1_Quantum theory_ introduction and principles

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1. Quantum theory: introduction and principles



= c

1.1 Wave
-
particle duality

1.2 The Schrödinger equation

1.3 The Born interpretation of the wavefunction

1.4 Operators and theorems of the quantum theory

1.5 The Uncertainty Principle

1.1 Wave
-
particle duality

A. The particle character of electromagnetic radiation

The photoelectric effect

The

photon

h

can

be

seen

as

a

particle
-
like

projectile

having

enough

energy

to

collide

and

eject

an

electron

from

the

metal
.

The

conservation

of

energy

requires

that

the

kinetic

energy

of

the

ejected

electron

should

obey
:

½mv
2

=

h

-

,

called

the

metal

workfunction
,

is

the

minimum

energy

required

to

remove

an

electron

from

the

metal

to

the

infinity
.

The

ejection

threshold

of

electrons

does

not

depend

on

the

intensity

of

the

incident

.

e
-

(E
k
)

h

metal

B. The wave character of the particles

Electron diffraction

Diffraction

is

a

characteristic

property

of

waves
.

With

X
-
ray,

Bragg

showed

that

a

constructive

interference

occurs

when

=
2
d

sin

.

Davidsson

and

Germer

showed

also

interference

phenomenon

but

with

electrons!

d

V

Particles are characterized by a wavefunction

An appropriate potential difference creates
electrons that can diffract with the lattice of nickel

(p=mv) and the wave (

)
natures

Example 1: Northern light

Magnetic field of the earth

The

sun

has

a

number

of

holes

in

its

corona

from

which

high

energy

particles

(e
-
,

p
+
,

n
0
)

stream

out

with

enormous

velocity
.

These

particles

are

thrown

out

through

our

solar

system,

and

the

phenomenon

is

called

solar

wind
.

A

part

of

this

solar

wind

meets

the

earth’s

magneto

sphere,

the

solar

wind

particles

are

accelerated

down

to

the

earth

along

the

open

magnetic

field

lines
.

The

field

lines

are

open

only

in

the

polar

regions
.

At

lower

latitudes

the

field

is

locked
.

That’s

why

we

have

the

Northern

Lights

only

in

the

polar

regions
.

When

the

solar

wind

particles

collide

with

the

air

molecules

(O
2
,

N
2
),

their

energy

is

transferred

to

excitation

energy

of

the

molecules
.

The

excited

molecules

come

back

in

their

ground

state

by

emitting

light

at

specific

frequencies
:

green
-
blue

color

from

N
2
,

red

and

green

from

O
2
.

It

is

billions

of

such

processes

occurring

simultaneously

that

produces

the

Northern

Lights
.

1.2 The Schrödinger Equation

From

the

wave
-
particle

duality,

the

concepts

of

classical

physics

(CP)

have

to

be

abandoned

to

describe

microscopic

systems
.

The

dynamics

of

microscopic

systems

will

be

described

in

a

new

theory
:

the

quantum

theory

(QT)
.

A

wave,

called

wavefunction

(r,t)
,

is

associated

to

each

object
.

The

well
-
defined

trajectory

of

an

object

in

CP

(the

location,

r,

and

momenta,

p

=

m
.
v,

are

precisely

known

at

each

instant

t)

is

replaced

by

(r,t)

indicating

that

the

particle

is

distributed

through

space

like

a

wave
.

In

QT,

the

location,

r,

and

momenta,

p,

are

not

precisely

known

at

each

instant

t

(see

Uncertainty

Principle
)
.

In

CP,

all

modes

of

motions

(rot,

trans,

vib)

can

have

any

given

energy

by

controlling

the

applied

forces
.

In

the

QT,

all

modes

of

motion

cannot

have

any

given

energy,

but

can

only

be

excited

at

specified

energy

levels

(see

quantization

of

energy
)
.

The

Planck

constant

h

can

be

a

criterion

to

know

if

a

problem

has

to

be

in

CP

or

in

QT
.

h

can

be

seen

has

a

quantum

of

an

action

that

has

the

dimension

of

ML
2
T
-
1

(E=

h

where

E

is

in

ML
2
T
-
2

and

is

in

T
-
1
)
.

