Commutator Algebra. The commutator of two operators is denoted by where.
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Transcript of Commutator Algebra. The commutator of two operators is denoted by where.
Commutator Algebra
Commutator Algebra
The commutator of two operators is denoted by where
,A B]ˆ,ˆ[ BA ABBABA ˆˆˆˆ]ˆ,ˆ[
Commutator Algebra
The commutator of two operators is denoted by where
The commutator indicates whether operators commute.
,A B]ˆ,ˆ[ BA ABBABA ˆˆˆˆ]ˆ,ˆ[
Commutator Algebra
The commutator of two operators is denoted by where
The commutator indicates whether operators commute.
Two observables A and B are compatible if their operators commute, ie
. Hence
,A B]ˆ,ˆ[ BA ABBABA ˆˆˆˆ]ˆ,ˆ[
0]ˆ,ˆ[ BA ABBA ˆˆˆˆ
Commutator Algebra
If A and B commute they can be measured simultaneously. The results of their measurements can be carried out in any order.
Commutator Algebra
If A and B commute they can be measured simultaneously. The results of their measurements can be carried out in any order.
If they do not commute, they cannot be measured simultaneously; the order in which they are measured matters.
Commutator Algebra Consider the following properties:
1. R
2. R
3. R
4. R
5. R
6. R
7. r
ABBABA ˆˆˆˆ]ˆ,ˆ[
0]ˆ,ˆ[]ˆ,ˆ[ ABBA0]ˆ,ˆ[ AA
]ˆ,ˆ[]ˆ,ˆ[]ˆˆ,ˆ[ CABACBA
]ˆ,ˆ[]ˆ,ˆ[]ˆ,ˆˆ[ CBCACBA
]ˆ,ˆ[ˆˆ]ˆ,ˆ[]ˆˆ,ˆ[ CABCBACBA
]ˆ,ˆ[ˆˆ]ˆ,ˆ[]ˆ,ˆˆ[ CBABCACBA
Commutator Algebra (in QM)
If and are Hermitian operators which do not commute, the physical observable A and B cannot be sharply defined simultaneously (cannot be measured with certainty).
A B
Commutator Algebra (in QM)
If and are Hermitian operators which do not commute, the physical observable A and B cannot be sharply defined simultaneously (cannot be measured with certainty). eg the momentum and position cannot be measured simultaneously in the same plane.
A B
ixPxPPx xxx ],[
Commutator Algebra
However,
0],[ xPxPPx yyy
Commutator Algebra
However,
The two observables can be measured independently.
0],[ xPxPPx yyy
Commutator Algebra Some examples of other cases are
shown below:
1. R
2. R
3. R
4. R
5. R
ziyzPyPyL yzx ],[],[
zyx PiPL ],[
0],[ xLx
0],[ zx PL
zyx LiLL ],[
irL
Commutator Algebra
The order of operators should not be changed unless you are sure that they commute.
Hermitian Operators & Operators in General
Operators
Operators act on everything to the right unless constrained by brackets.
Operators
Operators act on everything to the right unless constrained by brackets.
][ˆˆ xgxfPxgxfP
xgxfPxgxfP ˆˆ
Operators
Operators act on everything to the right unless constrained by brackets.
The product of operators implies successive operations.
][ˆˆ xgxfPxgxfP
xgxfPxgxfP ˆˆ
2
2ˆ
xfdxi
xfP
Hermitian Operators
Hermitian Operators
All operators in QM are Hermitian.
Hermitian Operators
All operators in QM are Hermitian. They are important in QM because the
eigenvalues Hermitian operators are real!
Hermitian Operators
All operators in QM are Hermitian. They are important in QM because the
eigenvalues Hermitian operators are real! This is significant because the results of an experiment are real.
Hermitian Operators
All operators in QM are Hermitian. They are important in QM because the
eigenvalues Hermitian operators are real! This is significant because the results of an experiment are real.
Hermiticity is defined as
dxAdxA nmnm ˆˆ
Hermitian Operators
Evaluate ],[ xPx
Hermitian Operators
Evaluate From definition xPxPPx xxx ],[
],[ xPx
Hermitian Operators
Evaluate From definition Replace by operators
][
],[
xxixi
x
xPxPPx xxx
],[ xPxxPxPPx xxx ],[
Hermitian Operators
Evaluate From definition Replace by operators
][
],[
xxixi
x
xPxPPx xxx
],[ xPxxPxPPx xxx ],[
][],[ xxixi
xPx x
Hermitian Operators
Evaluate From definition Replace by operators
][
],[
xxixi
x
xPxPPx xxx
],[ xPxxPxPPx xxx ],[
][],[ xxixi
xPx x
x
xixix
Hermitian Operators
x
x
ixix
xixPx x
],[
Hermitian Operators
Therefore
iPx x ],[
iPx x ],[
x
x
ixix
xixPx x
],[