The following is a
list of learning goals for the course. These are things you should know
or be able to do once we have covered each topic.
Broad
Goals:
After taking this course,
students should be able to:
- state the principle of relativity, and be able to
describe some of the basic implications of this that go against our usual
intuition (and explain how experimental
evidence supports these)
- analyze simple
dynamical processes using relativistic dynamics. This will include:
- knowing
how to relate physical quantities measured by observers moving at some
relative velocity and know which quantities observers at relative velocities
will agree upon.
- knowing
the limits of applicability of elementary formulae from mechanics and
know the more general formulae that replace these in situations where
velocities are not a negligible fraction of the speed of light
- describe and
predict basic behavior of light and electrons in terms of both classical mechanics
and quantum mechanics descriptions, and be able to specify differences
- argue how the assertions of quantum mechanics are inferred
from the experimental evidence
- calculate the
characteristics of quantum states and probabilities for outcomes of measurements
for a few very mathematically simple quantum systems, including ones that the
student has not seen before. This
requires understanding the basic mathematical framework well enough to apply it
to such novel systems.
- give qualitative predictions and explanations of the
behavior of simple quantum systems, such as the distribution of electrons in
atoms and the spectrum of light emitted and absorbed by atoms
- argue that physics goes beyond a collection of empirical
laws, and involves a deeper conceptual framework that is inferred from
experiment but is not at all obvious from our everyday experience
- better understand
popular science articles on current research in physics and answer questions
about modern physics from curious friends and relatives
- see value in achieving a deeper understanding of quantum
mechanics and learning more about modern physics
Topic
Specific Goals:
After taking the course,
students should be able to:
RELATIVITY
Newton’s Laws and Relativity
- explain what is meant by the principle of relativity
- explain what is meant by a “frame of reference” and an
“inertial frame”
- give
examples of how relativity manifests itself in ordinary
situations
- show that Newton’s
Laws obey the principle of relativity
- explain why the equivalence of physical laws in different
frames implies that it is impossible to set up an experiment to measure
an
absolute velocity
Puzzles
from Electromagnetism
- explain why the principle of relativity applied to
electromagnetism implies that the speed of light should be the same in
all frames of reference
- give
simple examples from electricity and magnetism to show
that either the principle of relativity or some basic notions of
distance,
time, and velocity must be abandoned
Einstein’s
Resolution
- describe experimental evidence that suggests the
velocity of light is
always independent of the of the motion of the source or of the observer
- state
Einstein’s principle relativity
- explain
how a given observer can set up a
coordinate system for making measurements of time and position
- be able to describe what is meant by and event or a trajectory
- be able to show how Einstein's postulates imply that observers at
large relative velocities will not agree on distances, time intervals
or
whether two events are simultaneous
- describe
qualitatively the meaning of length contraction, time dilation, and the relativity of simultenaity
- correctly
calculate the lengths and times differences that an observer will
measure, properly accounting for length contraction and/or time
dilation.
- use
Lorentz transformation
formulae to relate the measurements of observers moving at
relative
velocities
- determine the trajectory of an object in one frame of reference given its trajectory in another frame of reference
- analyze basic scenarios involving large velocities to calculate times and distances for various events,
or physically relevant time/distance intervals. Know when basic length
contraction and time dilation formulae are applicable and when Lorentz
transformations are needed.
- use
the velocity transformation formula to calculate the
observed velocity of an object in a new frame given the velocity in the
old
frame and the relative velocity of the two frames
- calculate
the Doppler shift of light frequencies for an
observer moving relative to the source and discuss the importance of
the
Doppler effect in astrophysics
- be
able to calculate relativistic effects in cases when velocities are
much smaller than the speed of light, using Taylor (binomial)
approximations to the exact formulae
Relativistic
Invariants
- describe the meaning of spacelike
separation, timelike
separation, proper length, and proper time
- be able to determine whether two given events are spacelike, timelike, or lightlike separated;
equivalently be able to determine whether there is a frame where two
events are simultaneous or a frame where two events are at the same
location
- know
how to calculate the proper length/time between two
events and the time elapsed on the clock of an observer on some general
trajectory
- represent
graphically simple scenarios on a spacetime
diagram
- use spacetime
diagrams to analyze simple processes involving relativistic velocities and
resolve apparent paradoxes (e.g. ladder passing through barn)
Relativistic
Energy and Momentum
- argue why classical formulae for momentum and energy must be modified
- state the
relativistic formulae for energy and momentum
- be able to determine the energy and momentum in one frame of reference given these quantities in another frame of reference
- explain the precise meaning of conservation of energy and conservation of momentum
- analyze
high-energy particle decay processes and scattering processes
using energy and momentum conservation
- provide
a definition for mass in terms of energy, and apply this to make
predictions about the masses of stable and unstable bound states
relative to the masses of their constituents
- determine the mass of an object given its energy and momentum
- explain why the conservation of mass can be violated in relativistic dynamics
- give evidence for and explain basic implications of the equivalence
between energy and mass
- state
