Transcript lect1-4

Syllabus for 1AMQ,
Atoms, Molecules and Quanta,
Dr. P.H. Regan, 31BC04, x6783,
[email protected]
Spring Semester 2001
Books, Modern Physics, K. Krane, Wiley
Quantum 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 Planck’s
Hypothesis
– Photoelectric effect and Einstein’s
Equation
– Compton Effect
See Krane, Chap. 3 & ERChaps. 1 and 2.
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Introduction
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`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
Relativity
New concepts
of time and
space
Measurements of
the speed of light
Quantum
mechanics
Spectrum of light,
from (a) hot, glowing
objects, (b) from electrical
breakdown in gases (atoms)
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New ideas about
determinism
and measurement
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Electromagnetic Waves
Light 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:
dsinq = l/2,
3l/2, 5l/2,
7l/2, nl/2
dsinq = path 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, 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.
From simple trigonometry,
(a/2) sinq = (l/2), thus, asinq = l
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. b=1).
Note, mo=0 if and only if v=c, since
E=pc=mvc and E=mc2
Einstein (Nobel prize, 1921).
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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 ->
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
lmax (metres) = 2.9 x 10-3 / T(K)
where lmax is the peak wavelength and T is the
absolute temperature of the surface.
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Since c=nl (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 is the AREA
under the spectral
function, Sl . Note
that the area under
the curve a T 4.
S = s T 4 Wm-2,
with
s =5.67 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, 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 = Boltzmann’s 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
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Note, in this
fig. the y-axis
is the TOTAL
energy emitted
per second, per
unit volume, at
a fixed
frequency, n
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
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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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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
De=hn, where h=const
and that each oscillator can only emit or
absosb energy is discrete amounts given by,
e=nhn, where n= integer.
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Result is that
eav(n) -> 0 as n -> 
Planck’s suggestion was that the average
energy per oscillator at a given
temperature was a function of oscillator
frequency such that,
h
e av ( ) =
e
h
kT
-1
Since in the high frequency limit,
e
h kT
h
h
 1
for
0
kT
kT
In the long-wavelength (UV) limit,
h
kT
 , e
h
kT
 , and thus, e av ( )  0
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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:
Sl =
2pc h
2
l
5
1
e
hc
lkT
-1
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
1 2
V = kx
2
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 eav(n)->0 as n-> 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 systematically
and 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, n).
• There is a threshold frequency, no such that
there is no emission for n<no. (Note no
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 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.
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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=hn
where h = Planck’s constant
and n 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, f.
The maximum kinetic energy for electrons
is then given by
K = hn - f
max
(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 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
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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 n,
•T he non-zero maximum velocity.
• The existence of n o. All arise directly
fromEinstein’s equation.
• No measurable time delay. Photons
can be absorbed instantaneously.
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The Compton Effect
The 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.
l (q )  l (0)
and l = l  Dl
'
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Compton scattering occurs in addition to the
classical process of Thomson scattering
(where l(q)=l(0o), ie absorption followed by
re-radiation).
The Compton shift, Dl , 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….
Comparing the x-ray photon energies with eenergies from the pe effect. For an x-ray photon
with l=0.07 nm (as used by Compton), from
E=hc/l, E=17.7 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 is
given by E=pc (mo=0 since v=c 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
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