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Standing Waves

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Title: Standing Waves


1
Something more about.
Standing Waves
Wave Function
Differential Wave Equation
Standing Waves Boundary Conditions
Separation of variables
XL
X0
Wave Function
2
Space f(x)
TIme f(t)
Equivalent to two ordinary (not partial)
differential equations
Space X(x)
Time T(t)
3
Eigenvalue Condition
n0, 1, 2, 3
Eigenfunctions
Since any linear Combination of the
Eigenfunctions would also be a solution
General solution Principle of superposition
Fourier Series
4
Fourier Series
Any arbitrary function f(x) of period L can be
expressed as a Fourier Series
REAL Fourier Series
COMPLEX Fourier Series
5
Wave Phenomena
Reflexion
Refraction
InterferenceDiffraction
Diffraction is the bending of a wave around an
obstacle or through an opening.
qi
Wavelenght dependence
n1 n2
qt
q
n1 sin (qi) n2 sin (qt)
w
pw sin(q)
ml
bright fringes
The path difference must be a multiple of a
wavelength to insure constructive interference.
q
d
ml
pd sin(q)
bright fringes
6
Interference and Diffraction Huygens construction
Intensity pattern that shows the combined effects
of both diffraction and interference when light
passes through multiple slits.
m2
m1
m0
7
Black-Body Radiation
A blackbody is a hypothetical object that absorbs
all incident electromagnetic radiation while
maintaining thermal equilibrium. 
8
Black-Body Radiation classical theory
Radiation as Electromagnetic Waves
1D
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
!!!Ultraviolet Catastrophe!!!
9
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
Energy Continuous
Planck Quanta
E hn
h 6.6262 x 10-34 Joule.sec
An oscillator could adquire Energy only in
discrete units called Quanta
!Nomenclature change! n ? f
10
Metal
Photoelectric Effect Einstein
The radiation itself is quantized
Fluxe
Fluxe
1
2
ngtno
I
n
no
  • 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 ngtno) photoelectrons are
    emited from the metal without any time lag.

11
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.

12
Light as a stream of Photons? E hn
discrete
Zero rest mass!!
Light as Electromagnetic Waves? E eo
Eelec2 (1/mo) Bmag2 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)
A smoothly varying Probability Density for
finding an atomistic packet of Energy. (Quantical)
Albert Einstein
13
The Wave Nature of Matter
All material particles are associated with
Waves (Matter waves
E hn E mc2
mc2 hn hc/l or mc h/l
De Broglie
A normal particle with nonzero rest mass m
travelling at velocity v
mv p h/l
14
Electron Diffraction
Electron Diffraction
Amorphous Material Crystalline Material
Source
Experimental
Source
Expected
Conclusion Under certain circunstances an
electron behaves also as a Wave!
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