3 The failures of classical physics
3.1Black body radiation
A ‘black body’ is an object capable of emitting and absorbing all
frequencies of radiation uniformly.
Example: a pinhole in an empty container maintained at a constant
temperature.
The energy density
r (=dE /dl) of the EM
field inside the
container increases
as the temperature T
is increased
3.1.1The energy density vs the wavelength
for black body radiation
Classical viewpoint: Electromagnetic
(EM) field as a collection of oscillators
of all possible frequencies (Lord
Rayleigh): Presence of frequency n
signifies that the EM oscillator of that
frequency has been excited.
This law is quite successful at long
wavelengths (low frequencies), but
fails badly at short wavelengths
(high frequencies).
(‘ultraviolet catastrophe’)
The explanation of black body radiation is beyond the capabilities
of classical physics!
3.2 The quantization of energy
Max Planck studied black-body radiation in 1900
from the viewpoint of thermodynamics.
He found that he could account for the experimental
observations by proposing that the
energy of each em oscillator is limited to discrete values
(and cannot be varied arbitrarily)  quantization of energy.
This is in contrast to classical physics!
Permitted energies of an em oscillator of frequency n are
integer multiples of hn:
E  nh
n=0, 1, 2, … h is a fundamental constant
(Planck’s constant)
3.3 Atomic and molecular spectra
Evidence for the quantization of energy comes also from the
observation of the discrete frequencies of radiation absorbed and
emitted by atoms and molecules:
He
H
C
If the energy of an atom decreases by DE, the energy is carried
away as radiation of frequency n= DE /h
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3.4 What is light?
3.5 What are Waves?
There are several (in fact two) ways to think of light:
1) Classical description: light is an electro-magnetic wave.
This means that it is a varying electric and magnetic field, which
spreads out or propagates from one place to another. It is not a
physical substance.
Waves are disturbance that travel through space with a finite
velocity.(e.g. ocean waves or sound waves).
2) Modern quantum mechanical description: light can also be
considered to be particles called ‘photons’. These carry energy
and momentum but have no mass (at rest).
Harmonic waves are waves with displacements that can be
expressed as sine and cosine functions.
Waves can be characterised by a wave equation, a differential
equation that describes the motion of the wave in space and time.
These concepts are used in classical physics to describe the wave
character of sound waves or electromagnetic radiation.
In both descriptions, the light energy is carried by a very real and
observable mechanism. Photons can be thought of as waves
and as ‘particles’.
3.5.1 Waves
(a) The wavelength, l, of a wave is
the peak-to-peak distance.
(b) The wave is shown travelling to
the right at a speed c. At a given
location, the instantaneous
amplitude of the wave changes
through a complete cycle (the four
dots show half a cycle) as it
passes a given point. The
frequency, n, is the number of
cycles per second that occur at a
given point. Wavelength and
frequency are related by l n =c.
3.5.2 Examples of waves
Longitudinal wave
http://www.glenbrook.k12.il.us/gbssci/phys/Class/waves/u10l1b.html
Transversal wave
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3.5.3 Parameters by which waves are
characterized
•
•
3.5.3 Parameters by which waves are
characterized
Amplitude (A): the maximum excursion of particles from their
original position. For example, how much back and forth the air
molecules in a sound wave vibrate, or how far up and down the
violin string vibrates.
• Angular frequency (w=2pf): also used as a `frequency',
Wavelength (l ): the distance over which the wave pattern
repeats itself.
• Velocity (c=l /T=l n)
Frequency (f or n): the number of times per unit time that the
same shape or disturbance passes by.
• Wavelength, l, and frequency, n (colour!)
measured in radians per second, with rad/s equivalent to 1
cycle per second.
• Wave number (k=2p/l )
For electromagnetic waves (such as light, X-rays etc):
•
• Velocity = speed of light, c = l n
• Energy E=h n
•
Period (T=1/n): the amount of time it takes before the wave
pattern repeats.
3.6 Wave-particle duality
3.6.1 The particle character of waves
The energy of the electro-magnetic (EM) field and of
oscillating atoms are quantized.
The particle character of EM radiation
EM radiation of frequency n can possess only the energies 0,
hn, 2hn, ...
(photoelectric effect)
This suggests that it can be thought of as consisting of 0, 1,
2, … particles, each particle having an energy hn. These
particles are called photons.
• EM waves are transversal waves!
3.6.2 The wave character of particles
Davisson and Germer observed (accidentally) a diffraction
pattern when bombarding a crystal with electrons in 1925.
Diffraction is a characteristic property of waves and occurs
when there is interference.
Diffraction pattern of crystalline aluminum
bombarded with electrons
The experiment which has been repeated with other particles
(including hydrogen molecules) shows clearly that particles
have wave-like properties.
On an atomic scale the classical concepts of particle and
wave melt together.
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3.6.3 The wavelength of particles–de Broglie relation
de Broglie suggested in 1924 that any particle,
not only photons, travelling with a linear
momentum p should have (in some sense) a
wavelength l given by:

3.7 Schematic of the diffraction of an
electron beam from a single crystal surface
h
p
The scattering of an
electron beam from a nickel
crystal shows a variation of
intensity characteristic of a
diffraction experiment in
which waves interfere
constructively and
destructively in different
directions.
What about macroscopic objects…?
Worked example: Diffraction experiment with tennis balls
Today the wave-like character of electrons is the basis of the
structural technique of electron diffraction. Electrons are
scattered strongly by their interaction with the charges of
electrons and nuclei. The application to surfaces is called
Low Energy Electron Diffraction (LEED).
3.7.1 Application of the wave character of particles
The example shows a
sequence of LEED
pattern of Fe and Fe-rich
alloys containing Mn and
Ni. The images in the left
column are at
thicknesses where the
films are uniformly
magnetized. These
pattern have a (nx1)
symmetry with n~5. At a
slightly larger coverage
this has changed to a
(2x1) pattern or to a
p(1x1) symmetry.
3.8 Conclusions
There is a need to describe ‘small’ objects in a different way
from what we are all familiar with on the macroscopic scale
(‘Classical Mechanics’).
The description of macroscopic objects within the new frame
of ‘Quantum Mechanics’ has of course to be consistent with
what is known from classical physics.
http://www.physik.fu-berlin.de/~schumann/LEED.htm
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