quanta of light (4 lectures) electromagnetic waves spectrum and generation two slit intereference

1AMQ P.H. Regan, Spring 2001

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Quanta of Light (4 lectures)

– Electromagnetic Waves

• Spectrum and generation

• Two slit intereference

• Single slit diffraction

– Black Body Radiation

– Energy Quanta and Planck’s Hypothesis

– Photoelectric effect and Einstein’s Equation

– Compton Effect

See Krane, Chap. 3 & ERChaps. 1 and 2.

Syllabus for 1AMQ, Atoms, Molecules and Quanta,

Dr. P.H. Regan, 31BC04, x6783, [email protected]

Spring Semester 2001Books, Modern Physics, K. Krane, Wiley

Quantum Physics, Eisberg & Resnick, Wiley


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Introduction

`Classical Physics’ -> before ~ 1900 Modern (quantum) Physics, after 1900 New theories arose from the ability to do

better measurements….ie. better technology

Allowed the exploration of 3 EXTREMES of nature, ie

– very fast - special relativity replaces Newtonian mechanics

– very small- Quantum mechanics replaces Newtonian mechanics

– very large- General relativity replaces Newtonian gravitation. (note I. Newton, b. 25 Dec. 1642)


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Modern physics theories are refinements of the old, classical ideas, but are CONCEPTUALLY RADICAL.

Classical theories still work (as good approximations) at everyday speeds and sizes.

The new ideas were discovered using advanced technology, therefore, become more important at extremes physical conditions. (key experiments were to do with light (very fast, c=3x108 ms-1 !) and atoms (very small, r~10-10m).

New experiments New theories New concepts

Measurements of the speed of light

New concepts of time and space

Relativity

Spectrum of light, from (a) hot, glowingobjects, (b) from electrical breakdown in gases (atoms)

New ideas about determinismand measurement

Quantum

mechanics


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Electromagnetic WavesLight often behaves like an electromagnetic wave, travelling with speed, c (in vacuum), predicted by Maxwell’s equations and exhibiting interference and diffraction effects. However, as we shall see, in some circumstances, the predictions of wave theory are wrong and it was the study of those cases which led to the development of the quantum theory.

The Intereference Theory of Light was a success for wave theory. The two slit experiment of Thomas Young (1803) shows wave-like intereference for light.

Condition for minima (destructive interference) is that:

dsinn

dsinpath difference


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Further successes of wave theory, diffraction from a single slit……….

If the size of the slit, a, is comparable with the wavelength of the light, then a diffraction pattern is observed, rather than a sharp image. There is a central maximum, the width of which is defined by the first minima on either side.

From simple trigonometry,

(a/2) sinthus, asin

ie. the wave-like nature of light was well established.


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Quanta of Light

Studying the speed of light led to the theory of special relativity.

Studying interference, diffraction and refraction of light showed its wave-behaviour. These phenomena can not be understood by a particle or `corpuscular’ model of light.

However…..

At the atomic level, some phenomena can NOT be understood if light acts as a wave!

…..but can be understood if we take light to be a stream of particles with

mass mo=0 and

speed, v=c (ie. .

Note, mo=0 if and only if v=c, since E=pc=mvc and E=mc2

Einstein (Nobel prize, 1921).


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The study of `black body radiation’ gave the first clues to the breakdown of classical laws which led to quantum theory.

Thermal radiation: heated objects emit e-m radiation as they cool.

Hot coals glow red, very hot surfaces eg. Solar surface, incandescent filaments glow white.

The wavelengths (colour):

•Depends on temperature, T.

•As T increases, decreases. red hot -> white hot -> blue hot, details don’t really depends on the actual material being heated.

• Have a continous spectrum

Expts. give Wien’s Law (Nobel Prize 1911)

I

max (metres) = 2.9 x 10-3 / T(K)

where max is the peak wavelength and T is the absolute temperature of the surface.

Black Body Radiation


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S is the AREA under the spectral function, S. Note that the area under

the curve

Since c= (where = frequency),

max(Hz) = 1.03 x 1011 x T(K)

Note that the power emitted (ie. energy radiated per unit time) increases rapidly with T

Power emitted per unit area is given by Stefan-Boltzman Law.

S = Wm-2, with5.67 x 10-8Wm2K-4


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The origin of these electromagnetic waves is the thermal motion (vibration) of the charged consituents of the atoms in the material.

A Blackbody is an idealised perfect absorber and perfect emitter of thermal radiation. (The surface does not affect the radiation, and the spectrum of the radiation only depends on T).

Examples of Blackbodies ?

•A lump of coal, which absorbs all incident light (ie. is apparently black in colour)

•Tbe sun (see spectrum)….note blackbodies are not black in colour when they are hot!

•Uniformly heated cavity with small exit/entrance hole. (see later)


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The Ultra-Violet Catastrophe !!!

aka `The Problem with Classical BB Theory

Under the classical wave theory, if a cavity of dimension, a, is filled with e-m radiation, (note the hole rather than the cavity is the BB here), the number of standing waves of frequency, is given by (see ER, p11)

N(da/c)3 . d

The average energy of each standing wave in the box is given by the classical equipartition law, (k = Boltzmann’s const.)

av = kT Result is that S is proportional to ie, infinite at large small corresponds to the UV regime).

Conclusions wrong! UV Catastrophe


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As the above figure (from ER p13) shows, although the classical theory works (approx) in the low frequency (long wavelength) region, it fails dramatically at higher frequencies.ax Planck (Nobel Prize, 1918) showed that the `mistake’ was in the assumption that the average energy, av was a constant. (Note, in this example, the frequency corresponds to that of the vibration of the atoms in the walls). This was derived from the assumptions:

•The energies followed a Boltzmann distribution and

•The range of possible energies was continuous.