With

the

specific

parameters

of

a

problem,

we

built

a

quantity

having

the

dimension

of

an

action

(ML
2
T
-
1
)
.

If

this

quantity

has

the

order

of

magnitude

of

h

(
~
10
-
34

Js),

the

problem

has

to

be

treated

within

the

QT
.

In

CP,

the

dynamics

of

objects

is

described

by

Newton’s

laws
.

Hamilton

developed

a

more

general

formalism

expressing

those

laws
.

For

a

conservative

system,

the

dynamics

is

described

by

the

Hamilton

equations

and

the

total

energy

E

corresponds

to

the

Hamiltonian

function

H=T+V
.

T

is

the

kinetic

energy

and

V

is

the

potential

energy
.

This

formalism

appears

to

be

close

to

that

in

which

the

dynamics

of

quantum

systems

is

developed
.

Because

of

this

similarity,

the

correspondence

principles

are

proposed

to

pass

from

the

CP

to

the

QT
:

)
,
,
,
(
2
2
t
z
y
x
V
m
p
H

)
,
,
,
(
)
,
,
,
(
2
2
2
2
2
2
2
2
2
2
t
z
y
x
V
t
z
y
x
V
t
i
E
z
y
x
p
z
y
x
i
i
p
x
x

)
,
,
,
(
2
2
2
t
z
y
x
V
m
H

E
V
T
H

t
i
H

Classical mechanics

Quantum mechanics

Schrödinger
Equation

)
,
,
,
(
2
2
2
t
z
y
x
V
m
H

t
i
H

The

Schrödinger

Equation

(SE)

shows

that

the

operator

H

and

/

t

give

the

same

results

when

they

act

on

the

wavefunction
.

Both

are

equivalent

operators

corresponding

to

the

total

energy

E
.

In

the

case

of

stationary

systems
,

the

potential

V(x,y,z)

is

time

independent
.

The

wavefunction

can

be

written

as

a

stationary

wave
:

(x,y,z,t)=

(x,y,z)

e
-
i

t

(with

E=ħ

)
.

This

solution

of

the

SE

to

a

density

of

probability

|

(x,y,z,t)|
2
=

|

(x,y,z)|
2
,

which

is

independent

of

time
.

The

Time

Independent

Schrödinger

Equation

is
:

)
,
,
(
)
,
,
(
)
,
,
(
2
2
2
z
y
x
E
z
y
x
z
y
x
V
m

E
H

or

The

Schrödinger

equation

is

an

eigenvalue

equation
,

which

has

the

typical

form
:

(operator)(function)=(constant)
×
(same

function)

The

eigenvalue

is

the

energy

E
.

The

set

of

eigenvalues

are

the

only

values

that

the

energy

can

have

(quantization)
.

The

eigenfunctions

of

the

Hamiltonian

operator

H

are

the

wavefunctions

of

the

system
.

To

each

eigenvalue

corresponds

a

set

of

eigenfunctions
.

Among

those,

only

the

eigenfunctions

that

fulfill

specific

conditions

have

a

physical

meaning
.

NB: In the following, we only
envisage the time independent
version of the SE.

1.3 The Born interpretation of the wavefunction

Example of a 1
-
dimensional system

Physical

meaning

of

the

wavefunction
:

If

the

wavefunction

of

a

particle

has

the

value

(r)

at

some

point

r

of

the

space,

the

probability

of

finding

the

particle

in

an

infinitesimal

volume

d

=dxdydz

at

that

point

is

proportional

to

|

⡲(|
2
d

|

(r)|
2

=

(r)

*
(r)

is

a

probability

density
.

It

is

always

positive!

Hence,

if

the

wavefunction

has

a

negative

or

complex

value,

it

does

not

mean

that

it

has

no

physical

meaning

because

what

is

important

is

the

value

of

|

(r)|
2

0
;

for

all

r
.

But,

the

change

in

sign

of

(r)

in

space

(presence

of

a

node)

is

interesting

to

observe

in

chemistry
:

antibonding

orbital

overlap

(see

chap

4
:

Electronic

structure

in

molecules)
.

Node

A. Normalization Condition

The solution of the differential equation of Schrödinger is defined
within a constant N
.
Indeed, if

’ is a known solution of H

’=E

’, then

=
N

’ is a also solution for the same E.

