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Quantum Mechanics for
Scientists and Engineers
David Miller
Operators and quantum mechanics
Operators and quantum mechanics
Hermitian operators in quantum mechanics
Commutation of Hermitian operators
For Hermitian operators  and B̂ representing
physical variables
it is very important to know if they
commute
ˆ ˆ  BA
ˆˆ ?
i.e., is AB
Remember that
because these linear operators obey the
same algebra as matrices
in general operators do not commute
Commutator
For quantum mechanics, we formally define
an entity
ˆ ˆ  BA
ˆˆ
 Aˆ , Bˆ   AB


This entity is called the commutator
ˆ ˆ  BA
ˆˆ
An equivalent statement to saying AB
is then  Aˆ , Bˆ   0
Strictly, this should be written  Aˆ , Bˆ   0 Iˆ
where Iˆ is the identity operator
but this is usually omitted
Commutation of operators
If the operators do not commute
then  Aˆ , Bˆ   0 does not hold


and in general we can choose to write
 Aˆ , Bˆ   iCˆ


where Ĉ is sometimes referred to as
the remainder of commutation or
the commutation rest
Commuting operators and sets of eigenfunctions
Operators that commute share the same set of
eigenfunctions
and
operators that share the same set of eigenfunctions
commute
We will now prove both of these statements
Commuting operators and sets of eigenfunctions
Suppose that operators  and B̂ commute
and suppose the  n are the eigenfunctions of Â
with eigenvalues Ai
ˆ ˆ   BA
ˆ   A Bˆ 
ˆ ˆ   BA
Then AB
i
i
i
i
i
i
i.e., Aˆ  Bˆ  i   Ai  Bˆ  i 
But this means that the vector Bˆ  i
is also the eigenvector  i or is proportional to it
i.e., for some number Bi
Bˆ  i  Bi  i
Commuting operators and sets of eigenfunctions
This kind of relation Bˆ  i  Bi  i
holds for all the eigenfunctions  i
so these eigenfunctions
are also the eigenfunctions of the operator B̂
with associated eigenvalues Bi
Hence we have proved the first statement that
operators that commute share the same set of
eigenfunctions
Note that the eigenvalues Ai and Bi are not in general
equal to one another
Commuting operators and sets of eigenfunctions
Now we consider the statement
operators that share the same set of eigenfunctions
commute
Suppose that the Hermitian operators  and B̂
share the same complete set  n of eigenfunctions
with associated sets of eigenvalues An and Bn
respectively
Then
ˆ ˆ   AB
ˆ   AB 
AB
i
and similarly
i
i
i
i
i
ˆ ˆ   BA
ˆ  BA 
BA
i
i
i
i i
i
Commuting operators and sets of eigenfunctions
Hence, for any function f
which can always be expanded in this complete set of
functions  n i.e., f   ci  i
i
we have
ˆ ˆ f  c A B   c B A   BA
AB
 i i i i  i i i i ˆˆ f
Since we have proved this for an arbitrary function
we have proved that the operators commute
hence proving the statement
operators that share the same set of
eigenfunctions commute
i
i
Operators and quantum mechanics
General form of the uncertainty principle
General form of the uncertainty principle
First, we need to set up the concepts of the mean and
variance of an expectation value
Using A to denote the mean value of a quantity A
we have, in the bra-ket notation
for a measurable quantity associated with the
Hermitian operator Â
when the state of the system is f
A  A  f Aˆ f
General form of the uncertainty principle
Let us define a new operator Â
associated with the difference between the measured
value of A and its average value
Aˆ  Aˆ  A
Strictly, we should write Aˆ  Aˆ  AIˆ
but we take such an identity operator to be
understood
Note that this operator is also Hermitian
General form of the uncertainty principle
Variance in statistics is the “mean square” deviation
from the average
To examine the variance of the quantity A
we examine the expectation value of the operator (Aˆ ) 2
Expanding the arbitrary function f
on the basis of the eigenfunctions  i of Â
i.e., f   ci  i
i
we can formally evaluate the expectation value
of (Aˆ ) 2 when the system is in state f
General form of the uncertainty principle
We have
2


 ˆ

2
2
ˆ
ˆ
(A)  f (A) f    ci  i  A  A   c j  j 
 i

 j




 ˆ

   ci  i  A  A   c j  Aj  A   j 
 i

 j






2



   ci  i    c j  Aj  A   j    ci
 i
 j
i

2
 A  A
i
2
General form of the uncertainty principle
Because the ci are the probabilities that the system is
found, on measurement, to be in the state  i
2
and  Ai  A  for that state simply represents
the squared deviation of the value of the quantity A
from its average value
then by definition
2
 A
2

