chem 373- lecture 6: operators and quantum mechanics

8/3/2019 Chem 373- Lecture 6: Operators and Quantum Mechanics

1/20

Lecture 6: Operators and Quantum

Mechanics

The material in this lecture covers the following in Atkins.

11.5 The informtion of a wavefunction

(c) Operators

Lecture on-line

Operators in quantum mechanics (PDF)

Operators in quantum mechanics (HTML)

Operators in Quantum mechanics (PowerPoint)

Handout (PDF)Assigned Questions


2/20

Tutorials on-line

Reminder of the postulates of quantum mechanicsThe postulates of quantum mechanics (This is the writeup for Dry-lab-II)(

This lecture has covered postulate 3)

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 Matricies

Basic math backgroundDifferential 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


3/20

Audio-visuals on-lineEarly 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 ,**)


4/20

(Ia) A Quantum mechanical system is specifiedby the statefunction ( )x

( ) (Ib x) contains all

information about the system we can know

The state function

We now have

Operators and Quantum Mechanics

(Ic) A system described by thestate function H ( (x) = E x)has exactly the energy E

Review


5/20

O

X

We have seen that a ' free' particle movingin one dimension in a constant (zero) potential

has the Hamiltonian

=h2 2

22m xE

((

x)x)

H

m x= h

2 2

22

The Schrodinger equation is

with the general solution:

( ) exp exp x A Bikx ikx= +

and energies E =

2h k

m

2

2


Review


6/20

How does the state function (x, t) give usinformation about an observable other thanthe energy such as the position or the momentum ?

Any observable ' ' can be expressed in classical physicsin terms of x,y, z and p .x

, ,p py z

Examples :

= x, p p T, V(x), Ex x2 , , ,vx


Good question


7/20

Classical Mechanics Quantum Mechanics

;

;

;

x p x x pi x

y p y y pi y

z p z z pi z

x x

y y

z z

> >

> >

> >

h

h

h

We can construct the corresponding operatorfrom the substitution:

as (x,y,z, i i i

, , )h h h

ddx

ddy

ddz

Such as:

= x, p p T, V(x), Ex x2

, , ,

vx

Operators and Quantum Mechanics Review


8/20

n n n=

For an observable with the corresponding

operator we have the eigenvalue equation :


(IIIa). The meassurement of the quantity represented byhas as the o n l y outcome one of the values

n n = 1,2,3 ....

(IIIb). If the system is in a state described by

a meassurement of will result in thevalue

n

n

Im tanpor t news


9/20

Quantum mechanical principle Operators..

we can solve the eigenvalue problem

n n n=

For any such operator

We obtain eigenfunctions and eigenvalues

The only possible values that can arise from measurementsof the physical observable are the eigenvalues n

Postulate 3

Im tanpor t news


10/20

The x - component 'p ' of the linear momentumx

r r r rp p e p e p ex x y y z z= + +

Is represented by the operator pi

x =h

x

With the eigenfunctions Exp ikx[ ] and eigenvalue kh

hh

i

Exp ikxExp ikx

x

= k[ ]

[ ]

We note that k can take any value

> k >

Operators and Quantum MechanicsIm tanpor t news


11/20

( ) exp exp x A Bikx ikx= + and energies E =

2h k

m

2

2

For A = 0

=( ) expx B

ikx

this wavefunction is also an eigenfunction to px

With eigenvalue for p of - kx h

Thus describes a particle of energy E =

and momentum p note E =

P

2m as it must be.

-

2

x

x2

( )

;

xk

m

k

h

h

2

2

=

This system corresponds to a particlemoving with constant velocity

v

p

m - k/mxx

= =h

We know nothing about its positionsince | (x)|2 = B

Operators and Quantum Mechanics New insight


12/20

( ) exp exp x A Bikx ikx= + and energies E =

2h k

m

2

2

For B = 0

+ =( ) expx A

ikx

this wavefunction is also an eigenfunction to px

With eigenvalue for p of kx h

Thus describes a particle of energy E =

and momentum p note E =P

2m

as it must be.

2

xx2

( )

;

xk

m

k

h

h

2

2

=

This system corresponds to a particlemoving with constant velocity

vp

m

k/mxx= = h We know nothing about its position

since | (x)|2 = B



13/20

What about : ?( ) exp expx A Bikx ikx = +

It is not an eigenfunction to p since :x

exp expexp exp

p (x) = A i B ix

h h

h h

d

dx

d

dxA k B k

ikx ikx

ikx ikx+

=

How can we find

p in this case ?x



14/20

A linear operator A will have a set of

eigenfunctions f {n = 1,2,3..etc}

and associated eigenvalues k such that :n

n

( )x

The set of eigenfunction {fis orthonormal :

f f

n

ispace

j

( ), ..}

( ) ( )*

x n

x x dxall

ij

=

=

1

( ) ( )Af fn nx k xn=

= o if i j

= 1 if i = j

Quantum mechanical principles..Eigenfunctions


15/20


e ei j ij =

ei

ei

ei

An example of an orthonormal set is the Cartesian unit vectors

An example of an orthonormal function set is

nL

n x

L(x) =

n = 1,2,3,4,5....

1sin

no

L

m nmx x( ) ( )* =


16/20

That is, any function g(x) thatdepends on the same variablesas the eigenfunctions can be written

g(x) = ani=1

allf xn( )

where

a f g(x)n n*

space

= ( )x dxall


The set of eigenfunction {fforms a complete set.

n( ), ..}x n = 1

ei

ei

ei

rei ; i = 1,2,3 form a complete set

For any vector vr

v v e e v e e v e e= + + ( ) ( ) ( )r r r r r r r r r

1 1 2 2 3 3


17/20

we can show that : a f g(x)n nV

= ( )*x dx

In the expansion: g(x) = a (1)ii=1

all

f xi( )

from the orthonormality : f fi

V

j( ) ( )*x x dx ij =


g(x) = a f (x) g(x)dx = aii=1

all

n

*

ii=1

all

f x f x f x dxi V V n i( ) ( ) ( )

*

A multiplication by f (x) on both sides followed by

integration affordsn

or : a g(x)fn nspace

= ( )x dxall ij

A multiplication by f (x) on both sides followed by

integration affordsn

or : a g(x)fn nspace

= ( )*x dxall


18/20

is a linear combinationof two eigenfunctions to px( ) exp exp

x A Bikx ikx= +

How can we find

p in this case ?x


p kx =h

p kx = h


19/20

1. Postulate 3

For an observable with the corresponding operatorwe have the eigenvalue equation :

( )

n n n

n

i

=The meassurement of the quantity represented by

has as the o n l y outcome one of the values n = 1,2,3 ....(ii) If the system is in a state described by

a meassurement of will result in the valuen

n

What you should learn from this lecture

Illustrations

x A

x A

mx

x

ikx

ikx

:

( ) exp

( ) exp

)

( )

( )

+

+

==

is an eigenfunction to p with eigenvalue k

is an eigenfunction to p with eigenvalue - k

Both are eigenfunctions to the Hamiltonian for a free particle

H = (p with eigenvalues E = k2m

represents a free particle of momentum k

represents a free particle of momentum - k

x

x

2 x 2

h

h

h h

h

h

2 2

2


20/20

What you should learn from this lecture

2. Postulate 4.

The set of eigenfunction {fforms a complete set.That is, any function g(x) that depends on the samevariables as the eigenfunctions can be written :

g(x) = a where

a g(x)f

n

ni=1

all

n n

space

( ), ..}

( )

( )

x n

f x

x dx

n

all

=

=

1

chem 373- lecture 6: operators and quantum mechanics

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