kb/data/en.wikipedia.org/wiki/Bra–ket_notation-0.md

title	chunk	source	category	tags	date_saved	instance
Bra–ket notation	1/8	https://en.wikipedia.org/wiki/Bra–ket_notation	reference	science, encyclopedia	2026-05-05T14:40:03.882193+00:00	kb-cron

Bra–ket notation or Dirac notation is a notation for linear algebra and linear operators on complex vector spaces together with their dual spaces both in the finite- and infinite-dimensional cases. It is specifically designed to ease the types of calculations that frequently arise in quantum mechanics. It is now of ubiquitous usage in that subject. Bra–ket notation was created by Paul Dirac in his paper, "A New Notation for Quantum Mechanics" from 1939. The name comes from the English word bracket.

== Quantum mechanics == In quantum mechanics and quantum computing, bra–ket 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 (complex) 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 (\cdot ,\cdot )}

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
    
    ,
    ⋅
    )
    ≡
    ⟨
    ϕ
    
      |
    
    .
  

{\displaystyle ({\boldsymbol {\phi }},\cdot )\equiv \langle \phi |.}

The correspondence between these notations is then

    (
    
      ϕ
    
    ,
    
      ψ
    
    )
    ≡
    ⟨
    ϕ
    
      |
    
    ψ
    ⟩
  

{\displaystyle ({\boldsymbol {\phi }},{\boldsymbol {\psi }})\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 forms 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 onto the state

      ϕ
    
    ,
  

{\displaystyle {\boldsymbol {\phi }},}

to find how linearly dependent two states are, etc. For the vector space

        C
      
      
        n
      
    
  

{\displaystyle \mathbb {C} ^{n}}

, kets can be identified with column vectors, and bras with row vectors. Combinations of bras, kets, and linear operators are interpreted using matrix multiplication. If

        C
      
      
        n
      
    
  

{\displaystyle \mathbb {C} ^{n}}

has the standard Hermitian inner product

    (
    
      v
    
    ,
    
      w
    
    )
    =
    
      v
      
        †
      
    
    w
  

{\displaystyle ({\boldsymbol {v}},{\boldsymbol {w}})=v^{\dagger }w}

, under this identification, the identification of kets and bras and vice versa provided by the inner product is taking the Hermitian conjugate (denoted

    †
  

{\displaystyle \dagger }

). It is common to suppress the vector or linear form from bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator

            σ
            ^
          
        
      
      
        z
      
    
  

{\displaystyle {\hat {\sigma }}_{z}}

on a two-dimensional space

    Δ
  

{\displaystyle \Delta }

of spinors has eigenvalues

    ±
    
      
        1
        2
      
    
  

{\textstyle \pm {\frac {1}{2}}}

with eigenspinors

        ψ
      
      
        +
      
    
    ,
    
      
        ψ
      
      
        −
      
    
    ∈
    Δ
  

{\displaystyle {\boldsymbol {\psi }}_{+},{\boldsymbol {\psi }}_{-}\in \Delta }

. In bra–ket notation, this is typically denoted as

        ψ
      
      
        +
      
    
    =
    
      |
    
    +
    ⟩
  

{\displaystyle {\boldsymbol {\psi }}_{+}=|+\rangle }

, and

        ψ
      
      
        −
      
    
    =
    
      |
    
    −
    ⟩
  

{\displaystyle {\boldsymbol {\psi }}_{-}=|-\rangle }

. As above, kets and bras with the same label are interpreted as kets and bras corresponding to each other using the inner product. In particular, when also identified with row and column vectors, kets and bras with the same label are identified with Hermitian conjugate column and row vectors. Bra–ket notation was effectively established in 1939 by Paul Dirac; it is thus also known as Dirac notation, despite the notation having a precursor in Hermann Grassmann's use of

    [
    ϕ
    
      ∣
    
    ψ
    ]
  

{\displaystyle [\phi {\mid }\psi ]}

for inner products nearly 100 years earlier.

== Vector spaces ==