Chem 373- Lecture 8: Hermetian Operators and the Uncertainty Principle

8/3/2019 Chem 373- Lecture 8: Hermetian Operators and the Uncertainty Principle

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Lecture 8: Hermetian Operators and

the Uncertainty Principle

The material in this lecture covers the following in Atkins.

11.6 The uncertainty principle

Lecture on-line

Hermetian Operators and the Uncertainty Principle (PDF)

Hermetian Operators and the Uncertainty Principle(PPT)handouts

Assigned problems


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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 4)

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 **You should read this**

Dirac Notation **You should read this**

Use of Matricies

Basic math background

Differential Equations

Operator Algebra

Eigenvalue EquationsExtensive account of Operators

Historic development of quantum mechanics from classical

mechanics

The Development of Classical Mechanics

Experimental Background for Quantum mecahnicsEarly Development of Quantum mechanics


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Audio-visuals on-line

Heisenberg Uncertainty principle (PDF)(simplified version from Wilson)

Heisenberg Uncertainty Principle (HTML)

(simplified version from Wilson)

Heisenberg Uncertainty Principle (PowerPoint

****)(simplified version from Wilson)

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)


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Hermetian Operators

Hermetian OperatorsConsider a system described by the state function .

Let F^

be the operator representing the observable F

The average value of F , or the expectation value is given by

=

* F^ d

A physical expectation value must be real

Thus :

=

* F^ d = (

* F^ d )

=

(F^ ) d =

An operator that satisfy this condition is Hermit ian

One Definition ofHermitian operator


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Hermetian Operators

We shall now show that a hermitian operatorA satisfy

f Agdx g(Af) dx ; if f and g are well behaved* *

=

We have from the difinition of a hermetian operator

* * ( )A dx A dx =

Let ( )x = f(x) + cg(x); c = constant

We have : [f(x) cg(x)] A[f(x) cg(x)]dt

[f(x) cg(x)] (A[f(x) cg(x)]) dt

*

*

+ + =

+ +

This equation must hold for any c

Alternative definition ofHermitian operator


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Hermetian Operators

We have : + + =

+ +

[ ( ) ( )] [ ( ) ( )]

[ ( ) ( )] ( [ ( ) ( )])

*

*

f x cg x A f x cg x d

f x cg x A f x cg x d

Expanding:

f A c f A

c g A cc g A

* *

* * * *

+ +

+ =

f d g d

f d g d

The first and last term on each side are the same as A

is hermetian


f A c g A

c f A c c g A

f) d ( f) d

g) d ( g) d

* *

* *

(

( * * +

+


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Hermetian Operators

c f A g dt c g A f dt

c g (A f) dt c f (A g) dt

* * *

* * *

+ =

+

hus:

This equation must be satisfied for all c

i i i if A g dt g A f dt g (A f) dt f (A g) dt* * * * = c = i

f A g dt g A f dt g (A f) dt f (A g) dt* * * * + = +

c = 1

f A g dt g A f dt g (A f) dt f (A g) dt* * * * =

after dividing with i



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Hermetian Operators

f A g dt g (A f) dt

* *

=

After adding the two equations from c = 1 and c = i :

f A g dt g A f dt g (A f) dt f (A g) dt* * * * + = +

f A g dt g A f dt g (A f) dt f (A g) dt* * * * =



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Let the linear operators A and B havea complete set of common eigenfunctions

The Uncertainty Principle

Ag = a gi i i Bg = b gi i i

Let A and B represent two observables

In this case if the system is described bygi

A meassurement of A and B willhave as the only outcome

< >=A ai and < >=B bi

Operators with commoneigenfunctions commute


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Let the linear operators A and B havea complete set of common eigenfunctions

Ag a gi i i= Bg b gi i i=

Then A and B must commute :

ABf BAf = [A,B]f = 0 for any function f

proof :f c gii

i= since g forms a complete set ofeigenfunctions

i



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[A,B]f = ? f c gii i= ( ) ( )

( ) ( )

ABf AB c g A c Bg

A c b g c b Ag c b a g

i i

i

i i

ii i i

ii i i

ii i i i

i

= =

= = =

( ) ( )

( ) ( )

BAf BA c g B c Ag

B c a g c b Bg c a b g c b a g

i ii

i ii

i i i

i

i i i

i

i i i i

i

i i i i

i

= =

= = = =

ABf - BAf = [A,B]f = 0Thus :



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On the other hand :

If A and B do not commute : [ , ]A B f 0

Than we can not find a commen set

of eigenfunctions g such that :iAg = a gi i i Bg = b gi i i

We can not find states such that themeassurement of A and B each time

have the same outcome a and bi i

Operators that commutehave common

eigenfunctions

Why ?


