15 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bra–ket notation | 5/8 | https://en.wikipedia.org/wiki/Bra–ket_notation | reference | science, encyclopedia | 2026-05-05T14:40:03.882193+00:00 | kb-cron |
=== Hermitian conjugate of kets === It is common to see the usage
|
ψ
⟩
†
=
⟨
ψ
|
{\displaystyle |\psi \rangle ^{\dagger }=\langle \psi |}
, where the dagger (
†
{\displaystyle \dagger }
) corresponds to the Hermitian conjugate. This is however not correct in a technical sense, since the ket,
|
ψ
⟩
{\displaystyle |\psi \rangle }
, represents a vector in a complex Hilbert-space
H
{\displaystyle {\mathcal {H}}}
, and the bra,
⟨
ψ
|
{\displaystyle \langle \psi |}
, is a linear functional on vectors in
H
{\displaystyle {\mathcal {H}}}
. In other words,
|
ψ
⟩
{\displaystyle |\psi \rangle }
is just a vector, while
⟨
ψ
|
{\displaystyle \langle \psi |}
is the combination of a vector and an inner product.
=== Operations inside bras and kets === This is done for a fast notation of scaling vectors. For instance, if the vector
|
α
⟩
{\displaystyle |\alpha \rangle }
is scaled by
1
/
2
{\displaystyle 1/{\sqrt {2}}}
, it may be denoted
|
α
/
2
⟩
{\displaystyle |\alpha /{\sqrt {2}}\rangle }
. This can be ambiguous since
α
{\displaystyle \alpha }
is simply a label for a state, and not a mathematical object on which operations can be performed. This usage is more common when denoting vectors as tensor products, where part of the labels are moved outside the designed slot, e.g.
|
α
⟩
=
|
α
/
2
⟩
1
⊗
|
α
/
2
⟩
2
{\displaystyle |\alpha \rangle =|\alpha /{\sqrt {2}}\rangle _{1}\otimes |\alpha /{\sqrt {2}}\rangle _{2}}
.
== Linear operators ==
=== Linear operators acting on kets === A linear operator is a map that inputs a ket and outputs a ket. (In order to be called "linear", it is required to have certain properties.) In other words, if
A
^
{\displaystyle {\hat {A}}}
is a linear operator and
|
ψ
⟩
{\displaystyle |\psi \rangle }
is a ket-vector, then
A
^
|
ψ
⟩
{\displaystyle {\hat {A}}|\psi \rangle }
is another ket-vector. In an
N
{\displaystyle N}
-dimensional Hilbert space, we can impose a basis on the space and represent
|
ψ
⟩
{\displaystyle |\psi \rangle }
in terms of its coordinates as a
N
×
1
{\displaystyle N\times 1}
column vector. Using the same basis for
A
^
{\displaystyle {\hat {A}}}
, it is represented by an
N
×
N
{\displaystyle N\times N}
complex matrix. The ket-vector
A
^
|
ψ
⟩
{\displaystyle {\hat {A}}|\psi \rangle }
can now be computed by matrix multiplication. Linear operators are ubiquitous in the theory of quantum mechanics. For example, observable physical quantities are represented by self-adjoint operators, such as energy or momentum, whereas transformative processes are represented by unitary linear operators such as rotation or the progression of time.
=== Linear operators acting on bras === Operators can also be viewed as acting on bras from the right hand side. Specifically, if A is a linear operator and ⟨φ| is a bra, then ⟨φ|A is another bra defined by the rule
(
⟨
ϕ
|
A
)
|
ψ
⟩
=
⟨
ϕ
|
(
A
|
ψ
⟩
)
,
{\displaystyle {\bigl (}\langle \phi |{\boldsymbol {A}}{\bigr )}|\psi \rangle =\langle \phi |{\bigl (}{\boldsymbol {A}}|\psi \rangle {\bigr )}\,,}
(in other words, a function composition). This expression is commonly written as (cf. energy inner product)
⟨
ϕ
|
A
|
ψ
⟩
.
{\displaystyle \langle \phi |{\boldsymbol {A}}|\psi \rangle \,.}
In an N-dimensional Hilbert space, ⟨φ| can be written as a 1 × N row vector, and A (as in the previous section) is an N × N matrix. Then the bra ⟨φ|A can be computed by normal matrix multiplication. If the same state vector appears on both bra and ket side,
⟨
ψ
|
A
|
ψ
⟩
,
{\displaystyle \langle \psi |{\boldsymbol {A}}|\psi \rangle \,,}
then this expression gives the expectation value, or mean or average value, of the observable represented by operator A for the physical system in the state |ψ⟩.
=== Outer products === A convenient way to define linear operators on a Hilbert space H is given by the outer product: if ⟨ϕ| is a bra and |ψ⟩ is a ket, the outer product
|
ϕ
⟩
⟨
ψ
|
{\displaystyle |\phi \rangle \,\langle \psi |}
denotes the rank-one operator with the rule
(
|
ϕ
⟩
⟨
ψ
|
)
(
x
)
=
⟨
ψ
|
x
⟩
|
ϕ
⟩
.
{\displaystyle {\bigl (}|\phi \rangle \langle \psi |{\bigr )}(x)=\langle \psi |x\rangle |\phi \rangle .}
For a finite-dimensional vector space, the outer product can be understood as simple matrix multiplication:
|
ϕ
⟩
⟨
ψ
|
≐
(
ϕ
1
ϕ
2
⋮
ϕ
N
)
(
ψ
1
∗
ψ
2
∗
⋯
ψ
N
∗
)
=
(
ϕ
1
ψ
1
∗
ϕ
1
ψ
2
∗
⋯
ϕ
1
ψ
N
∗
ϕ
2
ψ
1
∗
ϕ
2
ψ
2
∗
⋯
ϕ
2
ψ
N
∗
⋮
⋮
⋱
⋮
ϕ
N
ψ
1
∗
ϕ
N
ψ
2
∗
⋯
ϕ
N
ψ
N
∗
)
{\displaystyle |\phi \rangle \,\langle \psi |\doteq {\begin{pmatrix}\phi _{1}\\\phi _{2}\\\vdots \\\phi _{N}\end{pmatrix}}{\begin{pmatrix}\psi _{1}^{*}&\psi _{2}^{*}&\cdots &\psi _{N}^{*}\end{pmatrix}}={\begin{pmatrix}\phi _{1}\psi _{1}^{*}&\phi _{1}\psi _{2}^{*}&\cdots &\phi _{1}\psi _{N}^{*}\\\phi _{2}\psi _{1}^{*}&\phi _{2}\psi _{2}^{*}&\cdots &\phi _{2}\psi _{N}^{*}\\\vdots &\vdots &\ddots &\vdots \\\phi _{N}\psi _{1}^{*}&\phi _{N}\psi _{2}^{*}&\cdots &\phi _{N}\psi _{N}^{*}\end{pmatrix}}}
The outer product is an N × N matrix, as expected for a linear operator. One of the uses of the outer product is to construct projection operators. Given a ket |ψ⟩ of norm 1, the orthogonal projection onto the subspace spanned by |ψ⟩ is
|
ψ
⟩
⟨
ψ
|
.
{\displaystyle |\psi \rangle \,\langle \psi |\,.}
This is an idempotent in the algebra of observables that acts on the Hilbert space.
=== Hermitian conjugate operator ===