Quanta of Light (4 lectures)

1
Syllabus for 1AMQ, Atoms, Molecules and
Quanta, Dr. P.H. Regan, 31BC04, x6783,
p.regan_at_surrey.ac.uk Spring Semester
2001Books, Modern Physics, K. Krane,
WileyQuantum Physics, Eisberg Resnick, Wiley

• Quanta of Light (4 lectures)
• Electromagnetic Waves
• Spectrum and generation
• Two slit intereference
• Single slit diffraction
• Black Body Radiation
• Energy Quanta and Plancks Hypothesis
• Photoelectric effect and Einsteins Equation
• Compton Effect
• See Krane, Chap. 3 ERChaps. 1 and 2.

2
Introduction

• Classical Physics -gt 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)

3
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, c3x108 ms-1 !)
and atoms (very small, r10-10m).
4
(No Transcript)
5
Electromagnetic Waves
Light often behaves like an electromagnetic wave,
travelling with speed, c (in vacuum), predicted
by Maxwells 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 dsinq l/2, 3l/2, 5l/2, 7l/2, nl/2
dsinq path difference
6
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, l, 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.
ie. the wave-like nature of light was well
established.
7
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 mo0 and
speed, vc (ie. b1).
Note, mo0 if and only if vc, since
Epcmvc and Emc2 Einstein (Nobel prize, 1921).

8
Black Body Radiation

• 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, l decreases. red hot -gt white
  hot -gt blue hot, details dont really depends on
  the actual material being heated.
• Have a continous spectrum
• Expts. give Wiens Law (Nobel Prize 1911)
• I
• lmax (metres) 2.9 x 10-3 / T(K)
• where lmax is the peak wavelength and T is the
  absolute temperature of the surface.

9
Since cnl (where n frequency),
nmax(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 s T 4 Wm-2,
with s 5.67 x 10-8Wm2K-4
S is the AREA under the spectral function, Sl .
Note that the area under the curve a T 4.
10
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)

11
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, n, is
given by (see ER, p11) N(n) dn p .
(2a/c)3 . n2 dn The average energy of each
standing wave in the box is given by the
classical equipartition law, (k Boltzmanns
const.) eav kT
Result is that Sn is proportional to n2, ie,
infinite at large n (small l corresponds to the
UV regime).
Conclusions wrong! UV Catastrophe
12
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.

• Max Planck (Nobel Prize, 1918) showed that the
  mistake was in the assumption that the average
  energy, eav 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.

13
(No Transcript)
14
The classical theory, known and
Raleigh-Jeans prediction, clearly fails at long
(uv) wavelengths.
Planck guessed that the gaps between allowed
values of oscillator frequency, increased with
frequency, ie Dehn, where hconst
and that each oscillator can only emit or absosb
energy is discrete amounts given by,
enhn, where n integer.
15
Result is that eav(n) -gt 0 as n
-gt ?
Plancks suggestion was that the average energy
per oscillator at a given temperature was a
function of oscillator frequency such that,
Since in the high frequency limit,
In the long-wavelength (UV) limit,
16
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
This fits the data perfectly for a value of
h6.63x10-34 Js (Plancks constant)
Physical Picture of Plancks Hypothesis
The physical background behind Plancks proposal
was that the atomic oscillators behave like
simple (quantum) harmonic oscillators, which have
a potential energy given by
x displacement k constant
17
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 eav(n)-gt0 as n-gt infinity.
18
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
systematically and highlight the important
features. A suitable apparatus is shown below.
19

• The simple Wave Theory of light, (energy
  transmitted per unit time is proportional to Eo2)
  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.

20
Einsteins Photon Hypothesis
Einstein proposed that light (em-radiation)
consists of particle-like packets of
energy, called photons. Each photon carries an
energy,
This extends Plancks 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.
21
The maximum kinetic energy for electrons is
then given by
Kmax hn - f
(Note, KltKmax for more tightly bound e-s or from
e- collisions after emission.)
Note that KmaxVstope where e is the electron
charge. Therefore, plotting Vstop versus n 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
eV Recalling that 1eV 1.6x10-19J
22

• 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 n,
• T he non-zero maximum velocity.
• The existence of n o. All arise directly
  fromEinsteins equation.
• No measurable time delay. Photons can be
  absorbed instantaneously.

23
The Compton Effect
The Compton effect refers to collisions between
photons and electrons. Arthur Comptons
experiments (performed 18 years after Einsteins
PE theory) showed a definite, particle-like
(photon) behaviour for X-rays.
24
Comparing the x-ray photon energies with
e- energies from the pe effect. For an x-ray
photon with l0.07 nm (as used by Compton),
from Ehc/l, E17.7 keV, ie gtgt 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 is given by Epc (mo0
since vc for photon). Cons. of linear momentum,
mass energy and the energy-momentum relationship
can then calculate the scattered energies for any
incident photon energy and scattering angle .
25
(No Transcript)
26
(No Transcript)
27
(No Transcript)