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Lecture 7: Expectation Values
The material in this lecture covers the following in Atkins.
11.5 The informtion of a wavefunction
(d) superpositions and expectation values
Lecture on-line
Expectation Values (PDF)
Expectation value (PowerPoint)
handouts
Assigned problems for lecture 7
Tutorials on-line
Reminder of the postulates of quantum mechanics
The postulates of quantum mechanics
(This is the writeup for Dry-lab-II)( This
lecture has covered postulate 5)
Basic concepts of importance for the understanding of the postulates
Observables are Operators - Postulates of Quantum Mechanics
Expectation Values - More Postulates
Forming Operators
Hermitian Operators
Dirac Notation
Use of Matricies
Basic math background
Differential Equations
Operator Algebra
Eigenvalue Equations
Extensive account of Operators
Historic development of quantum mechanics from classical mechanics
The Development of Classical Mechanics
Experimental Background for Quantum mecahnics
Early Development of Quantum mechanics
Audio-visuals on-line
Postulates of Quantum mechanics
(PDF) (simplified version from Wilson)
Postulates of Quantum mechanics
(HTML) (simplified version from Wilson)
Postulates of quantum mechanics
(PowerPoint ****)(simplified version from
Wilson)
Slides from the text book (From the CD included in Atkins ,**)
Operators and Expectation Values
Re view of average
calculations
Consider a large number N of
identical boxes with identical
particles all described by the
same wavefunction (x,t) :
Let us for each system at the same time meassure the property F
let the outcome of this meassurement be
f1, f2 , f3 , ........, fN
the average value for F is given by
N
 fk
<F > = k
N
k runs over number of meassurements
Operators and Expectation Values
Re view of average
calculations
Since N is large many experiments might give the same result.
Let n i be the times f i was observed. In this case we might also
wrire < F > as :
<F> =
1
 fj
Nj
1
=  nifi
Ni
j runs over all values i runs over different values
We might also write:
ni
< F > =  ( )fi   Pifi
i N
i
ni
Here Pi = ( ) is the probability of measuring the
N
value f i for F
Operators and Expectation Values
New apl. of
Born interp.
Let us now consider the x - coordinate in our N systems.
We have from the Born interpretation
probability of finding particle
Pi  P(x )  (x,t)* (x,t)dx between x and x + x
Thus the average value of x is given by

 x  =  P(x)x =  (x, t)x * (x, t)dx
x
-
Operators and Expectation Values
New apl. of
Born interp.
For a physical property that depends on the x, y,x
coordinates only : F(x,y,z)
The average value is given by
  
 F  =     * (x,y, z,t)F(x,y,z)(x,y,z, t)dxdydz
- - -
This is a simple extension
of the Born postulate
which is part of
Operators and Expectation Values New postulate 5.
A general property will depend on x,y,z as well as
the linear momenta px , py , pz .
F = F(x,y, z,px ,py ,p z )
We postulate :
  
 F  =    * (x,y, z,t)Fˆ (x,y, z,t)dxdydz
- - -
ˆ x,p
ˆ y , pˆ z )
Where Fˆ = Fˆ (x,y, z, p
Note : operator Fˆ is "sandwiched" between
 * and  .
the average value < F > is also called
an expectation value
Operators and Expectation Values New postulate 5.
Consider the special case where (x) is a
ˆ and Fˆ
simultanious eigenfunction to H
ˆ (x) = E(x)
H
; Fˆ (x) = k(x)
In this case

<F> = 
* (x)Fˆ  * (x)dx
-

1
= k   * (x) * (x)dx
=k
-
In this case a meassurement of F will always give k as an answer
Operators and Expectation Values New postulate 5.
Consider next the more general case where
 (x) as a statefunction is an eigenfunction to
ˆ but not to Fˆ
H
ˆ  (x) = E (x) ; Fˆ  (x)  k (x)
H
In this case the meassurement of F will
give one of the eigenvalues of F
Fi  k i i
The average value from a large number
of meassurements will be

ni
 F   ( )fi    * (x )Fˆ (x )
N
i

statistics (logic)
Postulate 5
Operators and Expectation Values

ni
 F   ( )fi    * (x )Fˆ (x )
N
i

Good question
about postulate 5.
What is the probability
n
Pi  ( i )
N
That the meassurement will have the outcome f i ?
the eigenfunctions i (i = 1,2,..)
Fi  k i i
forms a complete set on which we can expand our
statefunction  (x) :