Note, in this fig. the y-axis is the TOTAL energy emitted per second, per unit volume, at a fixed frequency,


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2 new ideas were needed:

• assume the energy allowed per (atom) oscillator is NOT CONTINUOUS (ie energy is to have DISCRETE values.

• Assume that the gaps between allowed values are larger for higher frequency (atomic) oscillators.


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Planck guessed that the gaps between allowed values of oscillator frequency, increased with frequency, ie

hwhere h=const

and that each oscillator can only emit or absosb energy is discrete amounts given by,

nh, where n= integer.

The classical theory, known and Raleigh-Jeansprediction, clearly fails at long (uv) wavelengths.


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Result is that avas

Planck’s suggestion was that the average energy per oscillator at a given temperature was a function of oscillator frequency such that,

1)(

kThav

e

h

Since in the high frequency limit,

0for 1 kT

h

kT

he kTh

In the long-wavelength (UV) limit,

0)( thus,and , ,

avkT

hekT

h


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The result is that the spectral function,which corresponds to the product of the average emitted energy at a given frequency times the number of oscillators at that frequency, is given by:

1

125

2

kThc

e

hcS

This fits the data perfectly for a value of h=6.63x10-34 Js (Planck’s constant)

Physical Picture of Planck’s Hypothesis

The physical background behind Planck’s proposal was that the atomic oscillators behave like simple (quantum) harmonic oscillators, which have a potential energy given by

2

2

1kxV

x = displacement

k = constant


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The quantum energy hypothesis means that only certain amplitudes are allowed and there is a non-zero minimum. If the atom is not given enough energy in collisions with its neighbours, it will not oscillate at all. Higher frequencies need a greater amplitude to start vibrating. They carry more energy, but vibrate less often, with the result that as av(as infinity.


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The Photoelectric Effect

This is a quantum effect involving light and electrons. Light shining on a (metal) surface can cause electrons to be emitted.

Experiments should study the effect systematicallyand highlight the important features. A suitable apparatus is shown below.


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There are 5 main features to explain, namely:

•The number of electrons ejected per unit time is proportional to the intensity of light.

• The electrons are emitted with velocities up to a maximum velocity (Vstop<0).

•The maximum kinetic energy does NOT depend on intensity of light. (Vstop is the same regardless of intensity BUT increases linearly with frequency of light,

• There is a threshold frequency, o such that there is no emission for o. (Note o depends on the metal).

• There is no measurable time delay between the light striking the metal and the electron emission.

The simple Wave Theory of light, (energy transmitted per unit time is proportional to Eo

2) has the following problems…..• Intensity dependence is predicted incorrectly• Threshold effect is NOT predicted• Time delay should easily be several seconds.• No electron KE dependence on light intensity.


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Einstein’s Photon Hypothesis

Einstein proposed that light (em-radiation) consists of particle-like packets of energy,called photons. Each photon carries an energy,

E=hwhere h = Planck’s constant and is the classical, wave model light frequency.

This extends Planck’s ideas regarding emission/absorption so that they also apply to radiation as it is transmitted.

Quantum Theory of the Photoelectric Effect

(Einstein, Nobel prize, 1921, theory 1905).

The emission of electrons is caused by single photons which are completely absorbed by individual electrons. The electrons are initially energetically bound to the metal and need some minimum initial energy to overcome this binding.


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The minimum energy required for the electrons to escape is called the work function,

The maximum kinetic energy for electrons is then given by Kmax = h

(Note, K<Kmax for more tightly bound e-s or from e- collisions after emission.)

Note that Kmax=Vstope where e is the electron charge. Therefore, plotting Vstop versus has a slope of h/e. Thus h can be measured and compared with the value obtained from the Black Body spectrum.

h = 6.626 x 10-34 Js = 4.136 x 10-15 eVRecalling that 1eV = 1.6x10-19J


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The Einstein model accounts for all 5 features of the photoelectric effect, ie.

• The intensity dependence. Since the intensity is equal to the energy deposited per unit area per unit time, this means that the intensity is proportional to the number of incident photons.

• Proportionality of Kmax to •T he non-zero maximum velocity.• The existence of o. All arise directly fromEinstein’s equation.

• No measurable time delay. Photons can be absorbed instantaneously.


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The Compton EffectThe Compton effect refers to collisions between photons and electrons. Arthur Compton’s experiments (performed 18 years after Einstein’s PE theory) showed a definite, particle-like (photon) behaviour for X-rays.

'

)0()(

and


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Comparing the x-ray photon energies with e-

energies from the pe effect. For an x-ray photonwith 0.07 nm (as used by Compton), fromE=hc/keV, ie >> than e- bind. ene. Simplify situation by considering a collision between a photon and a free (unbound) electron,initially at rest

For the incoming photon, the momentum isgiven by E=pc (mo=0 since v=c for photon).

Cons. of linear momentum, mass energy and the energy-momentum relationship can thencalculate the scattered energies for any incident photon energy and scattering angle.

Compton scattering occurs in addition to the classical process of Thomson scattering (where ie absorption followed by re-radiation).

The Compton shift, , has a clear angular dependence, but does not depend on the material used for the scatterer. This suggests that the photons are colliding with something found in all materials, such as electrons….


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quanta of light (4 lectures) electromagnetic waves spectrum and generation two slit intereference

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