Transcript Commutator Algebra and Hermitian Operators

Commutator Algebra
Commutator Algebra

ˆ , Bˆ is
The commutator of two operators A
ˆ , Bˆ ] where [ Aˆ , Bˆ ]  Aˆ Bˆ  Bˆ Aˆ
denoted by [ A
Commutator Algebra


ˆ , Bˆ is
The commutator of two operators A
ˆ , Bˆ ] where [ Aˆ , Bˆ ]  Aˆ Bˆ  Bˆ Aˆ
denoted by [ A
The commutator indicates whether
operators commute.
Commutator Algebra
ˆ , Bˆ is
The commutator of two operators A
ˆ , Bˆ ] where [ Aˆ , Bˆ ]  Aˆ Bˆ  Bˆ Aˆ
denoted by [ A
 The commutator indicates whether
operators commute.
 Two observables A and B are compatible
if their operators commute, ie
[ Aˆ , Bˆ ]  0 . Hence Aˆ Bˆ  Bˆ Aˆ

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

1.
2.
3.
4.
5.
6.
7.
Consider the following properties:
[R
Aˆ , Bˆ ]  Aˆ Bˆ  Bˆ Aˆ
[R
Aˆ , Bˆ ]  [ Bˆ , Aˆ ]  0
[R
Aˆ , Aˆ ]  0
[R
Aˆ , Bˆ  Cˆ ]  [ Aˆ , Bˆ ]  [ Aˆ , Cˆ ]
[R
Aˆ  Bˆ , Cˆ ]  [ Aˆ , Cˆ ]  [Bˆ , Cˆ ]
[R
Aˆ , Bˆ Cˆ ]  [ Aˆ , Bˆ ]Cˆ  Bˆ[ Aˆ , Cˆ ]
[rAˆ Bˆ , Cˆ ]  [ Aˆ , Cˆ ]Bˆ  Aˆ[Bˆ , Cˆ ]
Commutator Algebra (in QM)

If Aˆ and Bˆ are Hermitian operators which
do not commute, the physical observable
A and B cannot be sharply defined
simultaneously (cannot be measured
with certainty).
Commutator Algebra (in QM)

If Aˆ and Bˆ 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.
[ x, Px ]  xPx  Px x  i
Commutator Algebra

However,
[ x, Py ]  xPy  Py x  0
Commutator Algebra

However,
[ x, Py ]  xPy  Py x  0

The two observables can be measured
independently.
Commutator Algebra

1.
2.
3.
4.
5.
Some examples of other cases are
shown below:
[R
Lx , y]  [ yPz  zPy , y]  iz
[R
Lx , Py ]  iPz
[R
Lx , x]  0
[R
Lx , Pz ]  0
[R
Lx , Ly ]  iLz
  
Lr 
i
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.
Pˆ f xg x  Pˆ[ f xg x]
Pˆ f xg x  Pˆ f x g x
Operators

Operators act on everything to the right
unless constrained by brackets.
Pˆ f xg x  Pˆ[ f xg x]
Pˆ f xg x  Pˆ f x g x

The product of operators implies
successive operations.
 

Pˆ 2 f x   
f x 
 i dx

2
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
ˆ    dx  

A



m



n


m
ˆ  dx
A
n
Hermitian Operators

Evaluate [ x, Px ]
Hermitian Operators


Evaluate [ x, Px ]
From definition [ x, Px ]  xPx  Px x
Hermitian Operators



Evaluate [ x, Px ]
From definition [ x, Px ]  xPx  Px x
Replace by operators
[ x, Px ]  xPx  Px x
 
 
x ]
 [x

i x
i x
Hermitian Operators



Evaluate [ x, Px ]
From definition [ x, Px ]  xPx  Px x
Replace by operators
[ x, Px ]  xPx  Px x
 
 
x ]
 [x

i x
i x
 
 
x ]
[ x, Px ]  [ x

i x
i x
Hermitian Operators



Evaluate [ x, Px ]
From definition [ x, Px ]  xPx  Px x
Replace by operators
[ x, Px ]  xPx  Px x
 
 
x ]
 [x

i x
i x
 
 
x ]
[ x, Px ]  [ x

i x
i x
 
 
 x 
x

i x
i x
Hermitian Operators
 
 
 x
[ x, Px ]  x
x

i x
i x
i x
Hermitian Operators
 
 
 x
[ x, Px ]  x
x

i x
i x
i x
 [ x, Px ]  i

Therefore
 [ x, Px ]  i