chem 373- lecture 7: expectation values

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



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



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 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



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



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.



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


This is a simple extension

of the Born postulatewhich is part of

New apl. of Born interp.



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.



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


(x) = k (x) Fψ ψ

= k (x) (x)dx-

ψ ψ * *

∞

∞

∫ = k

New postulate 5.



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 =


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.



< >= ∑ = ∫ −∞

∞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.



< >= ∑ = ∫ −∞

∞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.



< 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


good

answer to




< 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


good

answer to




< 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




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


good

answer to




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




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



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



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*

chem 373- lecture 7: expectation values

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