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
radiation
.
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
A link between the particle
(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
addressed
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
grad
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
iħ
/
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
leads
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
http://www.almaden.ibm.com/vis/stm
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
adatoms
.
The
barriers
were
assembled
by
individually
positioning
Fe
adatoms
using
the
tip
of
a
low
temperature
scanning
tunneling
microscope
(STM)
.
A
circular
corral
of
radius
71
.
3
Angstroms
was
constructed
in
this
way
out
of
48
Fe
adatoms
.
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
adatoms
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
about
the
position
of
the
particle
(
x=
0
),
then
there
is
a
complete
uncertainty
about
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
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