In quantum mechanics, bra–ket notation, or Dirac notation, is ubiquitous. The notation uses the angle brackets, "
⟨
{\displaystyle \langle }
" and "
⟩
{\displaystyle \rangle }
", and a vertical bar "
|
{\displaystyle |}
", to construct "bras" and "kets" .
A ket looks like "
|
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 looks like "
⟨
f
|
{\displaystyle \langle f|}
", and 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 on
V
{\displaystyle V}
exists an inner product
(
⋅
,
⋅
)
{\displaystyle (\cdot ,\cdot )}
with antilinear first argument, which makes
V
{\displaystyle V}
a Hilbert 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 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.
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 ({\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 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 {\hat {\sigma }}_{z}}
on a two dimensional space
Δ
{\displaystyle \Delta }
of spinors, has eigenvalues
±
{\displaystyle \pm }
½ with eigenspinors
ψ
+
,
ψ
−
∈
Δ
{\displaystyle {\boldsymbol {\psi }}_{+},{\boldsymbol {\psi }}_{-}\in \Delta }
. In bra-ket notation one typically denotes this as
ψ
+
=
|
+
⟩
{\displaystyle {\boldsymbol {\psi }}_{+}=|+\rangle }
, and
ψ
−
=
|
−
⟩
{\displaystyle {\boldsymbol {\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.)