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Standing Waves
Wave Function
Differential Wave Equation
Standing Waves
Boundary Conditions:
 ( x  0, t )   ( x  L, t )  0
Separation of variables:
X=0
sin(x/L)
sin(2x/L)
sin(3x/L)
X=L
1,0
Wave Function:
0,5
Y Axis Title
 ( x, t )  X ( x)T (t )
0,0
-0,5
-1,0
0
X Axis Title
L
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
 2
 2 X ( x)
 T (t )
2
x
x 2
 2
 2T (t )
 X ( x)
2
t
t 2
Particles and Waves
Space: f(x)
 2
1  2
 2 2
2
x
v t
TIme: f(t)
1  2 X ( x)
1  2T (t )
 2
 constant
2
2
X ( x) x
v T (t ) t
1 d 2 X ( x)
1 d 2T (t )
2
 2
  2  constant
2
2
X ( x) dx
v T (t ) dt
v
Equivalent to two ordinary (not partial) differential equations:
d 2 X ( x)
2
  2 X ( x)
2
dx
v
d 2T (t )
2



T (t )
2
dt
 nx 
 An sin( nt )  Bn cos(nt ) 
 L 
 ( x, t )  X ( x)T (t )  sin 
Space: X(x)
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Time: T(t)
Particles and Waves
n
Eigenvalue Condition:

2
L
n=0, ±1, ±2, ±3……
 nx 
 An sin( nt )  Bn cos(nt ) 
 L 
Eigenfunctions:  n ( x, t )  X ( x)T (t )  sin 
General solution: Principle of superposition


n 0
n 0
Since any linear Combination of
the Eigenfunctions would also be
a solution
 nx 
 An sin( nt )  Bn cos(nt ) 
 L 
   n ( x, t )   sin 
Fourier Series
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Particles and Waves
Fourier Series
Y Axis Title
Any arbitrary function f(x) of
period L can be expressed as a
Fourier Series
X Axis Title
2nx
2nx 

f ( x)  f ( x  L )    An sin(
)  Bn cos(
)
L
L 
n 0 


 2nx  
f ( x)  f ( x  L)    C n exp  i
 
 L 
n   

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REAL
Fourier Series
COMPLEX
Fourier Series
Particles and Waves
Wave Phenomena
Reflexion
Refraction
Diffraction is the bending
of a wave around an
obstacle or through an
opening.
qi
qi  qr
Interference
Diffraction
n1
qt
Wavelenght
dependence
Diffraction at Slits
n2
n1 sin (qi) = n2 sin (qt) w
q
p=w sinqm
bright fringes
Diffraction at a lattice
q
p=w sinqm
d
The path difference
must be a multiple
of a wavelength to
insure constructive
interference.
q
p=d sinqm
bright fringes
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Particles and Waves
Intensity pattern that shows the
combined effects of both diffraction
and interference when light passes
through multiple slits.
Interference and Diffraction: Huygens construction
m=2
m=1
m=0
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Particles and Waves
Black-Body Radiation
A blackbody is a hypothetical object
that
absorbs
all
incident
electromagnetic radiation while
maintaining thermal equilibrium.
E ( f )df 
4V 2
f U f df
c3
U f  kT
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
Black-Body Radiation:
classical theory

2L
n
 n
dn
2L
 2
d

2L

1D
Radiation as Electromagnetic Waves
v
 ;
f
4V
dn
 4
d

d
v
 2
df
f
dn d dn 4V 2

 3 f
d df df
v
3D
Since there are many more permissible high frequencies than low frequencies, and
since by Statistical Thermodynamics all frequencies have the same average Energy, it
follows that the Intensity I of balck-body radiation should rise continuously with
increasing frequency.
Breakdown of classical
mechanical
principles
when applied to radiation
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!!!Ultraviolet Catastrophe!!!
Particles and Waves
The Quantum of Energy – The Planck Distribution Law
Physics is a closed subject in
which new discoveries of any
importance could scarcely
expected….
However… He changed the World of Physics…
Nature does not
make a Jump
Matter
Discrete
Energy
Continuous
Classical Mechanics
Max Planck
3
8

