Transcript CHM 6470 - University of Florida

Commutation
The most important property of operators
PRODUCT is NOT ALWAYS COMMUTATIVE
ˆ  ˆ  A   ˆ ˆ  A
COMMUTATOR
ˆˆ
ˆ ˆ  
 ˆ , ˆ   
the value of a commutator is given by its application to a ket
ˆˆ A
ˆ , ˆ  A  
ˆ ˆ A  


ˆ ˆ A then we say that "ˆ and ˆ commute"  ˆ , ˆ   0
ˆ ˆ A  
if 


ˆ ˆ A then we say that "ˆ and ˆ do not commute"  ˆ , ˆ   0
ˆ ˆ A  
if 


the evaluation of an operator between a
A ˆ B  ?
and a
 number
B A  number
B A ?
applying
let's apply B A to a ket P
to a
or a
yields a new
or
 operator
|> and <| are vectors and represent STATES
Operators represent VARIABLES and OBSERVABLES (measurables)
there are operators for Energy
momentum
position, etc
In CM, observables commute  it doesn’t matter which one we measure first (the
order of the measurement is not important)
IN QM, observables NOT always commute it matters which one we measure first
(the order of the measurement is FUNDAMENTAL)
Adjoint operators
Adjoint = Complex Conjugate
we know that A  A and A  A
consider
P  ˆ B 

A B  B A
P  .........  B ˆ
 ˆ is the adjoint of ˆ
P A  A P 
ˆˆ B
and for Q  ˆ P  
Q A  AQ
and

which can also be written as
 B ˆ A  A ˆ B
Q  P ˆ  B ˆ ˆ
ˆˆ B
B ˆ ˆ A  A 
ˆˆ
ˆ ˆ  
Self-Adjoint operators
Self-Adjoint operators are also called HERMITIAN.
if ˆ A  ˆ A
then ˆ is self-adjoint (HERMITIAN)
C  B ˆ A  A ˆ B 

C '  B ˆ A  A ˆ B 
if ˆ  ˆ  C  C ' and C 
sometimes, the definition of Hermiticity is given by
B ˆ A  A ˆ B
and for A  B
A ˆ A  A ˆ A
it can be shown that A ˆ B  B ˆ A
is a consequence of
A ˆ A  A ˆ A (see your homework)
The eigenvalue-eigenvector equation
For some kets |>, when an operator is applied to the |>, the same ket is obtained,
multiplied by a number
 eigenvalue-eigenvector eq.
ˆ
operator
P 
eigenket
p
P
or
eigenvalue eigenket
( number )
Q
eigenbra
ˆ
operator

q
eigenvalue eigenbra
( number )
the eigenvalues (p) of hermitian operators (ˆ )correspond to the results
of a measurement of the state
A
Q
Theorems of Hermitian operators
Eigenvalues of Hermitian operators are real numbers
consider the eigenvalue-eigenvector equations
ˆ A  a A
A ˆ A  A a A  a A A  a
A ˆ A  A a A  a A A  a  a *
because ˆ is Hermitian, A ˆ A  A ˆ A  a  a*  a 
2
Two eigenvectors of an Hermitian operator with two  eigenvalues are orthogonal
consider
ˆ F  f F
and
ˆ G  g G where ˆ is Hermitian
F ˆ G  G ˆ F
F g G  G f F
g F G  f G F
and because ˆ is Hermitian, g and f are real numbers
g F G  f G F  f F G
if
f  g then F G  0  F and G are orthogonal
if f  g , then F and G are degenerate,
and 2 orthogonal functions can be constructed (Schmidt orthogonalization)
if an eigenvector of ˆ is multiplied by a number, it is still an eigenvector of ˆ
ˆ A  a A

