13 KiB
| title | chunk | source | category | tags | date_saved | instance |
|---|---|---|---|---|---|---|
| Bra–ket notation | 4/8 | https://en.wikipedia.org/wiki/Bra–ket_notation | reference | science, encyclopedia | 2026-05-05T14:40:03.882193+00:00 | kb-cron |
For instance, the momentum operator
p
^
{\displaystyle {\hat {\mathbf {p} }}}
has the following coordinate representation,
p
^
(
r
)
Ψ
(
r
)
=
def
⟨
r
|
p
^
|
Ψ
⟩
=
−
i
ℏ
∇
Ψ
(
r
)
.
{\displaystyle {\hat {\mathbf {p} }}(\mathbf {r} )~\Psi (\mathbf {r} )\ {\stackrel {\text{def}}{=}}\ \langle \mathbf {r} |{\hat {\mathbf {p} }}|\Psi \rangle =-i\hbar \nabla \Psi (\mathbf {r} )\,.}
One occasionally even encounters an expression such as
∇
|
Ψ
⟩
{\displaystyle \nabla |\Psi \rangle }
, though this is something of an abuse of notation. The differential operator must be understood to be an abstract operator, acting on kets, that has the effect of differentiating wavefunctions once the expression is projected onto the position basis,
∇
⟨
r
|
Ψ
⟩
,
{\displaystyle \nabla \langle \mathbf {r} |\Psi \rangle \,,}
even though, in the momentum basis, this operator amounts to a mere multiplication operator (by iħp). That is, to say,
⟨
r
|
p
^
=
−
i
ℏ
∇
⟨
r
|
,
{\displaystyle \langle \mathbf {r} |{\hat {\mathbf {p} }}=-i\hbar \nabla \langle \mathbf {r} |~,}
or
p
^
=
∫
d
3
r
|
r
⟩
(
−
i
ℏ
∇
)
⟨
r
|
.
{\displaystyle {\hat {\mathbf {p} }}=\int d^{3}\mathbf {r} ~|\mathbf {r} \rangle (-i\hbar \nabla )\langle \mathbf {r} |~.}
=== Overlap of states === In quantum mechanics the expression ⟨φ|ψ⟩ is typically interpreted as the probability amplitude for the state ψ to collapse into the state φ. Mathematically, this means the coefficient for the projection of ψ onto φ. It is also described as the projection of state ψ onto state φ.
=== Changing basis for a spin-1/2 particle === A stationary spin-1⁄2 particle has a two-dimensional Hilbert space. One orthonormal basis is:
|
↑
z
⟩
,
|
↓
z
⟩
{\displaystyle |{\uparrow }_{z}\rangle \,,\;|{\downarrow }_{z}\rangle }
where |↑z⟩ is the state with a definite value of the spin operator Sz equal to +1⁄2 and |↓z⟩ is the state with a definite value of the spin operator Sz equal to −1⁄2. Since these are a basis, any quantum state of the particle can be expressed as a linear combination (i.e., quantum superposition) of these two states:
|
ψ
⟩
=
a
ψ
|
↑
z
⟩
+
b
ψ
|
↓
z
⟩
{\displaystyle |\psi \rangle =a_{\psi }|{\uparrow }_{z}\rangle +b_{\psi }|{\downarrow }_{z}\rangle }
where aψ and bψ are complex numbers. A different basis for the same Hilbert space is:
|
↑
x
⟩
,
|
↓
x
⟩
{\displaystyle |{\uparrow }_{x}\rangle \,,\;|{\downarrow }_{x}\rangle }
defined in terms of Sx rather than Sz. Again, any state of the particle can be expressed as a linear combination of these two:
|
ψ
⟩
=
c
ψ
|
↑
x
⟩
+
d
ψ
|
↓
x
⟩
{\displaystyle |\psi \rangle =c_{\psi }|{\uparrow }_{x}\rangle +d_{\psi }|{\downarrow }_{x}\rangle }
In vector form, you might write
|
ψ
⟩
≐
(
a
ψ
b
ψ
)
or
|
ψ
⟩
≐
(
c
ψ
d
ψ
)
{\displaystyle |\psi \rangle \doteq {\begin{pmatrix}a_{\psi }\\b_{\psi }\end{pmatrix}}\quad {\text{or}}\quad |\psi \rangle \doteq {\begin{pmatrix}c_{\psi }\\d_{\psi }\end{pmatrix}}}
depending on which basis you are using. In other words, the "coordinates" of a vector depend on the basis used. There is a mathematical relationship between
a
ψ
{\displaystyle a_{\psi }}
,
b
ψ
{\displaystyle b_{\psi }}
,
c
ψ
{\displaystyle c_{\psi }}
and
d
ψ
{\displaystyle d_{\psi }}
; see change of basis.
== Pitfalls and ambiguous uses == There are some conventions and uses of notation that may be confusing or ambiguous for the non-initiated or early student.
=== Separation of inner product and vectors === A cause for confusion is that the notation does not separate the inner-product operation from the notation for a (bra) vector. If a (dual space) bra-vector is constructed as a linear combination of other bra-vectors (for instance when expressing it in some basis) the notation creates some ambiguity and hides mathematical details. We can compare bra–ket notation to using bold for vectors, such as
ψ
{\displaystyle {\boldsymbol {\psi }}}
, and
(
⋅
,
⋅
)
{\displaystyle (\cdot ,\cdot )}
for the inner product. Consider the following dual space bra-vector in the basis
{
|
e
n
⟩
}
{\displaystyle \{|e_{n}\rangle \}}
, where
{
ψ
n
}
{\displaystyle \{\psi _{n}\}}
are the complex number coefficients of
⟨
ψ
|
{\displaystyle \langle \psi |}
:
⟨
ψ
|
=
∑
n
⟨
e
n
|
ψ
n
{\displaystyle \langle \psi |=\sum _{n}\langle e_{n}|\psi _{n}}
It has to be determined by convention if the complex numbers
{
ψ
n
}
{\displaystyle \{\psi _{n}\}}
are inside or outside of the inner product, and each convention gives different results.
⟨
ψ
|
≡
(
ψ
,
⋅
)
=
∑
n
(
e
n
,
⋅
)
ψ
n
{\displaystyle \langle \psi |\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}}
⟨
ψ
|
≡
(
ψ
,
⋅
)
=
∑
n
(
e
n
ψ
n
,
⋅
)
=
∑
n
(
e
n
,
⋅
)
ψ
n
∗
{\displaystyle \langle \psi |\equiv ({\boldsymbol {\psi }},\cdot )=\sum _{n}({\boldsymbol {e}}_{n}\psi _{n},\cdot )=\sum _{n}({\boldsymbol {e}}_{n},\cdot )\,\psi _{n}^{*}}
=== Reuse of symbols === It is common to use the same symbol for labels and constants. For example,
α
^
|
α
⟩
=
α
|
α
⟩
{\displaystyle {\hat {\alpha }}|\alpha \rangle =\alpha |\alpha \rangle }
, where the symbol
α
{\displaystyle \alpha }
is used simultaneously as the name of the operator
α
^
{\displaystyle {\hat {\alpha }}}
, its eigenvector
|
α
⟩
{\displaystyle |\alpha \rangle }
and the associated eigenvalue
α
{\displaystyle \alpha }
. Sometimes the hat is also dropped for operators, and one can see notation such as
A
|
a
⟩
=
a
|
a
⟩
{\displaystyle A|a\rangle =a|a\rangle }
.