# Standing Waves - PowerPoint PPT Presentation

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## Standing Waves

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Title: Standing Waves

1
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
A blackbody is a hypothetical object that absorbs
maintaining thermal equilibrium.
8
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
!!!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
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!