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In quantum mechanics, bra–ket notation, or Dirac notation, is used ubiquitously to denote quantum states. The notation uses angle brackets, ⟨ {\displaystyle \langle } and ⟩ {\displaystyle \rangle } , and a vertical bar | {\displaystyle |} , to construct "bras" and "kets".
A ket is of the form | v ⟩ {\displaystyle |v\rangle }. Mathematically it denotes a vector, v {\displaystyle {\boldsymbol {v}}} , in an abstract vector space V {\displaystyle V} , and physically it represents a state of some quantum system.
A bra is of the form ⟨ f | {\displaystyle \langle f|}. Mathematically it denotes a linear form f : V → C {\displaystyle f:V\to \mathbb {C} } , i.e. a linear map that maps each vector in V {\displaystyle V} to a number in the complex plane C {\displaystyle \mathbb {C} }. Letting the linear functional ⟨ f | {\displaystyle \langle f|} act on a vector | v ⟩ {\displaystyle |v\rangle } is written as ⟨ f | v ⟩ ∈ C {\displaystyle \langle f|v\rangle \in \mathbb {C} }.
Assume that on V {\displaystyle V} there exists an inner product {\displaystyle } with antilinear first argument, which makes V {\displaystyle V} an inner product space. Then with this inner product each vector ϕ ≡ | ϕ ⟩ {\displaystyle {\boldsymbol {\phi }}\equiv |\phi \rangle } can be identified with a corresponding linear form, by placing the vector in the anti-linear first slot of the inner product: ≡ ⟨ ϕ | {\displaystyle \equiv \langle \phi |}. The correspondence between these notations is then ≡ ⟨ ϕ | ψ ⟩ {\displaystyle \equiv \langle \phi |\psi \rangle }. The linear form ⟨ ϕ | {\displaystyle \langle \phi |} is a covector to | ϕ ⟩ {\displaystyle |\phi \rangle } , and the set of all covectors form a subspace of the dual vector space V ∨ {\displaystyle V^{\vee }} , to the initial vector space V {\displaystyle V}. The purpose of this linear form ⟨ ϕ | {\displaystyle \langle \phi |} can now be understood in terms of making projections on the state ϕ {\displaystyle {\boldsymbol {\phi }}} , to find how linearly dependent two states are, etc.