ket

In quantum mechanics, bra–ket notation is a common notation for quantum states, i.e. vectors in a complex Hilbert space on which an algebra of observables acts. More generally the notation uses the angle brackets (the ⟨ and ⟩ symbols) and a vertical bar (the | symbol), for a ket (for example,




|

v
⟩


{\displaystyle |v\rangle }
) to denote a vector in an abstract (usually complex) vector space



V


{\displaystyle V}
and a bra, (for example,



⟨
f

|



{\displaystyle \langle f|}
) to denote a linear functional on



V


{\displaystyle V}
, i.e. a co-vector, an element of the dual vector space




V

∨




{\displaystyle V^{\vee }}
. The natural pairing of a linear functional



f
=
⟨
f

|



{\displaystyle f=\langle f|}
with a vector



v
=

|

v
⟩


{\displaystyle v=|v\rangle }
is then written as



⟨
f

|

v
⟩


{\displaystyle \langle f|v\rangle }
. On Hilbert spaces, the scalar product



(

,

)


{\displaystyle (\ ,\ )}
(with anti linear first argument) gives an (anti-linear) identification of a vector ket



ϕ
=

|

ϕ
⟩


{\displaystyle \phi =|\phi \rangle }
with a linear functional bra



(
ϕ
,

)
=
⟨
ϕ

|



{\displaystyle (\phi ,\ )=\langle \phi |}
. Using this notation, the scalar product



(
ϕ
,
ψ
)
=
⟨
ϕ

|

ψ
⟩


{\displaystyle (\phi ,\psi )=\langle \phi |\psi \rangle }
.
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 operators are interpreted using matrix multiplication. If





C


n




{\displaystyle \mathbb {C} ^{n}}
has the standard hermitian inner product



(
v
,
w
)
=

v

†


w


{\displaystyle (v,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



†


{\displaystyle \dagger }
.
It is common to suppress the vector or functional from the bra–ket notation and only use a label inside the typography for the bra or ket. For example, the spin operator




σ

z




{\displaystyle \sigma _{z}}
on a two dimensional space



Δ


{\displaystyle \Delta }
of spinors, has eigenvalues



±


{\displaystyle \pm }
½ with eigenspinors




ψ

+


,

ψ

−


∈
Δ


{\displaystyle \psi _{+},\psi _{-}\in \Delta }
. In bra-ket notation one typically denotes this as




ψ

+


=

|

+
⟩


{\displaystyle \psi _{+}=|+\rangle }
, and





ψ

−


=

|

−
⟩


{\displaystyle \psi _{-}=|-\rangle }
. Just 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 and is thus also known as the Dirac notation. (Still, the bra-ket notation has a precursor in Hermann Grassmann's use of the notation



[
ϕ

∣

ψ
]


{\displaystyle [\phi {\mid }\psi ]}
for his inner products nearly 100 years earlier.)

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