Μεταθέτης
commutator
Μεταθέτης
Μεταθέτης
Μεταθέτης
- Ένα Μαθηματικό Μέγεθος
Η ονομασία "Μεταθέτης " σχετίζεται ετυμολογικά με την λέξη "μετάθεση " .
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative.
There are different definitions used in group theory and ring theory .
[
A
^
,
B
^
]
=
A
^
B
^
−
B
^
A
^
{\displaystyle [\hat{A},\hat{B}] = \hat{A}\hat{B}-\hat{B}\hat{A}}
Ιδιότητες:
[
A
^
,
B
^
]
+
[
B
^
,
A
^
]
=
0
{\displaystyle [{\hat A}, {\hat B} ]+ [ {\hat B}, {\hat A} ] = 0 }
[
A
^
,
A
^
]
=
0
{\displaystyle [ {\hat A} , {\hat A} ]= 0
}
[
A
^
,
B
^
+
C
^
]
=
[
A
^
,
B
^
]
+
[
A
^
,
C
^
]
{\displaystyle [ \hat{A}, \hat{B} + \hat{C} ] = [ \hat{A}, \hat{B} ]
+ [ \hat{A}, \hat{C} ]}
[
A
^
+
B
^
,
C
^
]
=
[
A
^
,
C
^
]
+
[
B
^
,
C
^
]
{\displaystyle [ \hat{A} + \hat{B}, \hat{C} ]= [ \hat{A}, \hat{C} ]
+ [ \hat{B}, \hat{C} ]}
[
A
^
,
B
^
C
^
]
=
[
A
^
,
B
^
]
C
^
+
B
^
[
A
^
,
C
^
]
{\displaystyle [ \hat{A}, \hat{B} \hat{C} ]= [ \hat{A}, \hat{B} ]
\hat{C} + \hat{B} [\hat{A}, \hat{C} ]
}
[
A
^
B
^
,
C
^
]
=
[
A
^
,
C
^
]
B
^
+
A
^
[
B
^
,
C
^
]
{\displaystyle [ \hat{A} \hat{B}, \hat{C} ]= [ \hat{A}, \hat{C} ]
\hat{B} + \hat{A} [\hat{B}, \hat{C} ]}
[
A
^
,
[
B
^
,
C
^
]
]
+
[
C
^
,
[
A
^
,
B
^
]
]
+
[
B
^
,
[
C
^
,
A
^
]
]
=
0
{\displaystyle [ \hat{A}, [ \hat{B}, \hat{C} ] ]
+ [\hat{C}, [ \hat{A}, \hat{B}] ] +
[ \hat{B}, [ \hat{C}, \hat{A}] ] = 0 }
If
A
^
{\displaystyle \hat{A}}
B
^
{\displaystyle \hat{B} }
[
A
^
,
B
^
n
]
=
n
B
^
n
−
1
[
A
^
,
B
^
]
{\displaystyle [\hat{A}, \hat{B}^{n}]= n \hat{B}^{n-1} [\hat{A}, \hat{B}]
}
[
A
^
n
,
B
^
]
=
n
A
^
n
−
1
[
A
^
,
B
^
]
{\displaystyle [\hat{A}^{n}, \hat{B}]= n \hat{A}^{n-1} [\hat{A}, \hat{B}]
}
We also have the identity (useful for coupled-cluster theory)
e
A
^
B
^
e
−
A
^
=
B
^
+
[
A
^
,
B
^
]
+
1
2
!
[
A
^
,
[
A
^
,
B
^
]
+
1
3
!
[
A
^
,
[
A
^
,
[
A
^
,
B
^
]
]
]
+
⋯
{\displaystyle e^{\hat{A}} \hat{B} e^{-\hat{A}} = \hat{B} + [\hat{A},\hat{B}] +
\frac {1}{2!} [ \hat{A}, [ \hat{A},\hat{B}] + \frac {1}{3!} [ \hat{A}, [ \hat{A}, [ \hat{A}, \hat{B}] ] ] + \cdots
}
Finally, if
[
A
^
,
B
^
]
=
i
C
^
{\displaystyle [ \hat{A}, \hat{B} ] = i \hat{C}}
then the uncertainties in A and B, defined as:
Δ
A
2
=<
A
2
>
−
<
A
>
2
{\displaystyle \Delta A^2 = <A^2> - <A>^2}
obey the relation
(
Δ
A
)
(
Δ
B
)
≥
1
2
|
<
C
>
|
{\displaystyle
(\Delta A) (\Delta B) \geq \frac{1}{2} \vert<C>\vert
}
This is the famous Heisenberg uncertainty principle .
