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

],[