaystyle {\hat {p}}|\varpi \rangle =0} . Consequently, the corresponding wavefunction is a constant, ⟨ x | ϖ ⟩ 2 π ℏ = 1 {\displaystyle \langle x|\varpi \rangle {\sqrt {2\pi \hbar }}=1} , and | x ⟩ = δ ( x ^ − x ) | ϖ ⟩ 2 π ℏ , {\displaystyle |x\rangle =\delta ({\hat {x}}-x)|\varpi \rangle {\sqrt {2\pi \hbar }},} as well as | p ⟩ = exp ⁡ ( i p x ^ / ℏ ) | ϖ ⟩ . {\displaystyle |p\rangle =\exp(ip{\hat {x}}/\hbar )|\varpi \rangle .} Typically, when all matrix elements of an operator such as ⟨ x | A | y ⟩ {\displaystyle \langle x|A|y\rangle } are available, this resolution serves to reconstitute the full operator, ∫ d x d y | x ⟩ ⟨ x | A | y ⟩ ⟨ y | = A . {\displaystyle \int dx\,dy\,|x\rangle \langle x|A|y\rangle \langle y|=A\,.} == Notation used by mathematicians == The object physicists are considering when using bra–ket notation is a Hilbert space (a complete inner product space). Let ( H , ⟨ ⋅ , ⋅ ⟩ ) {\displaystyle ({\mathcal {H}},\langle \cdot ,\cdot \rangle )} be a Hilbert space and h ∈ H a vector in H. What physicists would denote by |h⟩ is the vector itself. That is, | h ⟩ ∈ H . {\displaystyle |h\rangle \in {\mathcal {H}}.} Let H* be the dual space of H. This is the space of linear functionals on H. The embedding Φ : H ↪ H ∗ {\displaystyle \Phi :{\mathcal {H}}\hookrightarrow {\mathcal {H}}^{*}} is defined by Φ ( h ) = φ h {\displaystyle \Phi (h)=\varphi _{h}} , where for every h ∈ H the linear functional φ h : H → C {\displaystyle \varphi _{h}:{\mathcal {H}}\to \mathbb {C} } satisfies for every g ∈ H the functional equation φ h ( g ) = ⟨ h , g ⟩ = ⟨ h ∣ g ⟩ {\displaystyle \varphi _{h}(g)=\langle h,g\rangle =\langle h\mid g\rangle } . Notational confusion arises when identifying φh and g with ⟨h| and |g⟩ respectively. This is because of literal symbolic substitutions. Let φ h = H = ⟨ h ∣ {\displaystyle \varphi _{h}=H=\langle h\mid } and let g = G = |g⟩. This gives φ h ( g ) = H ( g ) = H ( G ) = ⟨ h | ( G ) = ⟨ h | ( | g ⟩ ) . {\displaystyle \varphi _{h}(g)=H(g)=H(G)=\langle h|(G)=\langle h|{\bigl (}|g\rangle {\bigr )}\,.}