H

=E

H(N

’)= E(N

’)

N(H

’)=N(E

’)

H

’=E

The

mathematical

expression

of

the

eigenfunction

should

be

such

that

the

sum

of

the

probability

of

finding

the

particle

over

all

infinitesimal

volumes

d

of

the

space

is

1
.

That

insures

the

particle

to

be

present

in

the

space
:

Normalization

condition
.

We

have

to

determine

the

constant

N,

such

that

the

solution

=N

’of

the

SE

is

normalized
.


d
N
d
N
d
N
N
d
*
*
2
*
*
'
'
1
1
'
'
1
)
'
)(
'
(
1
B. Other mathematical conditions

(r)
≠∞

;

r

if not: no physical meaning for the normalization condition

(r) should be single
-
valued

r

if not: 2 probability for the same point!!

The SE is a second
-
order differential equation:

(r) and d

(r)/dr should be continuous

???
*


d
C. The kinetic energy and the wavefunction

V
x
m
V
T
H

2
2
2
2

d
x
m
d
x
m
T
2
2
*
2
2
2
2
*
2
2

T

The

kinetic

energy

is

then

a

kind

of

average

over

the

curvature

of

the

wavefunction
:

a

large

contribution

to

the

observed

value

originates

from

the

regions

where

the

wavefunction

is

sharply

curved

(

2

/

x
2

is

large)

and

the

wavefunction

itself

is

large

(

*

is

large

too)
.

A

particle

is

expected

to

have

a

high

kinetic

energy

if

the

average

curvature

of

its

wavefunction

is

high
.

Real part of the wavefunction for
valence electrons in the potential
created by the nuclei

r
E
r
r
V
m

2
2
2
Schrödinger:

periodic potential:

R
r
V
r
V

Bloch theorem:

r
u
e
r
k
r
k
i

periodic

Example 2: the wave function in a periodic system: electrons in a metal

Scientists

discovered

a

new

method

for

confining

electrons

to

artificial

structures

at

the

nanometer

lengthscale
.

Surface

state

electrons

on

Cu(
111
)

were

confined

to

closed

structures

(corrals)

defined

by

barriers

built

from

Fe

.

The

barriers

were

assembled

by

individually

positioning

Fe

using

the

tip

of

a

low

temperature

scanning

tunneling

microscope

(STM)
.

A

circular

corral

of

71
.
3

Angstroms

was

constructed

in

this

way

out

of

48

Fe

.

This

STM

image

shows

the

direct

observation

of

standing
-
wave

patterns

in

the

local

density

of

states

of

the

Cu(
111
)

surface
.

These

spatial

oscillations

are

quantum
-
mechanical

interference

patterns

caused

by

scattering

of

the

two
-
dimensional

electron

gas

off

the

Fe

and

point

defects
.

Example 3: Quantum corral created and observed with
Scanning Tunneling Microscopy (STM)

1.4 Operators and principles of quantum mechanics

A. Operators in the quantum theory (QT)

An

eigenvalue

equation,

f

=

f,

can

be

associated

to

each

operator

.

In

the

QT,

the

operators

are

linear

and

hermitian
.

Linearity
:

is

linear

if
:

(c

f)=

c

f

(c=constant)

and

(
f+

)
=

f+



NB
:

“c”

can

be

defined

to

fulfill

the

normalization

condition

Hermiticity
:

A

linear

operator

is

hermitian

if
:

where

f

and

are

finite,

uniform,

continuous

and

the

integral

for

the

normalization

converge
.

The

eigenvalues

of

an

hermitian

operator

are

real

numbers

(

=

*
)

When

the

operator

of

an

eigenvalue

equation

is

hermitian,

2

eigenfunctions

(
f
j
,

f
k
)

corresponding

to

2

different

eigenvalues

(

j
,

k
)

are

orthogonal
.

d
f
d
f
*
*
*
j
j
j
f
f

k
k
k
f
f

0
*

d
f
f
k
j
B. Principles of Quantum mechanics

1
.

To

each

observable

or

measurable

property

<

>

of

the

system

corresponds

a

linear

and

hermitian

operator

,

such

that

the

only

measurable

values

of

this

observable

are

the

eigenvalues

j

of

the

corresponding

operator
.

f

=

f

2
.

