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Commutators and the Correspondence Principle
Formal Connection
Q.M.
Classical Mechanics
Correspondence between
Classical Poisson bracket of
functions f ( x , p) and g( x , p)
And
Q.M. Commutator of
operators f and g .
Copyright – Michael D. Fayer, 2007
Commutator of Linear Operators
 A, B  A B  B A
(This implies operating on an arbitrary ket.)
If A and B numbers, = 0
Operators don’t necessarily commute.
A B C  A  B C 
AQ
 Z
B A C  B  A C 
B S
 T
In General
Z  T
A and B do not commute.
Copyright – Michael D. Fayer, 2007
Classical Poisson Bracket
 f , g 
 f g g f

x  p x p
f  f ( x , p)
g  g( x , p)
These are functions representing classical
dynamical variables
not operators.
Consider position and momentum, classical.
x and p
Poisson Bracket
 x p  x p

 x , p 
 x p  p x
 x, p  1
zero
Copyright – Michael D. Fayer, 2007
Dirac’s Quantum Condition
"The quantum-mechanical operators f
and g, which in quantum theory replace
the classically defined functions f and g,
must always be such that the commutator
of f and g corresponds to the Poisson
bracket of f and g according to
i
 f , g   f , g  ."
Dirac
Copyright – Michael D. Fayer, 2007
This means
i
 f ( x , p), g( x , p)
  f , g 
Poisson bracket of
classical functions
(commutator operates
on a ket)
Commutator of
quantum operators
Q.M. commutator of x and p.
 x , p   i
commutator
Therefore,
 x , p
Poisson bracket
x, p  1
 x , p   i
Remember, the relation implies operating on
an arbitrary ket.
This means that if you select operators for x and p such that they
obey this relation, they are acceptable operators.
The particular choice
a representation of Q.M.
Copyright – Michael D. Fayer, 2007
Schrödinger Representation

p  P  i
 x
momentum operator, –i times derivative with
respect to x
xx x
position operator, simply x
Operate commutator on arbitrary ket
 x, P  S

( xP  P x ) S 

 

x  i
x S
 S i

x

x


Using the product rule




 i  x
S  S x
S 
x
x


i
s .
Therefore,
 x, P  S
i
S
and
 x, P   i
because the two sides have the
same result when operating on
an arbitrary ket.
S
Copyright – Michael D. Fayer, 2007
Another set of operators – Momentum Representation

xxi
 p
position operator, i times derivative with
respect to p
p p
momentum operator, simply p
A different set of operators, a different representation.
In Momentum Representation, solve position eigenvalue problem. Get
x , states of definite position.
They are waves in p space. All values of momentum.
Copyright – Michael D. Fayer, 2007
Commutators and Simultaneous Eigenvectors
A S  S
B S  S
S are simultaneous Eigenvectors of operators A and B
with egenvalues  and  .
Eigenvalues of linear operators
observables.
A and B are different operators that represent different observables, e. g.,
energy and angular momentum.
If S are simultaneous eigenvectors of two or more linear operators
representing observables, then these observables can be
simultaneously measured.
Copyright – Michael D. Fayer, 2007
A S  S
B S  S
B A S  B S
AB S  A S
 B S
 A S
  S
  S
Therefore,
Rearranging
AB S  B A S
 A B  B A
 A B  B A is the commutator of
S 0
A and B, and since in general S  0,
 A, B  0
The operators A and B commute.
Operators having simultaneous eigenvectors commute.
The eigenvectors of commuting operators can always be constructed in such
a way that they are simultaneous eigenvectors.
Copyright – Michael D. Fayer, 2007
There are always enough Commuting Operators (observables) to
completely define a system.
Example:
Energy operator, H, may give degenerate states.
H atom 2s and 2p states have same energy.
J
2
 square of angular momentum operator
j  1 for p orbital
j  0 for s orbital
But px, py, pz
J z  angular momentum projection operator
2
H , J , J z all commute.
Copyright – Michael D. Fayer, 2007
Commutator Rules
 A , B    B , A
 A , BC    A , B C  B  A , C 
 AB , C    A , C  B  A B , C 
 A ,  B , C    B , C , A  C ,  A , B   0
 A, B  C    A, B   A, C 
Copyright – Michael D. Fayer, 2007
Expectation Value and Averages
A a  a
eigenvector
normalized
eigenvalue
If make measurement of observable A on state a will observe .
What if measure observable A on state not an eigenvector of operator A.
Ab ?
normalized
Expand b in complete set of eigenkets a  Superposition principle.
Copyright – Michael D. Fayer, 2007
b  c1 a1  c2 a2  c3 a3  
(If continuous range
integral)
b   ci a i
i
Consider only two states (normalized and orthogonal).
b  c1 a1  c2 a2
A b  A  c1 a1  c2 a2

