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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)
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
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
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
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   

PHYSICAL CHEMISTRY - ADVANCED MATERIALS
REAL
Fourier Series
COMPLEX
Fourier Series
Particles and Waves
Wave Phenomena
Reflexion
qi  qr
Refraction
Interference
Diffraction
Diffraction is the bending
of a wave around an
obstacle or through an
opening.
qi
n1
qt
Wavelenght
dependence
n2
n1 sin (qi) = n2 sin (qt) w
q
p=w sinqm
bright fringes
d
The path difference
must be a multiple
of a wavelength to
insure constructive
interference.
q
p=d sinqm
bright fringes
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
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
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
!!!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
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
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.
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
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.
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
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)
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
Particles and Waves
The Wave Nature of Matter
All material particles are
associated with Waves
(„Matter waves“
E = hn
mc2 = hn = hc/
E = mc2
or: mc = h/
De Broglie
A normal particle with nonzero rest
mass m travelling at velocity v
PHYSICAL CHEMISTRY - ADVANCED MATERIALS
mv = p = h/
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