10 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bra–ket notation | 2/8 | https://en.wikipedia.org/wiki/Bra–ket_notation | reference | science, encyclopedia | 2026-05-05T14:40:03.882193+00:00 | kb-cron |
=== Vectors vs kets === In mathematics, the term "vector" is used for an element of any vector space. In physics, however, the term "vector" tends to refer almost exclusively to quantities like displacement or velocity, which have components that relate directly to the three dimensions of space, or relativistically, to the four of spacetime. Such vectors are typically denoted with over arrows (
r
→
{\displaystyle {\vec {r}}}
), boldface (
p
{\displaystyle \mathbf {p} }
) or indices (
v
μ
{\displaystyle v^{\mu }}
). In quantum mechanics, a quantum state is typically represented as an element of a complex Hilbert space, for example, the infinite-dimensional vector space of all possible wavefunctions (square integrable functions mapping each point of 3D space to a complex number) or some more abstract Hilbert space constructed more algebraically. To distinguish this type of vector from those described above, it is common and useful in physics to denote an element
ϕ
{\displaystyle \phi }
of an abstract complex vector space as a ket
|
ϕ
⟩
{\displaystyle |\phi \rangle }
, to refer to it as a "ket" rather than as a vector, and to pronounce it "ket-
ϕ
{\displaystyle \phi }
" or "ket-A" for |A⟩. Symbols, letters, numbers, or even words—whatever serves as a convenient label—can be used as the label inside a ket, with the
|
⟩
{\displaystyle |\ \rangle }
making clear that the label indicates a vector in vector space. In other words, the symbol "|A⟩" has a recognizable mathematical meaning as to the kind of variable being represented, while just the "A" by itself does not. For example, |1⟩ + |2⟩ is not necessarily equal to |3⟩. Nevertheless, for convenience, there is usually some logical scheme behind the labels inside kets, such as the common practice of labeling energy eigenkets in quantum mechanics through a listing of their quantum numbers. At its simplest, the label inside the ket is the eigenvalue of a physical operator, such as
x
^
{\displaystyle {\hat {x}}}
,
p
^
{\displaystyle {\hat {p}}}
,
L
^
z
{\displaystyle {\hat {L}}_{z}}
, etc.
=== Notation === Since kets are just vectors in a Hermitian vector space, they can be manipulated using the usual rules of linear algebra. For example:
|
A
⟩
=
|
B
⟩
+
|
C
⟩
|
C
⟩
=
(
−
1
+
2
i
)
|
D
⟩
|
D
⟩
=
∫
−
∞
∞
e
−
x
2
|
x
⟩
d
x
.
{\displaystyle {\begin{aligned}|A\rangle &=|B\rangle +|C\rangle \\|C\rangle &=(-1+2i)|D\rangle \\|D\rangle &=\int _{-\infty }^{\infty }e^{-x^{2}}|x\rangle \,\mathrm {d} x\,.\end{aligned}}}
Note how the last line above involves infinitely many different kets, one for each real number x. Since the ket is an element of a vector space, a bra
⟨
A
|
{\displaystyle \langle A|}
is an element of its dual space, i.e. a bra is a linear functional which is a linear map from the vector space to the complex numbers. Thus, it is useful to think of kets and bras as being elements of different vector spaces (see below however) with both being different useful concepts. A bra
⟨
ϕ
|
{\displaystyle \langle \phi |}
and a ket
|
ψ
⟩
{\displaystyle |\psi \rangle }
(i.e. a functional and a vector), can be combined to an operator
|
ψ
⟩
⟨
ϕ
|
{\displaystyle |\psi \rangle \langle \phi |}
of rank one with outer product
|
ψ
⟩
⟨
ϕ
|
:
|
ξ
⟩
↦
|
ψ
⟩
⟨
ϕ
|
ξ
⟩
.
{\displaystyle |\psi \rangle \langle \phi |\colon |\xi \rangle \mapsto |\psi \rangle \langle \phi |\xi \rangle ~.}
=== Inner product and bra–ket identification on Hilbert space ===
Bra–ket notation is particularly useful in Hilbert spaces which have an inner product that allows Hermitian conjugation and identifying a vector with a continuous linear functional, i.e. a ket with a bra, and vice versa (see Riesz representation theorem). The inner product on Hilbert space
(
,
)
{\displaystyle (\ ,\ )}
(with the first argument anti linear as preferred by physicists) is fully equivalent to an (anti-linear) identification between the space of kets and that of bras in bra–ket notation: for a vector ket
ψ
=
|
ψ
⟩
{\displaystyle \psi =|\psi \rangle }
define a functional (i.e. bra)
f
ϕ
=
⟨
ϕ
|
{\displaystyle f_{\phi }=\langle \phi |}
by
(
ϕ
,
ψ
)
=
(
|
ϕ
⟩
,
|
ψ
⟩
)
=:
f
ϕ
(
ψ
)
=
⟨
ϕ
|
(
|
ψ
⟩
)
=:
⟨
ϕ
∣
ψ
⟩
{\displaystyle (\phi ,\psi )=(|\phi \rangle ,|\psi \rangle )=:f_{\phi }(\psi )=\langle \phi |\,{\bigl (}|\psi \rangle {\bigr )}=:\langle \phi {\mid }\psi \rangle }
==== Bras and kets as row and column vectors ==== In the simple case where we consider the vector space
C
n
{\displaystyle \mathbb {C} ^{n}}
, a ket can be identified with a column vector, and a bra as a row vector. If, moreover, we use the standard Hermitian inner product on
C
n
{\displaystyle \mathbb {C} ^{n}}
, the bra corresponding to a ket, in particular a bra ⟨m| and a ket |m⟩ with the same label are conjugate transpose. Moreover, conventions are set up in such a way that writing bras, kets, and linear operators next to each other simply imply matrix multiplication. In particular the outer product
|
ψ
⟩
⟨
ϕ
|
{\displaystyle |\psi \rangle \langle \phi |}
of a column and a row vector ket and bra can be identified with matrix multiplication (column vector times row vector equals matrix). For a finite-dimensional vector space, using a fixed orthonormal basis, the inner product can be written as a matrix multiplication of a row vector with a column vector: