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Quantum Theory of Light
A TimeLine
Light as an EM Wave
Light as an EM Wave (Maxwell
1865-1873)

Quantum theory did not begin with an
attempt to explain the behaviour of light.
Light as an EM Wave (Maxwell
1865-1873)


Quantum theory did not begin with an
attempt to explain the behaviour of light.
Scientists had already accepted
Maxwell’s description of light!
Light as an EM Wave (Maxwell
1865-1873)

Maxwell had proposed that light was an
electromagnetic disturbance created by
extremely high frequency oscillators.
Light as an EM Wave (Maxwell
1865-1873)

Maxwell had proposed that light was an
electromagnetic disturbance created by
extremely high frequency oscillators.
Light as an EM Wave (Maxwell
1865-1873)

It was assumed from this theory that
these oscillators (resonators) were able
to emit light of frequency equal to their
own.
Light as an EM Wave (Maxwell
1865-1873)

Hertz was successful in indirectly
proving (was no way frequencies higher
than 109 Hz) the theory by showing that
it described the properties of light.
Light as an EM Wave (Maxwell
1865-1873)


Hertz was successful in indirectly
proving (was no way frequencies higher
than 109 Hz) the theory by showing that
it described the properties of light.
I.e. Reflection, interference etc.
Light as an EM Wave (Maxwell
1865-1873)



Hertz was successful in indirectly
proving (was no way frequencies higher
than 109 Hz) the theory by showing that
it described the properties of light.
I.e. Reflection, interference etc.
(Hertz unknowingly discovered the
photoelectric effect during his
experimental verifications)
Light as an EM Wave (Maxwell
1865-1873)

The success of the theory lead to its
application to the blackbody radiation
problem.
Light as an EM Wave (Maxwell
1865-1873)


The success of the theory lead to its
application to the blackbody radiation
problem.
However all attempts failed. (see Wien,
Rayleigh-Jeans Law)
Planck (1900)


The discovery by Planck was the
beginning of quantum theory.
Interpolating Wein’s law and the
relationship at low frequency he showed
that
8hf 3
u( f , T ) 
c3
1


  hf kT

1
e
Planck (1900)

From his work he concluded that the
energy was quantised, where the
frequency f can only be an integral
multiple of hf.
Planck (1900)

That is the energy which an resonator
can lose is nhf, where n = 1,2,3… .
Planck (1900)


That is the energy which an resonator
can lose is nhf, where n = 1,2,3… .
The idea of quantised energy levels
(states) was the significant development.
Planck (1900)



That is the energy which an resonator
can lose is nhf, where n = 1,2,3… .
The idea of quantised energy levels
(states) was the significant development.
This assumption contradicted what was
accepted classically.
Planck (1900)

Using this assumption he was able to
correctly produce a theory which agreed
with experimental results.
Planck (Summary)



In deriving his formula he made two
assumptions:
The energy of an oscillator of frequency f
can only be nhf.
During an emission or absorption the
change in energy is hf.
Planck (Summary)

Despite the significance, it was left to
Einstein to develop these ideas to the
next step.
Photoelectric Effect(1905)
Photoelectric Effect (1905)

When a metallic surface is illuminated by
light can cause electrons to be emitted
from the surface.
Photoelectric Effect (1905)

The number of electrons ejected from
the metal surface per second depends
on the intensity of the light.
Photoelectric Effect (1905)

The number of electrons ejected from
the metal surface per second depends
on the intensity of the light. -expected
Photoelectric Effect (1905)


The number of electrons ejected from
the metal surface per second depends
on the intensity of the light.
The kinetic energy of the electrons did
not depend on intensity.
Photoelectric Effect (1905)



The number of electrons ejected from
the metal surface per second depends
on the intensity of the light.
The kinetic energy of the electrons did
not depend on intensity.
Kinetic energy of the electrons depends
on the wavelength of the light.
Photoelectric Effect (1905)



The number of electrons ejected from the
metal surface per second depends on the
intensity of the light.
The kinetic energy of the electrons did not
depend on intensity.
Kinetic energy of the electrons depends on
the wavelength of the light. And there was a
cut of wavelength where no electrons are
emitted.
Einstein - Photoelectric Effect
(1905)

Noted that although Maxwell’s classical
theory described the interaction of light
over a long period, a new description
was needed for the individual
interactions of light and matter.
Photoelectric Effect (1905)

He extended Planck’s ideas to include
that light was also quantized.
Photoelectric Effect (1905)



He extended Planck’s ideas to include
that light was also quantized.
Light and matter interactions occur in
discrete packets called photons.
The energy of a photon is
hf 
hc

Compton Effect (1922)
Compton Effect (1922)

The Compton effect shows that photons
behave like particles with momentum hf
c
Compton Effect (1922)


The Compton effect shows that photons
hf
behave like particles with momentum c
Classical theory predicted that incident
radiation of frequency f0 would cause
electrons to be acceleration in the
direction of the radiation
Compton Effect (1922)


The Compton effect shows that photons
hf
behave like particles with momentum c
Classical theory predicted that incident
radiation of frequency f0 would cause
electrons to be acceleration in the
direction of the radiation and that the
freq of the scattered radiation depend on
the intensity and time of exposure.
Compton Effect (1922)

Instead Compton showed that only on
the scattering angle.
Compton Effect (1922)


Instead Compton showed that only on
the scattering angle.
This result disagrees with classical
theory
Compton Effect (1922)


Instead Compton showed that only on
the scattering angle.
This result disagrees with classical
theory and is only explained if the
photons act as particles.
Wave-Particle Duality
Wave-Particle Duality

Various experiments have been shown
which highlight either the wave nature or
particle nature of light.
Wave-Particle Duality


Various experiments have been shown
which highlight either the wave nature or
particle nature of light.
Is light simultaneous a wave and a
particle?
Wave-Particle Duality



Various experiments have been shown
which highlight either the wave nature or
particle nature of light.
Is light simultaneous a wave and a
particle?
Many questions on the nature of light
arise from issues in classical mechanics.
Wave-Particle Duality


Classically the two have mutually
properties.
Both views are required to describe the
behaviour of light.
Wave-Particle Duality



Classically the two have mutually
properties.
Both views are required to describe the
behaviour of light.
Neither model can exclusively describe
radiation adequately.
Wave-Particle Duality




Classically the two have mutually
properties.
Both views are required to describe the
behaviour of light.
Neither model can exclusively describe
radiation adequately.
Therefore for now both must be used.
Bohr Atom
Particle Nature (1913)
Bohr Atom

The Bohr model was developed from the
work of Rutherford to describe a stable
atomic model.
Bohr Atom


The Bohr model was developed from the
work of Rutherford to describe a stable
atomic model.
He provided to first successful theory of
atomic line spectra.
Bohr Atom

For his model he postulated that
classical radiation theory did not hold at
the atomic level.
Bohr Atom


For his model he postulated that
classical radiation theory did not hold at
the atomic level.
He also used the work Planck and
Einstein for the idea of quantised energy
levels and quantisation of light.
Bohr Atom


From these ideas he proposed that
electrons generally remained in stable,
stationary states.
However when an electron moves
between states specific frequency
radiation is emitted.
Matter Waves
De Broglie (1923)
Matter Waves
By the early 1920s it was recognised that
the Bohr theory had many inadequacies:
Matter Waves
By the early 1920s it was recognised that
the Bohr theory had many inadequacies:
 It could not predict the observed
intensities of spectral lines.
Matter Waves
By the early 1920s it was recognised that
the Bohr theory had many inadequacies:
 It could not predict the observed
intensities of spectral lines.
 Limited success in predicting emission,
absorption wavelengths of multi-electron
atoms.
Matter Waves

Overemphasized the particle nature but
couldn’t explain the wave-particle duality.
Matter Waves


Overemphasized the particle nature but
couldn’t explain the wave-particle duality.
Didn’t supply a general scheme for
quantising other systems.
Matter Waves



Overemphasized the particle nature but
couldn’t explain the wave-particle duality.
Didn’t supply a general scheme for
quantising other systems.
Just a few of the problems.
Enter Louis deBroglie
Matter Waves

He proposed that all forms of matter
have wave
Matter Waves

He proposed that all forms of matter
have wave and particle properties.
Matter Waves


