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Modern Physics
Waves and Optics
Special Relativity
Quantum Mechanics
Wave, particles, and weirdness
Atoms, molecules, and nuclei
Particle physics
General Relativity
Prof. Rick Trebino
Georgia Tech
www.frog.gatech.edu
Cosmology
Modern Physics is 20th century physics.
By 1900, physicists thought
they had it all together. They
had Physics I and II
(“classical physics”) down
and thought that that was
about it. All that remained
was to dot the i’s and cross
the t’s.
Scanning-tunneling microscope image
of individual atoms
Man, were they in for a surprise! Several of them actually. Modern
physics is the story of these surprises (quantum mechanics and
special and general relativity), surprises—revolutions, actually—that
have changed the world beyond all recognition.
The purpose of this course is to introduce you to all this fun new stuff.
The Beginnings of Modern Physics
Special
relativity
19th-century
physics
General relativity
Quantum mechanics
The introduction (~1905) of the
modern theories of special
relativity and quantum
mechanics became the starting
point of this most fascinating
revision. General relativity
(~1915) continued it.
c
Speed
These new discoveries and the
many resulting complications
required a massive revision of
fundamental physical
assumptions and theories.
0
0
Size
Huge
In 1900, it was well-known that the
universe contained only particles.
Waves, on the other hand, were simply collective motions of
particles—a much less fundamental phenomenon.
A
human
wave
More differences between particles and
waves
Particles are highly localized in space
and time.
Particles have well-defined trajectories.
Particles are either there (1) or not (0).
Particles cannot cancel each other out.
Waves are extended in space and time.
Waves have poorly defined trajectories.
Waves can be kind of there (~1/2) or
even the opposite of there (<0).
Waves can cancel out.
We’ll begin our story with
the age-old subjects of
waves and optics, which
hold the key to it all.
“I procured me a
triangular glass
prism to try
therewith the
celebrated
phenomena of
colours.” Isaac
Newton, 1665
Isaac Newton
(1642-1727)
Is light a particle or a wave? After remaining ambivalent for many
years, Newton concluded that light was made up of particles.
While particles travel in straight lines,
waves bend around corners.
Ocean waves passing through wave-breaks in Tel Aviv, Israel:
This is diffraction, and it occurs for all types of waves—but not for
particles.
Light passing through a hole bends
around the edges.
Thomas Young
(1773-1829)
Light pattern after passing
through a small square hole
In 1803, Thomas Young showed that light diffracted precisely as
predicted by Fresnel’s wave theory.
In the mid-19th century, Maxwell unified
electricity and magnetism into a single force
with his now famous equations.
In free space:
 E  0
 B  0
B
 E  
t
1 E
 B  2
c t
where E is the electric field, B is the
magnetic field, and c is the velocity of
light.
James Clerk Maxwell
(1831-1879)
In addition, Maxwell showed that light
is an electromagnetic wave.
The electric (E) and magnetic (B)
fields obey the wave equation:
Magnetic
field (B)
Electric
field (E)
2 f
1 2 f
 2 2  0
2
x
v t
Wavelength (l)
Different wavelengths (distances between the peaks) or
frequencies (2p times the rate at which the peaks pass by)
correspond to different colors, many of which we can’t see.
y
x
z
But exactly what was waving?
It seemed that
electromagnetic waves
could propagate through
empty space!
Indeed, precisely what
was electromagnetically
waving was unknown at
the time. Scientists
decided to call it aether
and figure out what it
was later.
Waves also interfere.
The color you
see is the one
for which the
light reflected
from the front
and back of the
bubble surface
are in phase.
By the mid-19th century, light was well-known to be a wave.
Input
beam
The Michelson
Interferometer
The Michelson
Interferometer deliberately
interferes two beams and so
yields a sinusoidal output
intensity vs. the difference in
path lengths.
L2
Output
beam
Mirror
