chem 373- lecture 7: expectation values
TRANSCRIPT
8/3/2019 Chem 373- Lecture 7: Expectation Values
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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)
handoutsAssigned problems for lecture 7
8/3/2019 Chem 373- Lecture 7: Expectation Values
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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 PostulatesForming Operators
Hermitian Operators
Dirac Notation
Use of MatriciesBasic math background
Differential Equations
Operator Algebra
Eigenvalue Equations
Extensive account of OperatorsHistoric development of quantum mechanics from classical mechanics
The Development of Classical Mechanics
Experimental Background for Quantum mecahnics
Early Development of Quantum mechanics
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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 ,**)
8/3/2019 Chem 373- Lecture 7: Expectation Values
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Operators and Expectation Values Consider a large number N of identical boxes with identical
particles all described by thesame wavefunction Ψ( , ) : x t
the average value for F is given by
< F > =
f
N
kk
N∑
k runs over number of meassurements
Let us for each system at the same time meassure the property F
let the outcome of this meassurement bef f , f ........, f1 2 3 N, ,
Review of average
calculations
8/3/2019 Chem 373- Lecture 7: Expectation Values
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We might also write :
< F > = (n
N
P
i
i
i
i∑ = ∑)f f i i
Here P = (n
Nis the probability of measuring the
value f for F
ii
i
)
Operators and Expectation Values Review of average
calculationsSince N is large many experiments might give the same result.Let n be the times f was observed. In this case we might also
wrire < F > as :i i
< F > = =j i
1 1
Nf
Nn fj i i∑ ∑
j runs over all values i runs over different values
8/3/2019 Chem 373- Lecture 7: Expectation Values
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Let us now consider the x - coordinate in our N systems.
We have from the Born interpretation
Pi = =P x ( ) *Ψ Ψ(x, t) (x, t)dx
Thus the average value of x is given by
< > ∫ ∑∞
∞x = P(x)x =
-xΨ Ψ( (*x, t)x x, t)dx
Operators and Expectation Values
probability of finding particle
between x and x + x∆
New apl. of Born interp.
8/3/2019 Chem 373- Lecture 7: Expectation Values
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For a physical property that depends on the x,y,x
coordinates only : F(x,y,z)
The average value is given by
< > ∫ ∫ ∫ ∞
∞
∞
∞
∞
∞
F = -- - Ψ Ψ
*
(x,y,z, t)F(x,y,z) (x,y,z, t)dxdydz
Operators and Expectation Values
This is a simple extension
of the Born postulatewhich is part of
New apl. of Born interp.
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A general property will depend on x,y,z as well asthe linear momenta p , p , p .
F = F(x,y,z,p ,p ,p
x y z
x y z )
We postulate :
=
-- -
< > ∫ ∫ ∫ ∞
∞
∞
∞
∞
∞F Ψ Ψ* ˆ(x,y,z, t)F (x,y,z, t)dxdydz
Where F = F(x,y,z,p ,p ,px y zˆ ˆ ˆ ˆ ˆ )
No t e :
operator F is "sandwiched" betweenand
ˆ
.*Ψ Ψ
the average value < F > is also called
an expectation value
Operators and Expectation Values New postulate 5.
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Consider the special case where x) is a
simultanious eigenfunction to H and Fψ (
ˆ ˆ
Hψ ψ (x) = E (x)
In this case
< F > = (x)F (x)dx- ψ ψ
* * ˆ
∞
∞
∫
1
In this case a meassurement of F will always give k as an answer
Operators and Expectation Values
(x) = k (x) Fψ ψ
= k (x) (x)dx-
ψ ψ * *
∞
∞
∫ = k
New postulate 5.
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ˆ ˆH Fψ ψ ψ ψ (x) = E (x) ; (x) k (x)≠
Consider next the more general case wherex) as a statefunction is an eigenfunction to
H but not to F
ψ (
ˆ ˆ
In this case the meassurement of F will
give one of the eigenvalues of FF k i i i ξ ξ =
Operators and Expectation Values
The average value from a large number
of meassurements will be
< >= ∑ = ∫ −∞
∞F
n
N f x F x
i
i i ( ) ( ) ˆ ( )*ψ ψ
statistics (logic) Postulate 5
New postulate 5.
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< >= ∑ = ∫ −∞
∞F n N
f x F x i
i i ( ) ( ) ˆ ( )*ψ ψ
What is the probability
P
That the meassurement will have the outcome f ?
i
i
= ( )n N
i
F k i i i ξ ξ =
forms a complete set on which we can expand our
statefunction ψ (x):
the eigenfunctions (i = 1,2,..)ξ i
ψ ξ ξ (x) = aii
∑ = ∫ −∞
∞
i i x f x x ( ) ( ) ( )*: ai
Operators and Expectation Values Good question
about postulate 5.
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< >= ∑ = ∫ −∞
∞F
n
N f x F x dx
i
i i ( ) ( ) ˆ ( )*ψ ψ
Now substituting the expression for the
expansion of the state function in
terms of the eigenfunctions to F
ψ
ξ
( )
ˆ
x
i
< F > =−∞
∞
∫ ∑ ∑( ) ˆ ( )* *a F a dx i i i
j j j
ξ ξ
Or after working with F on the sum to the right of F,and remember that F
ˆ ˆˆ ξ ξ j j j k =
< F > =−∞
∞
∫ ∑ ∑( )( )* *a a k dx i i i j
j j j
ξ ξ
Operators and Expectation Values Long
good
answer to
questionabout postulate 5.