 
Aˆ
2


Aˆ  A

2
 f

Aˆ  A

2
f
is the mean squared deviation for the quantity A
on repeatedly measuring the system prepared in state f
General form of the uncertainty principle
In statistical language
the quantity  A  is called the variance
and the square root of the variance
2
which we can write as A 
is the standard deviation
In statistics
the standard deviation gives a
well-defined measure of
the width of a distribution
 A
2
General form of the uncertainty principle
We can also consider some other quantity B
associated with the Hermitian operator B̂
B  B  f Bˆ f
and, with similar definitions
 B 
2

 
Bˆ
2


Bˆ  B

2

 f Bˆ  B

2
f
So we have ways of calculating the uncertainty in the
measurements of the quantities A and B
when the system is in a state f
to use in a general proof of the uncertainty principle
General form of the uncertainty principle
Suppose two Hermitian operators  and B̂ do not commute
and have a commutation rest Ĉ
as defined above in  Aˆ , Bˆ   iCˆ


Consider, for some arbitrary real number  , the number
G    Aˆ  iBˆ f Aˆ  iBˆ f  0

By Aˆ  iBˆ  f
 



we mean the vector Aˆ  iBˆ f
written this way to emphasize it is simply a vector
so it must have an inner product with itself
that is greater than or equal to zero
General form of the uncertainty principle
So

 Aˆ  iBˆ  f   0
f Aˆ  iBˆ Aˆ  iBˆ  f
f Aˆ  iBˆ Aˆ  iBˆ  f
f   Aˆ    Bˆ   i  Aˆ Bˆ  Bˆ Aˆ  f
f   Aˆ    Bˆ   i  Aˆ , Bˆ  f
f   Aˆ    Bˆ    Cˆ f
G    f Aˆ  iBˆ
By Hermiticity





†
2
2
2
†
†
2
2
2
2
2
2
General form of the uncertainty principle
So
G    f 
2
   
2
2
Aˆ  Bˆ   Cˆ f
 0
  2  A    B    C
2
2
where C  C  f Cˆ f
By a simple (though not obvious) rearrangement
2


C

C
2
2
   B  
G     A   
0
2
2


2

A
4 A 




2
General form of the uncertainty principle
But
2


C

C
2
2
   B  
0
G     A   
2
2

2 A  
4 A 

must be true for arbitrary 
C
so it is true for   
2
2 A 
which sets the first term equal to zero
2
C

so  A   B  
4
2
2
2
or AB 
C
2
General form of the uncertainty principle
So, for two operators  and B̂
corresponding to measurable quantities A and B
for which  Aˆ , Bˆ   iCˆ


in some state f for which C  C  f Cˆ f
we have the uncertainty principle
AB 
C
2
where A an B are the standard deviations of
the values of A and B we would measure
General form of the uncertainty principle
Only if the operators  and B̂ commute
i.e.,  Aˆ , Bˆ   0 (or, strictly,  Aˆ , Bˆ   0 Iˆ )




or if they do not commute, i.e.,  Aˆ , Bˆ   iCˆ
but we are in a state f for which f Cˆ f  0
is it possible for both A and B simultaneously
to have exact measurable values
Operators and quantum mechanics
Specific uncertainty principles
Position-momentum uncertainty principle
We now formally derive the position-momentum relation
Consider the commutator of pˆ x and x
(We treat the function x as the operator for position)
To be sure we are taking derivatives correctly
we have the commutator operate on an arbitrary function
d 
d
d
d

 pˆ x , x  f  i  x  x  f  i   x f   x f 
dx 
dx
 dx
 dx

d
d


 i  f  x
f x
f   i  f
dx
dx


Position-momentum uncertainty principle
In  pˆ x , x  f  i f
since f is arbitrary
we can write  pˆ x , x   i
and the commutation rest operator Ĉ
is simply the number Ĉ  
Hence C  
and so, from AB  C / 2

we have px x 
2
Energy-time uncertainty principle
The energy operator is the Hamiltonian Ĥ

and from Schrödinger’s equation Ĥ   i 
t
so we use Hˆ  i / t
If we take the time operator to be just t
then using essentially identical algebra as used for
the momentum-position uncertainty principle


 Hˆ , t   i  t  t   i


t 
 t
so, similarly we have

E t 
2
Frequency-time uncertainty principle
We can relate this result mathematically to
the frequency-time uncertainty principle
that occurs in Fourier analysis
Noting that E   in quantum mechanics
we have
1
t 
2