 c1 A a1  c2 A a2
 1c1 a1   2c2 a2
Left multiply by

b .
b A b  c1* a1  c2* a2
  c
1 1
a1   2c2 a2

 1c1*c1   2c2*c2
  1 c1   2 c2
2
2
Copyright – Michael D. Fayer, 2007
The absolute square of the coefficient ci, | ci|2, in the expansion
of b in terms of the eigenvectors ai of the operator (observable)
A is the probability that a measurement of A on the state b
will yield the eigenvalue i.
If there are more than two states in the expansion
b   ci a i
i
b A b    i ci
2
i
eigenvalue
probability of eigenvalue
Copyright – Michael D. Fayer, 2007
Definition: The average is the value of a
particular outcome times its
probability, summed over all
possible outcomes.
Then
b A b   ci  i
2
i
is the average value of the observable when many measurements are made.
Assume: One measurement on a large number
of identically prepared non- interacting systems
is the same as the average of many repeated
measurements on one such system prepared
each time in an identical manner.
Copyright – Michael D. Fayer, 2007
b Ab 
Expectation value of the operator A.
In terms of particular wavefunctions

b Ab 


 b A bd

Copyright – Michael D. Fayer, 2007
The Uncertainty Principle - derivation
Have shown -
 x , P  0
and that
x p 
Want to prove:
Given A and B,
Hermitian with
 A , B  i C
C another Hermitian operator (could be number –
special case of operator, identity operator).
Then
1
 AB 
C
2
with
C  SC S
short hand for expectation value
S and S
arbitrary but normalized.
Copyright – Michael D. Fayer, 2007
Consider operator
D  A  B  i  B
arbitrary real numbers
D S  Q
Q Q  S DD S  0
Since Q Q is the scalar product of vector
with itself.
Q Q  S DD S  A2   2   2  B 2   C '   C  0
(derive this in home work)
C '  AB  BA
is the anticommutator of
A and B .
AB  BA   A , B
2
A
 S A S
anticommutator
2
Copyright – Michael D. Fayer, 2007
B S 0
for arbitrary ket S .
Can rearrange to give

1 C'
2
2 
A  B 

2 B2

2
2


2
  B   1 C


2 B2



  1 C'
 4 B2

2
2
1 C

0
2
4 B
Holds for any value of  and .
Pick  and  so terms in parentheses are zero.
Then A2
B
2


1
C
4
2
 C'
2


1
C
4
2
(Have multiplied through by
2
B and transposed.)
Positive numbers because square of real numbers.
The sum of two positive
numbers is  one of them.
Thus,
A
2
B
2
1
 C
4
2
Copyright – Michael D. Fayer, 2007
A
2
B
Define
2
1
 C
4
  A
  B
2
2
 A  A
2
 B
2
2
2
 B
Second moment of distribution
- for Gaussian
standard deviation squared.
2
For special case
A  B 0
1
 AB 
C
2
Average value of the observable is zero.
C 
1
2 2
 C
square root of the
square of a number
Copyright – Michael D. Fayer, 2007


Have proven that for A , B  i C
1
 AB 
C
2
Example
 x, P   i
Number, special case of an operator.
Number is implicitly multiplied by the
identity operator.
x  P 0
Therefore
 x  p  / 2.
Uncertainty comes from superposition principle.
The more general case is discussed in the book.
Copyright – Michael D. Fayer, 2007