The outline below
should help give you the big picture of what we'll discuss in the
course. You may want to refer back to this during the course to see how
the current topic fits into the big picture, and also see where we're
going. Some of the sections will probably make more sense after we have
begun discussing a topic.
PART I: SPECIAL RELATIVITY
Newton’s laws
and
relativity
To
begin, we’ll review Newton’s laws and point out that if these
are true for one
person, they are equally true for anyone else moving at a constant
velocity
relative to that person. So two people moving at some constant relative
velocity will have exactly the same laws of mechanics. This means that
if the
two people set up identical (mechanical) experiments, they will get
identical
results. It is then impossible to come up with any mechanical
experiment to
measure absolute velocity, since such an experiment would have to give
different results for people moving at different velocities. Thus, at
least
from the point of view of mechanics, only relative velocities have any
practical
meaning, and this is what is meant by the principle of “relativity.”
Puzzles
from electromagnetism
We’ll
then review some basic
electricity and magnetism and recall how light arises as an
electromagnetic
wave. Since the equations of electromagnetism (Maxwell’s Equations)
predict a
specific value for the speed of light, it sounds like the principle of
relativity must be violated for electricity and magnetism: according to
the
usual rules of adding velocities, if the speed of light is X kilometers
per hour
as measured by one person, it would be X + v kilometers per hour as
measured by
a person moving toward the light source with velocity v kilometers per
hour. If
this is correct, only one of these two people could possibly observe
the value
predicted by electromagnetism. We could then define a “stationary”
observer as
one for whom Maxwell’s Equations (and their prediction for the speed of
light)
hold. This would give a practical distinction between moving observers
and
stationary observers, and an absolute notion of velocity.
Einstein’s
resolution: special relativity
Einstein
didn’t like this
conclusion, and various experiments, notably one performed by Michelson
and
Morley, could find no evidence that the speed of light was different
for
different observers. So Einstein proposed that the principle of
relativity does
hold for electricity and magnetism. He realized that this postulate is
self-consistent, but that it requires some profound revision of our
understanding of space and time. Einstein’s theory, now known as
Special
Relativity, implies that two observers moving at some large relative
velocity
will not agree on the time intervals or distances between events, or
even
whether two events occur at the same time or at different times.
To
understand this, we’ll
have to think carefully about how distances and times are measured and
how the
measurements as performed by one observer are related to those
performed by
another observer moving at some constant relative velocity. With the
basic
assumption that all observers should agree on the speed of light (in a
vacuum),
we’ll see that there are unique, mathematically precise, rules for
relating
measurements made by observers moving at different velocities. These
agree with
our ordinary intuition in the case where all velocities are much less
than the
speed of light, but give rise to the startling consequences we have
mentioned
in cases where velocities become large. Despite these counterintuitive
results,
the new rules provide a consistent framework for physics involving
arbitrary
velocities, and are now supported by compelling experimental evidence
that we
will discuss.
Relativistic invariants
In
order to talk about
physics in the new framework, we will want to understand which
quantities two
observers moving with some relative velocity will agree upon, since
these are
the quantities that they can sensibly compare with each other. We will
see that
there are invariant
(i.e. the same for all observers) notions of
distance, time and simultaneity (called proper
distance, proper time, and
spacelike
separation respectively) that generalize our usual notions.
We’ll see that
many of the new concepts can be understood most clearly in a pictorial
way
using spacetime diagrams.
Relativistic
energy and momentum
After
understanding the new
framework for measuring and comparing lengths and times we’ll see that
the
usual definitions of momentum and energy will have to be modified in
order that
the conservation of energy and momentum still hold.
We’ll see that the correct definition of
energy includes a term that is non-zero even for zero momentum (and
zero
potential), namely the mass
times the
speed of light squared. It is only the combination of this mass energy
and the
energy associated with momentum that is conserved in relativistic
processes, so
it is possible to convert mass energy to kinetic energy and vice-versa.
With
this observation, we can understand why the mass of a hydrogen atom is
less
than the mass of an electron plus the mass of a proton, and why nuclear
reactions
can be used to produce enormous amounts of energy.
Finally,
we’ll see that with
the new definitions, energy and momentum can be non-zero and finite
even in a
limit where the mass is taken to zero, assuming that the velocity is
taken to
be that of light. Furthermore, the relationship between energy and
momentum for
massless particles is exactly the same as for classical electromagnetic
waves.
We will soon see that the similarities between massless particles
traveling at
the speed of light and electromagnetic waves are not a coincidence.
PART II: QUANTUM MECHANICS
Light as a particle
To introduce quantum
mechanics, we’ll begin by pointing out a few simple phenomena that
classical
mechanics and classical electromagnetism cannot seem to explain. One of
these,
the photoelectric effect (in which
electromagnetic radiation liberates electrons from a metal), suggests
strongly
that light comes in discrete bundles or quanta
of energy, with the energy in each quantum proportional to
the frequency.
We can think of these quanta as particles of light, called photons, which together make up the
electromagnetic wave.
