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

1

18),(

3

3

kThfec

hfTfu

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

hc

hf

Compton Effect (1922)

Compton Effect (1922)

The Compton effect shows that photons behave like particles with momentum

c

hf

Compton Effect (1922)

The Compton effect shows that photons behave like particles with momentum

Classical theory predicted that incident radiation of frequency f0 would cause electrons to be acceleration in the direction of the radiation

c

hf

Compton Effect (1922)

The Compton effect shows that photons behave like particles with momentum

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.

c

hf

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:

p

h

h

Ef and

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:

p

h

h

Ef and

Matter Waves

Recall that the relativistic momentum is

22420

222 cmcmcpE

mvvmp 0

and

Matter Waves

Recall that the relativistic momentum is

22420

222 cmcmcpE

mvvmp 0

andwhere pcEphoton (these equations were

originally applied to photons )0 om

Matter Waves

Recall that the relativistic momentum is

Solving for the velocity gives

22420

222 cmcmcpE

v

c

mv

mc

p

h

h

Efvp

22

mvvmp 0

andwhere pcEphoton (these equations were

originally applied to photons )0 om

Matter Waves

Recall that the relativistic momentum is

Solving for the velocity gives

22420

222 cmcmcpE

v

c

mv

mc

p

h

h

Efvp

22

mvvmp 0

andwhere pcEphoton (these equations were

originally applied to photons )0 om

Where is the ‘velocity’ of the matter wave and is the velocity of material particle

vpv

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

k is the wave numberk

vp

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

kvp

Wave Packets

The main property of the wave packet is that it has time duration and space .xt

Wave Packets

The main property of the wave packet is that it has time duration and space .

In general the larger the spatial width, the larger the wave numbers required.

xt

Wave Packets

The main property of the wave packet is that it has time duration and space .

In general the larger the spatial width, the larger the wave numbers required.

This relationship is represented mathematically as

xt

1 kx

Wave Packets

Similarly to produce a small duration the range of frequencies must be increased. ie. 1 t

Wave Packets

This preceding analysis is used to show that a wave group can represent an electron.