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The Uncertainty PrincipleWe have shown

( , ]ddx

x)f - (x ddx

) f = [ ddx

x f f= 1

Consider now :

[x,p x,

i i

x,

i

x ] [ ] [ ]= = = =h h h

hd

dx

d

dx

i

[ , ] ; [ , ]d

dx

x xd

dx

= = 1 1

Also :

[x,p p p x

x2

x2

x2 ] x = p p x +p p p p

x2

x2

x x x x x x x

[ , ] x p px x [ , ]p x px x

[x,p p p

x

2

x x ] = + =i i

d

dxh h h2 2

Important operators

that don't commute


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For a particle in 3 -D[x,H] = [x,(T + V)] = [ , ] [ , ( , , )]x T x V x y z +

o

= [ , ]x T

= + +[ , ( )]xm

p p px y z1

2

2 2 2 = + +1

2

1

2

1

2

2 2 2

mx p

mx p

mx px y z[ , ] [ , ] [ , ]

oo

=1

2

2

mx px[ , ]

=

1

22

mi px( )h

=

i

mpx

h

We might also show :

[ , ]( , , )

p Hi

dV x y z

dxx =

h

Important operatorsthat don't commute


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The two operators p and x do not commutex

[x,px ] = ih

Thus we can not simultaniously findeigenfunctions to both operators

Consider a statefunction that is an eigenfunction to H

Since : [ , ]x H i= h

The statefunction is not an eigenfunction to x

Thus a state described by will nothave a sharp value for x

The meassurement of x can have different outcomes



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[ , ] ( , , )p H i d V x y z dxx =

h

Also since :

A system described by the statefunction

will not in general have a sharp value for px

That is the meassurement of p will have

more than one outcome x

Exceptions ?



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What you should learn from this lecture

2. A hermitian operator A satisfy

A dx (A ) dx or f Agdx g(Af) dx

for the "well behaved functions , f, and g

* * * *

= =

1.

| | *

Dirac notation

m n m n m n mnF d F F m F n F = = ( ) = =

3.

[A,B] = 0

If the linear operators A and B have

a complete set of common eigenfunctions

4.

.

[A,B] = 0 ,

.In this case one can find states such that a meassurementof A and B will give the same outcomes a and b

the meassurements are carried out. The two values

a and b are eigenvalues to

n m

n m

If the linear operators A and B have

a complete set of common eigenfunctions

each

time

A and B


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What you should learn from this lecture

5.

The uncertainty relation of quantum mechanics :

If do not commute, [A,B] 0 , meassurements

of A and B will give different values each time.

If the standard deviation in the meassurements of A and B

are A and B than :A B =

1

2i[A,B]

on the state we might have A B or A < B.However A B must be constant.

*

A and B

d

Depending

6. commutation relations :

[x,px

Im tan

[ , ] [ , ]; ] ;[ , ] ( , , )

por t

x H x T i p Hi

dV x y zdx

x= = =hh


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The Dirac Notation

We shall often be working with integrals of the form

m * F n d

where F^

is an operator

We shall introduce the DIRAC bracet notation orabbreviation

m * F

^ n d = = (m|F

^|n ) = < m|F

^|n>

We might also write

m * F

^ n d = Fmn

Dirac notationAppendix A


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For the special case in which F = 1 one has

m * n d = =

We might refer to as an overlap integral

The special overlap integral

m * m d = =

is refered to as the norm of m

We have

* = (

m * n d ) =

m n d

In particular < m|m> =

Dirac notationAppendix A


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Consider a large number N of

identical boxes with identical

particles all described by the

same statefunction ( , , ) :x y z

Consider the observable A represented by the operator A

Let [A, H] 0

Thus the system described by do not

have sharp value for A.

The average (expectation ) value is defined by :

< A >= A*

d

Appendix B


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A1

A2 An


The measurement of A on each of the n identicalsystems will give a different outcome A i

We define the variance as :

1

nii < >( ) = =A A AA

2 2 2 ( )

A A d2 2= < >

*( )A = < > + < > *( )A2 2 2A A A d

= < > + < >

* * * A

2

d A A d A d 22

= < > = < > < > *A2 d A A A 2 2 2

Appendix B


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We define : A = A2 as the standard deviation

e shall later show that two for two observables A and B

A B = 12i

[A,B]* d

Consider as an example x and px

[x,px ] = ihSince :

x =1

2i [x,p ]* xp dx = 1

2 h

We can not simultaniously obtain sharp values

for x and px

Appendix B


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( ) expx ikx=

p kx = h p kx = h

( ) expx ikx=

< < x < < x

xpx

= = 0

x

px

=

= 0

Appendix B


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We can write as a superposition of cosnxn = 1,2,3,4,5,6,7,8...

=

= +

C nx

C e e

nn

n

nn

ninx inx

0

0

cos

( )

Now x decreasesand p increasesx

Appendix B


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The Uncertainty Principle Appendix B

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Chem 373- Lecture 8: Hermetian Operators and the Uncertainty Principle

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