Modern Physics for Frommies IV

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Fromm Institute for Lifelong Learning University
of San Francisco
Modern Physics for Frommies IV The Universe -
Small to Large Lecture 2
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Agenda

• Administrative Matters
• Clouds on the Horizon
• Review of Relativity and Quantum Mechanics

Administrative Matters

• This is Lecture 2, Lecture 8 will be Wed. 7
  March
• Physics and Astronomy near future colloquia
  (P and A colloq 1 on Wiki)
• Fri, 27 Jan, KA 163, 1115-1230 Shining a
  light on the next generation of solar cells
• Tues, 31 Jan, HR 127, 1130-1240 "Lasers,
  needles, and clocks  toward seeing the nanoscale
  world through ultrafast eyes
• Fri, 2 Feb, CO 413, 1145-100 Rydberg Atoms
  and Quantum Computing

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Clouds on the Horizon
Beauty and clearness of theory Overshadowed by
two clouds Lord Kelvin Baltimore Lectures Johns
Hopkins University 1900
The two clouds
Failure of the Michelson Morley experiment
? Einsteins Relativity
Failure of classical electrodynamics to describe
thermal radiation
? Quantum Mechanics
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Troubles with Electrodynamics
Galilean Relativity Addition of velocities
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time
Note The path followed by the object is
different as viewed from different frames.
However the same laws of motion and the same law
of gravitation apply in both frames.
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Where is the luminiferous ether?
Whats waving?
Ocean waves ? H2O Sound waves ? air, liquid, solid
What supports the vibration of electromagnetic
waves?
Postulate the existence of an invisible,
weightless, tasteless etc. substance which
permeates all of space. Referred to as the
luminiferous ether.
It is assumed that the propagation velocity is
measured w.r.t the ether.
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Galilean Relativity and Maxwells Equations
2
1
v is velocity of frame 2 w.r.t. frame
1 (x-direction). v const.
y2
y1
v
x2
x1
z1
z2
This works fine in classical mechanics. Newtons
3 laws are invariant under these kind of Galilean
transformations.
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What about Maxwells elegant electromagnetic
theory?
The equations predict the velocity of light in
vacuo to be If this is true in Frame 1 (see
previous slide) then we would predict that the
velocity in Frame 2 would be c-v.
However, Maxwells equations do not differentiate
between the 2 systems.
Proposed a preferred inertial frame in which
velocity is c. But this is just the old
luminiferous ether frame.
The Michelson-Morley experimental results can be
interpreted as saying the velocity of light is c
in any inertial frame.
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Use of the Lorentz-Fitzgerald postulate can make
Maxwells equations invariant. Strictly an ad hoc
or empirical fix.
Albert Einstein (1905) presented an explanation
that was much more fundamental. It required a
complete rethinking of our conceptions of space
and time.
Einsteins Theories of Relativity
Albert Einstein (1879-1955)
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Einsteins Postulates
(1) Relativity Principle The laws of physics
have the same form in all inertial reference
frames.
Inertial frame ? one in which Newtons laws are
valid. i.e. one in which an object subject to no
external force moves in a straight line with
constant velocity.
(2) Constancy of the speed of light Observers in
all inertial frames measure the same value for
the speed of light in a vacuum. Light propagates
through empty space with a definite speed, c,
independent of the velocity of source or
observer.
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3
1
2
observer
source
light beam
c
0.5c
0.5c
According to Galilean relativity, the observer
should measure the velocity of the light beam to
be 2c.
2
1
y2
y1
v
v
x2
x1
z1
In fact, the observer measures the velocity to be
c.
Something is wrong with our velocity addition
rule.
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Our common sense is based on a lifetime of
experience in which we deal with velocities that
are very small in comparison to c.
Light travels approximately 1 foot in 1 nsec (1 x
10-9 sec).
Lorentz velocity addition
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Example Velocity Addition
Find speed of 2 w.r.t. Earth
Earth is frame F. 2 is frame F. x, x are along
flight
Galilean transform ? vx 1.20c
There is no addition of velocities that will
result in v gt c TRY IT
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Example Constancy of c
Replace 2 with light from 1s headlight in
previous example
Find speed of light w.r.t. Earth
c w.r.t. 1
Earth is frame F. 1 is frame F. x, x are along
flight
Try it for 1 traveling at 0.99c w.r.t. Earth
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Consequences of Einsteins Postulates(for more
detail see Wiki for course I)