h
n
dn
Energy
Continuous E (n )dn 
Planck: Quanta
dn
3
hn / kT
c
e
1
8n 2
-34
h 6.6262 x 10 Joule.sec
E = hn
hn  kT ,  E (n )dn  3 kT dn
c
An oscillator could adquire Energy only in discrete units called Quanta
!Nomenclature change!: n → f
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Particles and Waves
Photoelectric Effect: Einstein
The radiation
itself is
quantized
Fluxe
1 Fluxe 2
Metal
n>no
no
n
I
• Below a certain „cutoff“ frequency no of incident light, no photoelectrons are
ejected, no matter how intense the light.
1
• Above the „cutoff“ frequency the number of photoelectrons is directly proportional
to the intensity of the light.
2
• As the frequency of the incident light is increased, the maximum velocity of the
photoelectrons increases.
• In cases where the radiation intensity is extremely low (but n>no photoelectrons
are emited from the metal without any time lag.
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Particles and Waves
Photon
Energy of light:
E = hn
Kinetic Energy = Energy of light – Energy needed to escape surface (Work Function):
½ mev2= hn  hno
Fo : It depends on the Nature
of the Metal
• Increasing the intensity of the light would correspond to increasing the
number of photons.
• Increasing the frequency of the light would correspond to increasing the
Energy of photons and the maximal velocity of the electrons.
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Particles and Waves
Light as a stream of Photons?
E = hn discrete
Zero rest mass!!
Light as Electromagnetic Waves?
E = eo |Eelec|2 = (1/mo |Bmag|2 continuous
The square of the electromagnetic wave at
some point can be taken as the Probability
Density for finding a Photon in the volume
element around that point.
Energy having a definite and smoothly varying
distribution. (Classical)
Albert Einstein
A smoothly varying Probability Density for finding an
atomistic packet of Energy. (Quantical)
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Particles and Waves
The Wave Nature of Matter
All material particles are
associated with Waves
(„Matter waves“
De Broglie
A central concept of Quantics:
wave–particle duality is the
concept that all matter and
energy exhibits both wave like and particle -like
properties.
E = hn
mc2 = hn = hc/
E = mc2
or: mc = h/
A normal particle with nonzero rest
mass m travelling at velocity v
Then, every particle with nonzero
rest mass m travelling at velocity v
has an related wave 
mv = p = h/
 = h/ mv
1. The particle property is caused by their mass.
2. The wave property is related with particles' electrical charges.
3. Particle-wave duality is the combination of classical mechanics and electromagnetic field
theory.
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Particles and Waves
Electron Diffraction
Electron Diffraction
Crystalline Material
Expected
Source
Experimental
Source
Amorphous Material
Conclusion: Under certain circunstances an electron behaves also as a Wave!
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
The Waves and the Incertainty Principle of Heisenberger
„The measurement of particle position
leads to loss of knowledge about particle
momentum and visceversa.“
y
v
Dy
p
q
m
2p sin q =
Dpy2p/Dy
sin q = ±/Dy
Dpy . Dy ≈
2p = 2h
x
The momentum of the incoming beam is all in the x direction. But as a result of
diffraction at the slit, the diffracted beam has momentum p with components on
both x and y directions.
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Particles and Waves
Schrödinger's cat
It is a „Gedanken“ (thought experiment) often described as a paradox
I don‘t like it and I regret
that I got involved in it….
Superposition of two States:
Broadly stated, a quantum superposition is
the combination of all the possible states of a
system.
Alife+ Dead
Schrödinger
Miau
!
Alife
Dead
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Particles and Waves
Schrödinger wrote:
„One can even set up quite ridiculous cases. A cat is penned up in a steel
chamber, along with the following device (which must be secured against direct
interference by the cat): in a Geiger counter there is a tiny bit of radioactive
substance, so small, that perhaps in the course of the hour one of the atoms
decays, but also, with equal probability, perhaps none; if it happens, the counter
tube discharges and through a relay releases a hammer which shatters a small
flask of hydrocyanic acid. If one has left this entire system to itself for an hour,
one would say that the cat still lives if meanwhile no atom has decayed. The Y
function of the entire system would express this by having in it the living and dead
cat (pardon the expression) mixed or smeared out in equal parts.
It is typical of these cases that an indeterminacy originally restricted to the atomic
domain becomes transformed into macroscopic indeterminacy, which can then be
resolved by direct observation. That prevents us from so naively accepting as
valid a "blurred model" for representing reality. In itself it would not embody
anything unclear or contradictory. There is a difference between a shaky or outof-focus photograph and a snapshot of clouds and fog banks.“
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Particles and Waves
Schrödinger's famous tought experiment poses the question: when does a
quantum system stop existing as a mixture of states and become one or the
other? (More technically, when does the actual quantum state stop being a linear
combination of states, each of which resemble different classical states, and
instead begin to have a unique classical description?) If the cat survives, it
remembers only being alive. But explanations of experiments that are consistent
with standard microscopic quantum mechanics require that macroscopic objects,
such as cats and notebooks, do not always have unique classical descriptions.
The purpose of the thought experiment is to illustrate this apparent paradox: our
intuition says that no observer can be in a mixture of states, yet it seems cats, for
example, can be such a mixture. Are cats required to be observers, or does their
existence in a single well-defined classical state require another external
observer?
An interpretation of quantum mechanics. A key feature of quantum
mechanics is that the state of every particle is described by a wavefunction,
which is a mathematical representation used to calculate the probability for it to
be found in a location, or state of motion. In effect, the act of measurement
causes the calculated set of probabilities to "collapse" to the value defined by the
measurement. This feature of the mathematical representations is known as
wave function collapse.
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Particles and Waves