ˆ CA  Cˆ A  Ca A  ˆ CA
 C A is an eigenvector of ˆ with eigenvalue a
Remember that the eigenvalue of an hermitian operator corresponds to the result
of measuring the state. Since A and CA have the same eigenvalue, they
correspond to the same state
 only direction of the ket matters, length is irrelevant
Expansion of a vector
Complete set: any vector obeying the same boundary conditions can be expressed
by using this set of vectors
given the complete set
G 
G1 , G2 ..... Gn where ˆ Gi  gi Gi ,
any F can always be expanded as a linear combination of eigenkets that
form a complete set  F  c1 G1  c2 G2  ...   ci Gi
i
Expectation value
If we know the eigenket of an operator, it is easy to know what will be the outcome
of a measurement, but what happens if the state (ket) is not and eigenstate of that
operator?
for an eigenket
G i ˆ Gi  Gi gi Gi  gi
Gi Gi
 gi expectation value
1,because the kets
are normalized
when the state is not an eigenket of the operator,
we can do the same , but using the superposition of eigenkets:
what is the expected value of a measurement of the observable ˆ in a state
which is not an eigenstate?
F ˆ F  ?
F  c1 G1  c2 G2
Lets first do it for a superposition of 2 states


F ˆ F  c1 G1  c2 G2 ˆ  c1 G1  c2 G2

 c1c1 G1 ˆ G1  c1c2 G1 ˆ G2  c2c1 G2 ˆ G1  c2c2 G2 ˆ G2
G1 ˆ G1  g1 G1 G1  g1 because G1 is normalized
G2 ˆ G2  g 2 G2 G2  g 2 because G2 is normalized
G1 ˆ G2  G2 aˆ G1  0
because G1 and G 2 are eigenvectors of the same
operator with  eigenvalue
F ˆ F  c1 g1  c2 g2
2
2
the expectation value for the state F is given by the sum of the eigenvalues
of the eigenkets, weigthed by their relative contribution to the superposition
every time there is a measurement, one of the eigenvalues will be obtained (g1 or g 2 ),
after many measurements, we will obtain c1
the g 2 value
2
2
times the g1 eigenvalue and c2 times
for F   ci Gi
We can now do the generalization
i
F ˆ F  ?

 c jG j
j
ˆ
 ciGi
 cj Gj
i
j
 ciˆ
i
Gi   c j G j
j
 ci gi
Gi
i
  ci c j gi G j Gi   ci gi
2
i
j
i
as with the polarization experiment, it is not that each measurement yields
an average value. The inquiry of an observable can only yield EIGENVALUES,
but if the state is not an eigenstate, then each measurement will yield a different
eigenvalue, corresponding to the eigenkets of the superposition that describes that state.
Superposition of states
Any superposition of independent eigenvectors having the same eigenvalue is also
an eigenvector of the operator with the same eigenvalue
ˆ G1  p G1 
  T  c1 G1  c2 G2
ˆ G2  p G2 
ˆ T  ˆ  c1 G1  c2 G2
 c1 p G1  c2 p G2
 p  c1 G1  c2 G2
pT



then
ˆ T  p T
Eigenfunctions of commuting operators
If there is a complete set {Gi} which are eigenvectors of two operators, then the two
operators commute
if ˆ G  a G
and
ˆ G  b G
i
i
i
i
i
i
 Gi is simultaneously an eigenvector of ˆ and ˆ
measuring the property ˆ  ai


and measureing the property ˆ  bi 
let F   ci Gi
 ˆ  ˆ   ? 
i

ˆˆ
ˆ ˆ  
 
 c
i

  c  b  ˆ G
ˆˆ G
ˆ ˆ Gi  
  ci 
i
i
i
i
i
i
i
i

 ai  ˆ Gi
  ci  bi  ai  ai  bi  Gi



ˆˆ G
ˆ ˆ  
  ci 
i
Gi
i

ˆˆ F
ˆ , ˆ  F  
ˆ ˆ  



=  ci ˆ  bi Gi  ˆ  ai Gi

i

  ci  bi  ai Gi  ai  bi Gi
i
0

If two HERMITIAN operators commute, we can select a common complete set of
eigenvectors for them
consider ˆ Pi  ai Pi
if we now operate with ˆ
ˆ ˆ P  ˆ a P

i
i
i
 ai ˆ Pi
ˆˆ
ˆ ˆ  
and since ˆ ,ˆ   0 (they commute)  
ˆ ˆ Pi  ai ˆ Pi

or ˆ ˆ Pi  ai ˆ Pi


 Q  ˆ Pi



Qi

Qi
is an eigenvector of ˆ with eigenvalue ai
if Pi and Qi  ˆ Pi are non-degenerate, then Qi must be number x Pi
ˆ Pi  bi Pi  Pi is also an eigenvector of ˆ !
We will not prove the degenerate case