 (x) =  aii (x ) : ai   f * (x )i (x )
i

Operators and Expectation Values

ni
 F   ( )fi    * (x )Fˆ (x )dx
N
i

Long answer to
good question
about postulate 5.
Now substituting the expression for the
expansion of the state function  (x ) in
terms of the eigenfunctions i to Fˆ

< F > =  (  ai*i* )Fˆ (  a j j )dx

i
j
Or after working with Fˆ on the sum to the right of Fˆ ,
and remember that Fˆ j  k j j

< F > =  (  ai*i* )(  a j k j j )dx

i
j
Operators and Expectation Values

< F > =  (  ai*i* )(  a j k j j )dx

i
Long answer to
good question
about postulate 5.
j
Now multiply each term in the right hand sum
with each term in the left hand sum

< F > =    (ai* i*a j k j j )dx
 i j
Interchanging next order of integration and summation,
which is allowed for 'well behaved sums' :

< F > =    (ai* i*a j k j j )dx
i j 
Operators and Expectation Values
Long answer to
good question
about postulate 5.

< F > =    (ai* i*a j k j j )dx
i j 
Taking constant factors outside integration sign
<F > =

*
  ai a j k j  i* j dx
i j

Making use of th orthonormality of
eigenfunctions

< F > =   ai*a j k j  i* j dx
i j

< F > =  ai*ai k i   | ai |2 k i
i
i
ij

*
 i j dx  ij

Operators and Expectation Values
By comparing
< F > =  ai*ai k i   | ai |2 k i
i
Long answer to
good question
about postulate 5.
i
with

ni
 F   ( )fi    * (x )Fˆ (x )
N
i

ni
we note that | ai | 
N
2
probability of obtaining ki from
a meassurement of F in state
with state function  (x)

We have that ai    * (x)i (x)dx

Thus the chance of obtaining k i from a meassurement
of F for a system with state function  (x) is large
if the 'overlap' between (x) and i (x) is large
Operators and Expectation Values
We have that (x) is normalized

Long answer to
good question
about postulate 5.

* *

(x
)

(x
)dx

[
a



i i (x ) ][  a j j (x )]dx  1
*
- i
-
j
or after multiplying out the sum and interchange
summation and integration


* *

(x
)

(x
)dx

a



 i  i (x )a j j (x )dx  1
-
*
i
j
-
Operators and Expectation Values
finally using the orthonormality properties of the set {i ,i  1,2..}

  
i
j
ai* * i (x )a j j
-
(x )dx   
i
j
ai*a j

*

 i (x )j (x )dx  1
-
or :  | ai |2  1  sum of all probabilities
i
ij
Thus the sum of the individual probabilities ai (i = 1,2,..)for
obtaining the values fi (i = 1,2,..) in a meassurement of F
for a system with the statefunction (x) is one as it should;
if  (x) is normalized
Operators and Quantum Mechanics
ikx
ikx
 (x)  exp  exp
is a linear combination
of two eigenfunctions to pˆ x
px  k
How can we find
px in this case ?
50 % chance to
measure p = k
50 % chance to
measure p = - k
 Px  0
px   k
2k 2
p2
E

2m
2m
What you should learn from this lecture
1. Postulate 2 (Review)
For any observable (x,y, x , px , py ,pz ) that can
be expressed in classical physics in terms of x, y, x
and px , py , pz . We can construct the corresponding
ˆ ( xˆ , yˆ , xˆ , pˆ x , pˆ y , pˆ z )
quantum mechanical operator operator 
from the substitution:
Classical Mechanics Quantum Mechanics
x
px
y
py
z
pz
ˆ (x,y,z,
as 

xˆ   x ; pˆ x  
i x

yˆ   y ; pˆ y  
i y

zˆ   z ; pˆ z  
i z
d
d
d
,
,
)
i dx i dy i dz
What you should learn from this lecture
2. Postulate 3 (Review)
ˆ
The meassurement of the quantity represented by 
has as the o n l y outcome one of the eigenvalues  n n = 1,2, 3 ....
ˆ  n  n  n
to the eigenvalue equation : 
3. Postulate 5.
For a system in a state described by (x, y, z, t)
the average value meassured for  will be
  
ˆ
ˆ (x,y, z, t)dxdydz
 
=    * (x,y, z, t)
- - -
We call that the expectation value.
4. For a system in a state described by (x, y, z, t) the
probability to obtain the value  n in a meassurement of 
  
2
is | a n | where a n =    * (x,y, z, t) ndxdydz
- - -
ˆ n   n n and  n
Here  n is an eigenvalue to 
the corresponding eigenfunction