It is easy to derive the well-known relation
(
Δ
x
)
(
Δ
p
x
)
≥
ℏ
2
{\displaystyle
(\Delta x) (\Delta p_x) \geq \frac{\hbar}{2}
}
from this generalized rule.
i
ℏ
d
A
^
d
t
=
i
ℏ
∂
⟨
A
^
⟩
∂
t
+
[
A
^
,
H
^
]
{\displaystyle i \hbar \frac{\mathrm{d} \hat{A}}{\mathrm{d}t} = i\hbar \frac{\partial \langle\hat{A}\rangle}{\partial t} +[\hat{A},\hat{H}]}
[
L
^
2
,
H
^
]
=
0
{\displaystyle [\hat{L}^2,\hat{H}] = 0 }
[
L
→
^
,
H
^
]
=
0
{\displaystyle [\hat{\vec{L}},\hat{H}] = 0 }
[
H
^
,
r
→
^
]
=
i
ℏ
m
P
→
^
{\displaystyle [\hat{H},\hat{\vec{r}}] = \frac{i\hbar} {m} \hat{\vec{P}}}
[
L
→
^
2
,
L
^
x
]
=
0
^
{\displaystyle [\hat{\vec{L}}^2,\hat{L}_x] = \mathit{\hat {0}} }
[
L
→
^
2
,
L
^
y
]
=
0
^
{\displaystyle [\hat{\vec{L}}^2,\hat{L}_y] = \mathit{\hat {0}} }
[
L
→
^
2
,
L
^
z
]
=
0
^
{\displaystyle [\hat{\vec{L}}^2,\hat{L}_z] = \mathit{\hat {0}} }
[
x
^
,
P
^
x
]
=
i
ℏ
1
^
{\displaystyle [\hat{x},\hat{P}_x] = i\hbar \mathit{\hat {1}} }
[
y
^
,
P
^
y
]
=
i
ℏ
1
^
{\displaystyle [\hat{y},\hat{P}_y] = i\hbar \mathit{\hat {1}} }
[
z
^
,
P
^
z
]
=
i
ℏ
1
^
{\displaystyle [\hat{z},\hat{P}_z] = i\hbar \mathit{\hat {1}} }
[
x
^
,
y
^
]
=
0
^
{\displaystyle [\hat{x},\hat{y}] = \mathit{\hat {0}} }
[
P
^
x
,
P
^
y
]
=
0
^
{\displaystyle [\hat{P}_x,\hat{P}_y] = \mathit{\hat {0}} }
[
x
^
,
P
^
x
]
=
i
ℏ
1
^
{\displaystyle [\hat{x},\hat{P}_x] = i\hbar \mathit{\hat {1}} }
[
x
^
,
y
^
]
=
0
^
[
P
^
x
,
P
^
y
]
=
0
^
[
x
^
,
P
^
x
]
=
i
ℏ
1
^
{\displaystyle \begin{align}
\; [\hat{x},\hat{y}] = \mathit{\hat {0}} \; \\
\; [\hat{P}_x,\hat{P}_y] = \mathit{\hat {0}} \; \\
\; [\hat{x},\hat{P}_x] = i\hbar \mathit{\hat {1}}
\end{align}}
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>>Διαμαρτυρία προς την wikia - Όχι, στις διαφημίσεις που περιέχουν απαράδεκτο περιεχόμενο (άσεμνες εικόνες, ροζ αγγελίες κλπ.)
![{\displaystyle [{\hat {A}},{\hat {B}}]={\hat {A}}{\hat {B}}-{\hat {B}}{\hat {A}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/4e493b232beb5146fefec511c607b5e15d6fc378)
![{\displaystyle [{\hat {A}},{\hat {B}}]+[{\hat {B}},{\hat {A}}]=0}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/af481e1af0eaa7a2245d87b3bdbf9025b4f42de8)
![{\displaystyle [{\hat {A}},{\hat {A}}]=0}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/c5960e2d7edee6659220f3c1e64210344a855390)
![{\displaystyle [{\hat {A}},{\hat {B}}+{\hat {C}}]=[{\hat {A}},{\hat {B}}]+[{\hat {A}},{\hat {C}}]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/a637114315b8ae138bb7d3a4c4e1c6082a79d0ff)
![{\displaystyle [{\hat {A}}+{\hat {B}},{\hat {C}}]=[{\hat {A}},{\hat {C}}]+[{\hat {B}},{\hat {C}}]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/f1952809110a86b04c17f5367d52dc2aa6bc52e9)
![{\displaystyle [{\hat {A}},{\hat {B}}{\hat {C}}]=[{\hat {A}},{\hat {B}}]{\hat {C}}+{\hat {B}}[{\hat {A}},{\hat {C}}]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/47972ecc187073c5270bf0de14be20199f7a1636)
![{\displaystyle [{\hat {A}}{\hat {B}},{\hat {C}}]=[{\hat {A}},{\hat {C}}]{\hat {B}}+{\hat {A}}[{\hat {B}},{\hat {C}}]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/6be409f6a5f8832b56de6289b016be03260260cf)
![{\displaystyle [{\hat {A}},[{\hat {B}},{\hat {C}}]]+[{\hat {C}},[{\hat {A}},{\hat {B}}]]+[{\hat {B}},[{\hat {C}},{\hat {A}}]]=0}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/171e078b8c33c05a18799ce17117e6a01b7a31dd)