He proposed that all forms of matter
have wave and particle properties.
But couldn’t be confirmed at the time.
Matter Waves



He proposed that all forms of matter
have wave and particle properties.
But couldn’t be confirmed at the time.
According to de Broglie, electrons also
had a particle and wave nature.
Matter Waves

He proposed that each electron was
accompanied by a wave
Matter Waves

He proposed that each electron was
accompanied by a wave (not an EM
wave)
Matter Waves

He proposed that each electron was
accompanied by a wave (not an EM
wave) which piloted the electrons
through space.
Matter Waves


He proposed that each electron was
accompanied by a wave (not an EM
wave) which piloted the electrons through
space.
The proposed relationship frequency and
wavelength of a matter associated with a
particle is:
h
E
and f 

h
p
Matter Waves


He proposed that each electron was
accompanied by a wave (not an EM
wave) which piloted the electrons through
space.
The proposed relationship frequency and
wavelength of a matter associated with a
particle is:
h
E
and f 

h
p
Matter Waves

Recall that the relativistic momentum is
p  m0v  mv
and
E 2  p 2 c 2  m02c 4  m 2c 2
Matter Waves

Recall that the relativistic momentum is
p  m0v  mv
and
E  p c m c m c
2
2 2
2 4
0
2 2
whereE photon  pc
(these equations were
originally applied to
photons mo  0 )
Matter Waves

Recall that the relativistic momentum is
p  m0v  mv
and
E  p c m c m c
2

2 2
2 4
0
2 2
whereE photon  pc
Solving for the velocity gives
E h mc2 c 2
v p  f   

h p
mv
v
(these equations were
originally applied to
photons mo  0 )
Matter Waves

Recall that the relativistic momentum is
p  m0v  mv
and
E  p c m c m c
2

2 2
2 4
0
2 2
whereE photon  pc
(these equations were
originally applied to
photons mo  0 )
Solving for the velocity gives
E h mc2 c 2
v p  f   

h p
mv
v
Where v p is the ‘velocity’ of
the matter wave and v is the
velocity of material particle
Matter Waves

Where this velocity is the phase velocity
or velocity of a wave crest.
Matter Waves


Where this velocity is the phase velocity
or velocity of a wave crest.
However from the expression, this gives
a phase velocity greater than c.
Matter Waves



Where this velocity is the phase velocity
or velocity of a wave crest.
However from the expression, this gives
a phase velocity greater than c.
The problem arises because a single
matter wave can not properly represent
the localized particle.
Matter Waves


Instead a superposition of many waves
is needed. These waves interfere to form
a wave group.
This wave group has a group velocity.
Wave Packets
Representing particles by
finite wave groups
Wave Packets

We determined that properly define a
particle by a wave a superposition of
waves (wave group) is needed.
Wave Packets


We determined that properly define a
particle by a wave a superposition of
waves (wave group) is needed.
The velocity of the wave group now is
equal to the velocity of the particle.
Wave Packets

A wave packet is set of waves with
different wavelengths
Wave Packets

A wave packet is set of waves with
different wavelengths, amplitudes and
phases
Wave Packets

A wave packet is set of waves with
different wavelengths, amplitudes and
phases which interfere constructively
over a small region of space.
Wave Packets


A wave packet is set of waves with
different wavelengths, amplitudes and
phases which interfere constructively
over a small region of space.
Outside the region they interfere
destructively so that they have zero
amplitude.
Wave Packets

The wave may be described by the
formula: y = A cos(kx-ωt) where
vp 


k
k is the wave number
Wave Packets

The wave equation for the wave packet
can be constructed from the
superposition of each wave.
Wave Packets

In general the velocity is given by

vp 
k
Wave Packets

The main property of the wave packet is
that it has time duration t and space x .
Wave Packets


The main property of the wave packet is
that it has time duration t and space x .
In general the larger the spatial width, the
larger the wave numbers required.
Wave Packets



The main property of the wave packet is
that it has time duration t and space x .
In general the larger the spatial width, the
larger the wave numbers required.
This relationship is represented
mathematically as
x k  1
Wave Packets

Similarly to produce a small duration the
range of frequencies must be increased.
ie. t   1
Wave Packets

This preceding analysis is used to show
that a wave group can represent an
electron.