• The rules for transforming from one reference
  frame to another change.
• Spatial and temporal measurements are linked as
  space-time
• Lorentz transforms locations and velocities
• Moving rulers shrink
• Moving clocks slow down
• Velocities add such that light speed is c in all
  frames

• Energy and momentum are redefined to keep them
  conserved
• Energy and momentum are linked
• Energy and mass are related

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is called the Lorentz factor.
is called the speed
Lorentz contraction
Time dilation
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Relativistic Kinematics and Dynamics
Relativistic momentum Newtons 2nd law,
, is invariant under Gallilean transforms
What about Lorentz transforms?
If we use our accepted definiition of linear
momentum,

• 2nd law ? no limit to velocity attained under the
  application of an external force.
• Either the 2nd law or the definition of momentum
  needs modification at high velocity.
• And, what about conservation of momentum?

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It can be shown that for relativistic collisions
the classical momentum is not conserved.
This is disconcerting. Momentum conservation is
one of the rocks upon which the church was
founded 1.
To avoid throwing out the baby with the bath
water it was less disturbing to define
relativistic momentum as
With this definition, momentum is conserved.
Note that for vltltc the relativistic momentum
becomes the usual old momentum. (g ? 1)
1 See Mathew 1617, New Testament, Holy Bible,
King James Translation.
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Be careful
is not true
The correct, relativistic formulation of the 2nd
law is
Relativistic energy Classically
Use our relativistic momentum, the work-energy
theorem and some integration and algebra and we
get the relativistic kunetic energy
And the total energy of a free (no potential
energy) is
Rest energy
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Binding Energy Consider a bound state and its
constituents M ? ? m
stable
unstable
Example 1, Nuclear Fission
Example 2, Thermonuclear Fusion (D-T reaction)
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Example 3, Combustion
Mass is just another form of energy
Energy-Momentum
are Lorentz invariants
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More Troubles with Electrodynamics (? Quantum
Mechanics)
Black Body Radiation
StefanBoltzman law P/A sT4 s 5.67 x 10 8
W/m2 K4 Wiens displacement law lpeak k / T
k 2.9 x 10-3 K-m
Max Planck (1858-1947)
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The ultraviolet catastrophe
Treating the black body as a collection of
oscillators (atoms) and applying Maxwells
equations. I ? ? as f gets large (l gets
small) We know this doesnt happen
The quantum hypothesis Max Planck (ca. 1900)
Oscillators are not allowed to emit radiation
continuously. They are only allowed to emit (and
absorb) energy in discrete packets. E hf
where, h 6.63 x 10-34 Jsec (Plancks
constant)
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Often see this written as E hw, where h h/2p
When this condition is imposed the experimental
distribution is matched and the U. V. catastrophe
is averted.
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The Photoelectric Effect
Light frequency f
Light falling on the photocathode -- causes
emission of electrons. Electrons are collected
and a current is measured.
e-
I
A
V
Studying the measured photocurrent as a function
of V, the light intensity and the light frequency
yields results which have striking
inconsistencies with those expected from the
classical wave picture of E. M. radiation.
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Enter Albert Einstein and the Quantum (ca. 1905)
Einstein thought about Plancks quantization of
absorption and emission of radiation.
What if this is not just a mathematical trick?
Suppose the radiation field itself is quantized.
Energy of a beam of light is not continuously
distributed in space but consists of a finite
number of localized lumps (photons). Each photon
carries an energy E hf
These lumps propagate without dividing and can
only be emitted or absorbed as complete units
(no change given).
Conservation of energy then yields
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Wave Particle Duality
We seem to have developed a rather schizophrenic
description or set of descriptions for E.M.
radiation.
P. E. effect, Compton scattering etc. gt a
particle theory of light.
Interference and diffraction experiments of Young
and others gt a wave theory of light.
Incompatible but both shown to have validity ?
Nature of light is dual, more complex than a
simple wave or a simple beam of particles
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Principle of complementarity Niels Bohr
(1885-1962)
To understand any given experiment, we must use
either the wave or the photon theory, but not
both.
Duality cannot be visualized. The 2 aspects of
light are different faces that light shows to
experimenters.
Difficulty arises from thought process In the
mundane world we see energy propagated by the two
methods
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However, we cannot see directly whether light is
a wave or a particle, so we do indirect
experiments.
To explain the experiments we apply either the
wave or the particle model.
There is no reason why light should conform to
these models (visual images) taken from the
mundane world.
The true nature of light, what ever that means,
is not possible to visualize. The best we can do
is to realize that we are limited to indirect
experiments and that in mundane terms, light