Properties
of quanta
If
the photon description of
light is correct, we should be able to explain wavelike phenomena via
the
behavior of individual photons. We will quickly realize that this is
only
possible with some drastic departures from the rules of classical
physics.
To begin, we’ll discuss the
photon interpretation of familiar experiments involving polarizers. For
photons
of light polarized in the same direction as a polarizer or
perpendicular to the
polarizer, it must be that all the photons pass through or none of the
photons
pass through respectively. However, in order to explain the partial
attenuation
observed for any other polarization, we will be forced to conclude that
for a
stream of these identically polarized photons, some pass through the
polarizer
and some do not. Even with complete knowledge about the initial
polarization,
we cannot predict the fate of any individual photon, only the
probability that
it will pass through the polarizer.
This
indeterminacy is a
central difference between quantum mechanics and classical mechanics:
whereas
in classical mechanics, we could hope to predict the precise future
evolution
of a system, in quantum mechanics, we can only predict the
probabilities for
the various possible outcomes of an experiment.
In
the classical picture,
light waves with general polarizations are superpositions of waves with
orthogonal linear polarizations. To quantitatively explain the
polarizer
experiments based on photons, we will argue
that the individual photons with general polarizations
should still be
viewed mathematically as superpositions of the two special photon
states, but
one in which the overall amplitude has no physical meaning.
Thus,
we’ll arrive at one of
the most basic rules for quantum systems: for each possible result of a
given
measurement, there are special states, called eigenstates
for which that result will definitely be obtained. More
general states (for which the result is not predetermined) are
superpositions of
these eigenstates, and the amount of each eigenstate in the
superposition
determines the probability of obtaining the corresponding outcome.
Wave
properties of particles
With the understanding that
classical electromagnetic waves are comprised of photon particles, one
might
wonder whether other kinds of particles give rise to wavelike
phenomena. While
we don’t see any classical electron waves (this has to do with the
Pauli exclusion
principle) it turns out that a beam of electrons at some fixed momentum
does
exhibit diffraction phenomena, with a wavelength inversely proportional
to
momentum, just as for photons.
By
discussing a very simple
diffraction experiment, known as the double-slit experiment, we’ll
argue that
states of individual electrons with a given momentum do
not have well defined positions, and propose that these and
more
general states of the electrons are superpositions of eigenstates where
the
electrons do have definite positions, motivated by our discussion of
polarization experiments.
The
information about the
amount of each position eigenstate in a given superposition is known as
the wavefunction, and this
information
determines the probabilities of the possible outcomes when the position
is
measured. The wavefunction gives a complete description of the state of
a
particle at a given time and replaces classical description in terms of
instantaneous position and velocity.
The
description of general
electron states as superpositions of states with definite positions
provides
another example of our general rules for quantum mechanics. For any
given
question that we might ask about a particle (in this case “what is the
position?” or “what is the momentum?”), there are some states for which
the
answer is predetermined, while general states are superpositions of
these
states.
An
important point is that an
eigenstate for one physical quantity (e.g. an electron with some
definite
position) is usually not an eigenstate for another physical quantity.
For
example, there are no states that are eigenstates of both position and
momentum, and we’ll see that the more certain we are about the position
of a
given particle, the more uncertainty there is in the momentum. This is
known as
the Heisenberg Uncertainty Principle.
The
Schrödinger Equation
Since
the classical description of a particle in terms of position and
velocity have
been replaced by the idea of a quantum state described by a
wavefunction, we’ll
need to understand what replaces Newton’s Second Law and determines how
the
wavefunction evolves with time. The diffraction phenomena observed for
electrons suggests that electron states with definite momenta should
behave
like propagating waves with wavelength inversely proportional to the
momentum
and frequency proportional to the kinetic energy. By postulating that
the
wavefunctions for momentum eigenstates behave in this way, we will
arrive at an
evolution equation for wavefunctions known as the Schrödinger
equation.
This is the general equation for time evolution of quantum states, and
the
remainder of the course will be spent exploring its consequences.
Bound states
and atomic
spectra
We
will first study the Schrödinger equation for particles which are
classically
trapped in a finite region of space due to external forces. An example
is an
electron in an atom whose energy is less than the amount required to
overcome
the Coulomb attraction of the nucleus. We will see that in these cases,
the Schrödinger
equation implies that the particle can exist only at certain specific
energies.
This explains the discrete nature of atomic spectra: electrons can
absorb or
emit energy only in the precise amounts that allow them to jump between
their
allowed energies.
Tunnelling (?)
One
of the surprising consequences of the Schrödinger equation is that
particles
have some probability of being found in places where the classical
potential
energy is greater than the total energy of the particle. A result of
this is
that particles can pass through barriers created by external forces
which would
classically block them completely. This phenomenon can be used to
understand
certain types of radioactive decay and is central to a number of
important
technological applications, such as scanning-tunneling electron
microscopes
that are able to “see” individual atoms.
POSSIBLE
SPECIAL LECTURES
Nuclear and particle physics,
statistical mechanics and condensed matter physics, general relativity
and
cosmology