15 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bra–ket notation | 7/8 | https://en.wikipedia.org/wiki/Bra–ket_notation | reference | science, encyclopedia | 2026-05-05T14:40:03.882193+00:00 | 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 )}\,.}