reveals both wave and particle properties.
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de Broglies Hypothesis
Major symmetry fan Waves sometimes act like
particles (given) Particles sometimes act like
waves (hypothesis)
Sometimes called the de Broglie wavelength of a
particle
Louis de Broglie (1892-1987)
Sounds nuts, but remember h is very small (?
10-34 Jsec) A couple of examples may restore
your gullibility.
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Wavelength of a ball
0.20 kg moving with speed of 15 m/sec
Very small, something like 20 Planck lengths
l of any ordinary object is much to small to be
detected. Interference and diffraction are
significant only when the sizes of objects or
slits are not much larger than l
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Wave Interference
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D ? 0.3 nm
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Atomic Instability - Another UV Catastrophe
Classical Rutherford atom (1911)
e- are accelerated and should radiate. e- lose
energy and spiral into nucleus Atoms should be
unstable and the universe should only have lasted
a small fraction (10-9) of a second
Oh, and there are other problems, line spectra.
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Absorption in rarified gasses
Emission from rarified gasses
Line spectra cannot be explained by a classical
model
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Neils Bohr (1885-1962)
Bohrs Postulates
Electrons in atoms cannot lose energy
continuously, but must do so in quantum jumps.
Electrons move about the nucleus in circular
orbits, but only certain orbits are allowed.
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An electron in an orbit has a definite energy and
moves in the orbit without radiating energy. The
allowed orbits are referred to as stationary
states.
Emission and absorption of radiation can only
occur in conjunction with a transition between 2
stationary states. This results in an emitted or
absorbed photon of frequency such that hf ?E1 -
E2?
What makes an orbit allowed?
Maybe energy is not the only quantized quantity.
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de Broglie Waves and Bohrs Quantization
Bohrs model was largely ad hoc. Assumptions were
made so that theory would agree with experiment.
No reason why orbits should be quantized. No
reason why ground state should be stable
De Broglie proposed that an electron in a stable
orbit is actually a circular standing matter wave.
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The circumference of the wave must contain an
integral number of wavelengths
Bohr published his model in 1913. de Broglie did
not propose matter waves until 1923. Bohr tried
many quantization conditions in attempting to
explain the experimental data. Quantization of
angular momentum worked.
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Quantum Mechanics
Erwin Schrödinger 1887 - 1961
Werner Heisenberg 1901 - 1967
de Broglie proposes matter waves in 1923
Less than two years later two comprehensive
theories were independently developed by
Schrödinger and Heisenberg.
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Now recall the discussion of the double slit when
we made the light beam very weak so that we were
counting one photon at a time.
Over time, as we accumulate counts a distribution
of detected photons builds up which is identical
to the intensity distribution obtained from the
wave picture. Thus, E2 is a measure of the
probability of finding a photon at that location.
Now consider matter waves where
The displacement is described by a wave function,
y(x,t), as a function of time and position.
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If we treat particles (including photons) as
waves then Y (or E or B) represents the wave
amplitude. If we treat them as particles we must
do so on a probabalistic basis
We cannot predict, or even follow, the path of a
single particle through space and time
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Inside well Sinusoid Ground state E lt ?
well Outside well exponential decay of y y ? 0
outside is classically forbidden.
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Tunneling
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A model for a - decay a is 2p and 2n so q
2e 4He nucleus
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Heisenberg Uncertainty Principle
Expectation By using more precise
instrumentation, the uncertainty in a measurement
can be made indefinitely small. Quantum
Mechanics There is a limit to the accuracy of
certain measurements. Not an instrumental
restriction but an inherent fact of nature.
Feeling pool ball in the dark
g-Ray Microscope and some estimates
Try to measure the position of an e-
Requires the scattering of at least one g,
transferring some momentum to the e-.
Greater precision (smaller D x) requires shorter
l Shorter l gt higher momentum and hence the
higher the possible momentum transfer to the
electron (higher Dp).
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The act of observing produces an uncertainty in
either or both the position or momentum of the
electron.
Heisenberg 1927
Lets make an estimate of the magnitude of this
effect.
Suppose the e- can be detected by a single g
having momentum
Some or all of this momentum will be transferred,
but we cant tell beforehand how much. Therefore
the final e- momentum is uncertain by
Multiplying we obtain
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More careful calculations, e.g. using Fourier
analysis show that the very best we can do is
Another form of the uncertainty principle relates
energy and time
DX
x
ltxgt