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< F > =−∞
∞∫ ∑ ∑( )( )* *a a k dx i i
i j j j
j
ξ ξ
Now multiply each term in the right hand sum
with each term in the left hand sum
< F > =
−∞
∞
∫ ∑∑ ( )* *a a k dx i i
j
j j j
i
ξ ξ
Interchanging next order of integration and summation,
which is allowed for 'well behaved sums' :
< F > =−∞
∞∫ ∑∑ ( )* *a a k dx i i
j j j j
i
ξ ξ
Operators and Expectation Values Long
good
answer to
questionabout postulate 5.
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< F > =−∞
∞∫ ∑∑ ( )* *a a k dx i i
j j j j
i
ξ ξ
Taking constant factors outside integration sign
< F > = a a k dx i j j i j
j i
* *
−∞
∞
∫ ∑∑ ξ ξ
Making use of th orthonormality ofeigenfunctions −∞
∞∫ =ξ ξ δ i j dx *ij
< F > = a a k dx i j j i j j i
* *
−∞
∞
∫ ∑∑ ξ ξ
δ ij
< F > = a a k a k i i i
i i i
i
* | |∑ = ∑ 2
Operators and Expectation Values Long
good
answer to
questionabout postulate 5.
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< F > = a a k a k i i i i
i i i
* | |∑ = ∑2By comparing
ith
< >= ∑ = ∫ −∞
∞F
n
N f x F x
i
i i ( ) ( ) ˆ ( )*ψ ψ
we note that | ai |2
=
n
N i
probability of obtaining k from
a meassurement of F in statewith state function
i
ψ (x)
We have that ai = ∫ −∞
∞
ψ ξ
*
((x) x)dxi
Thus the chance of obtaining k from a meassurement
of F for a system with state function is large
if the 'overlap' between
i
ψ
ψ ξ
(x)
(x) and (x) is largei
perators and Expectation Values Long
good
answer to
questionabout postulate 5.
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We have that ψ (x) is normalized
ψ ψ ξ ξ
ψ ψ ξ ξ
* * *
* * *
( ) ( ) [ ( )][ ( )]
( ) ( ) ( ) ( )
x x dx a x a x dx
or
x x dx a x a x dx
i i
i j j
j
i j i i j j
- -
- -
after multiplying out the sum and interchange
summation and integration
∞
∞
∞
∞
∞
∞
∞
∞
∫ = ∑∫ ∑ =
∫ = ∑ ∑ ∫ =
1
1
Operators and Expectation Values Long
good
answer to
questionabout postulate 5.
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finally using the orthonormality properties of the set ξ i , , ..i = 1 2
i j i i j j
i j i j i j a x a x dx a a x x dx ∑ ∑ ∫ = ∑ ∑ ∫ =
∞
∞
∞
∞
- -* * * *( ) ( ) ( ) ( )ξ ξ ξ ξ 1
δ ij
or : | sum of all probabilitiesi
∑ = ⇒a i
|2 1
Thus the sum of the individual probabilities a (i = 1,2,..)for
obtaining the values f (i = 1,2,..) in a meassurement of F
for a system with the statefunction
i
i
ψ ψ
(x) is one as it should;if (x) is normalized
Operators and Expectation Values
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is a linear combinationof two eigenfunctions to pxψ ( ) exp exp
ˆx ikx ikx= + −
How can we findp in this case ?x
Operators and Quantum Mechanics
p kx =h
p kx = −h
50 % chance tomeasure p = kh
50 % chance tomeasure p = - kh
E
p
m
k
m= =
2 2 2
2 2
h
< >=Px 0
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1. Postulate 2 (Review)For any observable x, y, x , p ) that can
be expressed in classical physics in terms of x,y,xand p .
x,y,x ,p
x
x
x
Ω
Ω
( , ,
, ,
ˆ ( ˆ ˆ ˆ ˆ , ˆ , ˆ )
p p
p p
p p
y z
y z
y z
We can construct the corresponding
quantum mechanical operator operator
from the substitution :Classical Mechanics Quantum Mechanics
;
;
;
x p x x pi x
y p y y p
i y
z p z z pi z
x x
y y
z z
ˆ ˆ
ˆ ˆ
ˆ ˆ
− > − >
− > − >
− > − >
h
h
h
δ δ
δ
δ δ
δ
as (x,y,z, i i i ˆ , , )Ω
h h h d
dx
d
dy
d
dz
What you should learn from this lecture
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What you should learn from this lecture2. Postulate 3 (Review)
eigenvalue equation :
The meassurement of the quantity represented byhas as the o n l y outcome one of the eigenvalues n = 1,2,3 ....
to the
ˆ
ˆ
Ω
Ωϖ
ψ ϖ ψ n
n n n=
4. For a system in a state described by (x,y,z, t) theprobability to obtain the value in a meassurement of
is | a where a (x,y,z, t) dxdydz
Here is an eigenvalue to and
the corresponding eigenfunction
n
n n
n
=-- -
ΨΩ
Ψ
Ω
ϖ
ψ
ϖ ψ ϖ ψ ψ
|
ˆ
*2
∞
∞
∞
∞
∞
∞∫ ∫ ∫
=
n
n n n n
3. Postulate 5.
For a system in a state described by (x,y,z, t)the average value meassured for will be
(x,y,z,t) (x,y,z,t)dxdydz
We call that the expectation value.
=
-- -
ΨΩ
Ω Ψ ΩΨ< > ∫ ∫ ∫ ∞
∞
∞
∞
∞
∞ ˆ ˆ*