![{\displaystyle [{\hat {A}},{\hat {B}}^{n}]=n{\hat {B}}^{n-1}[{\hat {A}},{\hat {B}}]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/8a90e4269f784909bd07eb835c14d032a9128c34)
![{\displaystyle [{\hat {A}}^{n},{\hat {B}}]=n{\hat {A}}^{n-1}[{\hat {A}},{\hat {B}}]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/42ed08d3f307f7b69740a5da7df7296b0b2655d2)
![{\displaystyle e^{\hat {A}}{\hat {B}}e^{-{\hat {A}}}={\hat {B}}+[{\hat {A}},{\hat {B}}]+{\frac {1}{2!}}[{\hat {A}},[{\hat {A}},{\hat {B}}]+{\frac {1}{3!}}[{\hat {A}},[{\hat {A}},[{\hat {A}},{\hat {B}}]]]+\cdots }](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/eb08e68126436b267f62f75e46a4e4af434a312c)
![{\displaystyle [{\hat {A}},{\hat {B}}]=i{\hat {C}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/791d0c10e653e85d39f97f6dc3294c3b92064faa)
![{\displaystyle i\hbar {\frac {\mathrm {d} {\hat {A}}}{\mathrm {d} t}}=i\hbar {\frac {\partial \langle {\hat {A}}\rangle }{\partial t}}+[{\hat {A}},{\hat {H}}]}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/9a56955f0399739aa8d60dd9a6617e1069cd5a7a)
![{\displaystyle [{\hat {L}}^{2},{\hat {H}}]=0}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/d7ef2e3e22aa1110c433bca7c165300e97239d0c)
![{\displaystyle [{\hat {\vec {L}}},{\hat {H}}]=0}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/a2e3f42a8ede682ef7485602689ba81ac6fe9f79)
![{\displaystyle [{\hat {H}},{\hat {\vec {r}}}]={\frac {i\hbar }{m}}{\hat {\vec {P}}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/7a4a500b3e42092fe295da4e7417f8065ddff2e7)
![{\displaystyle [{\hat {\vec {L}}}^{2},{\hat {L}}_{x}]={\mathit {\hat {0}}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/d89f177599ad7f43c4799522a7068691ca567df8)
![{\displaystyle [{\hat {\vec {L}}}^{2},{\hat {L}}_{y}]={\mathit {\hat {0}}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/ff473cbcf6f8449dfbcdf41199113c2996ba41cd)
![{\displaystyle [{\hat {\vec {L}}}^{2},{\hat {L}}_{z}]={\mathit {\hat {0}}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/9653ecb683055341da25107782fa650382d69a86)
![{\displaystyle [{\hat {x}},{\hat {P}}_{x}]=i\hbar {\mathit {\hat {1}}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/4eab9fda37f9eab597adcdf365c89c7da4d8ab7e)
![{\displaystyle [{\hat {y}},{\hat {P}}_{y}]=i\hbar {\mathit {\hat {1}}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/b9f40d27636d62eba54cda92ac7688fc4143f4d2)
![{\displaystyle [{\hat {z}},{\hat {P}}_{z}]=i\hbar {\mathit {\hat {1}}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/423245df4d7daaa11a0028478b2d27fb898ea706)
![{\displaystyle [{\hat {x}},{\hat {y}}]={\mathit {\hat {0}}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/cfa4d55b19789e055400a2d8eabc92f7286b3dfa)
![{\displaystyle [{\hat {P}}_{x},{\hat {P}}_{y}]={\mathit {\hat {0}}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/fa92b3b47687bbb90debb56e891accb937f60883)
![{\displaystyle {\begin{aligned}\;[{\hat {x}},{\hat {y}}]={\mathit {\hat {0}}}\;\\\;[{\hat {P}}_{x},{\hat {P}}_{y}]={\mathit {\hat {0}}}\;\\\;[{\hat {x}},{\hat {P}}_{x}]=i\hbar {\mathit {\hat {1}}}\end{aligned}}}](https://services.fandom.com/mathoid-facade/v1/media/math/render/svg/a65d113bf60a36fa06eea7f299a938789b4fa7fe)
Κίνδυνοι Χρήσης