Beamsplitter
L1
Delay
Mirror
Output beam intensity vs. relative path length
l
I
DL = 2(L2 – L1)
It can also measure velocity.
Michelson & Morley
In 1887 Michelson and Morley
attempted simply to measure
the earth's velocity with
respect to the aether and found
it always to be zero—no matter
which direction the earth was
moving—effectively disproving
the existence
of the aether
and providing
a great crack
in the foundations of
physics.
Albert Michelson Edward Morley
(1852-1931)
(1838-1923)
In 1905, Einstein had a very good year.
That year, Einstein explained
Michelson’s and Morley’s experiment:
he realized that light didn’t need a
medium and was a property of free
space. It’s a wave—but not collective
motion of particles!
And light has the odd property that it
travels at the same velocity no
matter what speed you’re going.
This is Special Relativity.
Albert Einstein (1879-1955)
Oh, and he graduated from grad
school that year, too.
Before
Special
Relativity
One frame
moving at
velocity v with
respect to
another
z
x  x  vt
y  y
z  z
t  t
y
x
Basically, this seems so obvious that we
almost shouldn’t even have to say it.
Unfortunately, it’s wrong.
With Special Relativity
x 
1  v2 / c2
y  y
z  z
t 
y
x  vt
t  vx / c 2
1  v2 / c 2
z
x
The Lorentz transformations follow
directly from the constant-speed-of-light
assumption and are the correct way to
transform from one frame to the other.
They yield the speed of light in all
frames and are NOT at all obvious!
Lorentz himself didn’t believe them.
Measurements
of time
confirm
Special
Relativity
In Special Relativity, time
passes at a rate that
depends on your velocity.
Two airplanes traveled east and west around Earth as it rotated.
Atomic clocks on the airplanes were compared with similar clocks
kept at the observatory to show that the moving clocks in the
airplanes ticked at different rates.
When matter
is heated, it
not only
absorbs light;
it also emits
it.
Blackbody Radiation
A blackbody
is a medium
that’s black
when it’s cool
and so can
absorb and
emit all
colors.
Blackbodies are interesting because their emitted light spectra are
independent of the material and depend only on their temperature.
The Ultraviolet Catastrophe
In 1900, Lord Rayleigh used the classical theories of electromagnetism
and thermodynamics to show that the blackbody spectrum
should be:
UV
Visible
IR
2p ckT
I l (l , T ) 
l4
Rayleigh-Jeans Formula
This worked at longer wavelengths but deviated badly at short ones.
This problem became known as the ultraviolet catastrophe and
was one of many effects that classical physics couldn’t explain.
Shortly afterward, Max Planck found that
he could obtain the correct blackbody
result if light was actually a particle.
8p hc 2 / l 5 
I l (l , T ) 
exp hc / lkBT  1
where h is a constant now known
as Planck’s constant.
But, of course, he didn’t really
believe such a crazy idea.
Max Planck
(1858–1947)
No one else did either.
It’s now easy to see that light also
behaves like a particle.
Photographs taken in dimmer light look grainier.
Very very dim
Bright
Very dim
Very bright
Dim
Very very bright
When we detect very weak light, we find that it’s made up of
particles—photons.
19th-century scientists also could not
explain spectra of light emitted by gases.
Wavelength
Spectra could be partially explained by
the planetary model for the atom.
The electron orbital frequency should be the light frequency.
But from classical
electromagnetic theory, an
accelerated electric charge
radiates energy
(electromagnetic radiation),
which means that its energy
must decrease.
Electron
So the radius of its orbit around
the nucleus must decrease.
Why doesn’t the electron crash into the nucleus?
Nucleus
Fourier decomposing
functions plays a big
role in physics.
a1sin(t)
Here, we write a square wave as a sum
of sine waves of different frequencies.
a3sin(3t)
Fourier
developed the
Fourier
transform to
model heatflow problems.
Joseph Fourier
1768 - 1830
a5sin(5t)
Fourier extended the idea to a
continuous range of frequencies.
The Fourier transform converts a function of time to one of frequency:

F ( ) 

f (t ) exp(i t ) dt

and converting back uses almost the same formula:

f (t )  21p

F ( ) exp(i t ) d

The spectrum of a wave is given by:
F ( )
2
The Uncertainty
Principle is a
simple classical
property of the
Fourier
transform.
If Dt is the width of
a wave in time, and
D is its spectral
width, then:
D Dt 
F()
f(t)
Short
pulse
Dt
D
t

t

t

Mediumlength
pulse
1
2
This relation will play
an important role in
modern physics!
Long
pulse
If a light-wave also acted like a
particle, why shouldn’t matterparticles also act like waves?
In his thesis in 1923, Prince Louis V. de Broglie
suggested that mass particles should have wave
Louis de Broglie
properties similar to those of light. The wavelength
(1892-1987)
of a matter wave is called
the de Broglie
wavelength:
where h = Planck’s constant and
p is the particle’s momentum.
They would also have frequency:

E where E is the
And the matter-particles would be subject to
their own Uncertainty Principle!
particle’s energy.
The Schrödinger Equation
At about the same time, Schrödinger
introduced his Wave Equation, which
nicely explained atoms and their
properties and is the fundamental equation
of Quantum Mechanics. For a particle
moving in a potential V in one dimension,
it’s:
2
Y  x, t 
 2 Y  x, t 
i

 V Y  x, t 
2
t
2m
x
where:
Erwin Schrödinger
(1887-1961)
And Y is called the particle’s
wave function.
What on earth is Y?
Indeed, what is waving?
Probability! The probability P(x) dx of a particle being between x
and x + dx is:
P( x)  Y ( x)
2
x2
The probability of the particle
being between x1 and x2 is:

Y ( x) dx
2
x1
And quantum mechanics says that particles can remain in stationary
states forever without emitting any energy! Quantum mechanics has
its own laws, which need only approach classical laws as the system
increases in size to classical dimensions.
Y yields probability distribution functions
The probability density for the hydrogen atom for three different
stationary electron states.
Quantum mechanics is essential to
understand semiconductors.
Essentially all modern
technology is a direct result
of semiconductors and so is
due to quantum mechanics.
Economists estimate that
quantum mechanics is
responsible for ~80% of the
entire US economy.
Elementary
Particle
Physics
If nuclei are made up
of protons and
neutrons, what are
protons and
neutrons made of?
Physicists have
discovered a zoo of
elementary particles,
including quarks of
1/3 the charge of a
proton.
While there were
clearly some
problems in 19thcentury physics,
everyone remained
happy with Newton’s
Law of Gravitation.
Except Einstein.
Einstein was also unsatisfied with his Theory of Special Relativity; it
didn’t include acceleration. And because acceleration seemed
similar to gravity, in 1915 he lost interest in the quantum mechanical
revolution he had begun, and decided to pursue a geometrical
theory of gravity, in which acceleration and gravity were equivalent.
Gravitational lensing by galaxies
When light from a
distant object like a
quasar passes by a
nearby galaxy on its
way to us on Earth,
the light can be bent
multiple times as it
passes in different
directions around
the galaxy.
The Cosmic Horseshoe
General Relativity also predicts Black Holes
While a star is burning, the heat and pressure produced by the
thermonuclear reactions balance its gravity. When the star’s fuel is
depleted, gravity dominates. The star’s mass can collapse into a black
hole that warps space-time enough to not allow light to escape.
A star greater than 25 solar masses will collapse to a black hole.
Karl Schwarzschild determined the radius of a black hole, now known
as the event horizon.
General Relativity models the entire
universe.
The density, r, of
matter in the
universe
determines its
shape and future.
Closed
Open
W0 ≡ r / rcrit
where rcrit =
3H2/8pG is the
critical density
for which the
universe is flat.
Flat