Each

hermitian

operator

representing

a

physical

property

is

complete

.

Def
:

An

operator

is

complete

if

any

function

(finite,

uniform

and

continuous)

(x,y,z)

can

be

developed

as

a

series

of

eigenfunctions

f
j

of

this

operator
.

j
j
j
z
y
x
f
C
z
y
x
)
,
,
(
)
,
,
(

3
.

If

(x,y,z)

is

a

solution

of

the

Schrödinger

equation

for

a

particle,

and

if

we

want

to

measure

the

value

of

the

observable

related

to

the

complete

and

hermitian

operator

(that

is

not

the

Hamiltonian),

then

the

probability

to

measure

the

eigenvalue

k

is

equal

to

the

square

of

the

modulus

of

f
k
’s

coefficient,

that

is

|C
k
|
2
,

for

an

othornomal

set

of

eigenfunctions

{
f
j
}
.

Def
:

The

eigenfunctions

are

orthonormal

if

NB
:

In

this

case
:

ij
k
j
d
f
f

*

j
j
C
1
2

4
.

The

average

value

of

a

large

number

of

observations

is

given

by

the

expectation

value

<

>

of

the

operator

corresponding

to

the

observable

of

interest
.

The

expectation

value

of

an

operator

is

defined

as
:

5
.

If

the

wavefunction

=
f
1

is

the

eigenfunction

of

the

operator

(

f

=

f
),

then

the

expectation

value

of

is

the

eigenvalue

1
.

1
*
1
1
*
*

d
d
d
For normalized
wavefunction

d
d
*
*

j
j
j
C
d

2
*

6
.

Two

operators

having

the

same

eigenfunctions

are

commutable

.

Reciprocally,

if

two

operators

commute,

they

have

a

common

“complete”

set

of

eigenfunctions
.

Def
:

If

the

product

of

two

operators

is

commutative,

1

2

-

2

1

=

(

1

2
-

2

1
)

=
0
,

then

the

operators

are

commutable
.

In

this

case,

the

commutator

(

1

2
-

2

1
),

also

written

[

1
,

2
],

is

equal

to

zero
.

1.5 The Uncertainty Principle

1
.

When

two

operators

are

commutable

(and

with

the

Hamiltonian

operator),

their

eigenfunctions

are

common

and

the

corresponding

observables

can

be

determined

simultaneously

and

accurately
.

2
.

Reciprocally,

if

two

operators

do

not

commute
,

the

corresponding

observable

cannot

be

determined

simultaneously

and

accurately
.

If

(

1

2
-

2

1
)

=

c,

where

“c”

is

a

constant,

then

an

uncertainty

relation

takes

place

for

the

measurement

of

these

two

observables
:

where

2
2
1
c

2
/
1
2
1
2
1
1

Uncertainty
Principle

Example 4: the Uncertainty Principle

1
.

For

a

free

atom

and

without

taking

into

account

the

spin
-
orbit

coupling,

the

angular

orbital

moment

L
2

and

the

total

spin

S
2

commute

with

the

Hamiltonian

H
.

Hence,

an

exact

value

of

the

eigenvalues

L

of

L
2

and

S

of

S
2

can

be

measured

simultaneously
.

L

and

S

are

good

quantum

numbers

to

characterize

the

wavefunction

of

a

free

atom

see

Chap

3

“Atomic

structure

and

atomic

spectra”
.

2
.

Position

x

and

momentum

p
x

(along

the

x

axis)
.

According

to

the

correspondence

principles,

the

quantum

operators

are
:

x

and

ħ/i(

/

x)
.

The

commutator

can

be

calculated

to

be
:

i
x
x
i

,
2

x
p
x
The

consequence

is

a

breakdown

of

the

classical

mechanics

laws
:

if

there

is

a

complete

certainty

the

position

of

the

particle

(

x=
0
),

then

there

is

a

complete

uncertainty

the

momentum

(

p
x
=

)
.

3.
The time and the energy
:

If a system stays in a state during a time

t, the energy of this system cannot be determined more accurately than with an error

E.

This incertitude is of major importance for all spectroscopies
:

see Chap 7

2

E
t