the relation between energy and momentum for massless particles and use
this in analyzing dynamical processes involving light or other massless
particles
QUANTUM
MECHANICS
The electromagnetic description of light
- describe the electric and
magnetic fields inside a beam of light; explain what is meant by the
wavelength, period and frequency in this description
- state the relation between frequency and wavelegth for light and explain why this must be true
- explain
how basic observable properties of light (colour, brightness,
polarization) are related to details of the mathematical description as
an electromagnetic wave (e.g. amplitude, wavelength)
- describe how the energy density and momentum density in a light beam are related to the wavelegth and the amplitude
- explain
precisely what is meant by the intensity of a light beam and how this
is related to the other quantities associated with the classical
electromagnetic description
- describe the basic mechanism for producing electromagnetic radation
Problems with classical physics
- Explain
why the ordinary mechanics and electromagnetism fail to explain the
stability of atoms and the spectrum of light from a heated atomic gas
Light as a
Particle
- Describe the photon model of light and explain how the
wavelength/frequecy and amplitude/intensity of a light beam are related
to the underlying properties of the photons
- qualitatively
describe what is measured in the photoelectric effect experiment and
explain how this implies a quantum picture of light, including
explaining what results the classical interpretation of light would
predict for this experiment
- explain why there is a maximum wavelength above which light cannot eject electrons from a metal regardless of intensity
- explain
the relation between the maximum kinetic energy of the ejected
electrons and the frequency of the incident light, as predicted by the
photon model
- quantitatively
analyze photoelectric data to deduce the
relationship between energy of photons and frequency of light
Properties of
Quanta of Light ("Photons")
- describe what is meant by polarized light and predict the
reduction in intensity that results from sending polarized light
through a polarizer at some angle
- argue why the photon explanation of the polarizer experiments demands a probabilistic ( i.e. non-deterministic) behavior
of photons
- predict the likelihood of various outcomes in simple experiments governed by probabilistic behavior
- use
the mathematical model of photon polarizations as unit vectors to
predict probabilities for photons being transmitted through
polarizers
- define what is meant
by an "eigenstate" for a given measurement in the context of photons passing through polarizers
- explain
how the probabilities for transmission are related to the details of
how a photon state decomposes into a superposition of eigenstates
for the polarizer
- compare and contrast
classical superposition and quantum superposition
- describe how a photon's polarization state changes when it passes through a polarizer
- explain how the simple model for calculating transmission probabilities has features that generalize to all quantum systems
The quantum description of electrons
- describe the double slit experiment for light or electrons
and explain why this provides evidence that quantum particles do not
have deflnite positions and can exist in quantum superpositions
- explain why the results of the double slit experiment imply
that the initial electrons do not have
well defined positions
- explain
why the double slit experiment suggests that the behavior of
single particles is probabalistic and how the classical intensity
pattern is related to the relative probability for hitting various
points on the screen
- Describe what is meant by
probability density and evaluate whether or not a given function is a
valid probability density for finding a particle
- Explain what is meant by a position eigenstate and what is meant by a quantum superposition of position eigenstates.
- explain
how the process of describing general quantum superpositions of
position eigenstates leads to the concept of a wavefunction
- Use
the wavefunction to determine the probability for finding a particle in
a given region of space. Be able to determine the proper normalization
of the wavefunction if it is not given.
- State
what is meant by the expected value (or expectation value) in a
measurement of position. Be able to calculate the expected value given a particle's wavefunction
- Explain what happens to the wavefunction after a measurement of the particle's position
Momentum eigenstates, wavepackets, and uncertainty- explain why the double slit experiment suggets that we can associate a wavelength to electrons with a particular momentum
- state de Broglie’s relationship between wavelength and
momentum of an electron or other particle
- write
down the mathematical description of the wavefunction for a momentum
eigenstate; explain how this can have a wavelength even though the
probability density is a constant in space
- explain what is meant by a wavepacket and why the wavefunction for a real travelling electron should take this form
- give a simple explanation for why particles with well defined momentum cannot have a definite position
- explain
the physical interpretation of the mathematical fact that wavepackets
and other wavefunctions can be written as a sum of pure waves
- use
the amplitude function describing the superposition of pure waves
contributing to a given wavefunction to predict the relative
probability for possible outcomes in a measurement of momentum
- explain
qualitatively how the width of a wavepacket and its wavelength relate
to the combination of pure waves (momentum eigenstates) that it is
built from
- state the Heisenberg Uncertainty Principle and explain how it is related to the previous mathematical fact
- explain what is meant by uncertainty in position and uncertainty in momentum for a state
- explain
why we can't violate the Heisenberg Uncertainty Principle simply by
making a measurement of momentum immediately after a measuement of
position
- use the Heisenberg Uncertainty Principle to predict how quickly a given wavepacket will spread out with time
The
Schrödinger
Equation
- predict the velocity of a given wavepacket and how fast it will spread out by looking at the wavepacket at some initial time
- explain the
difference between phase and group velocities for a wavepacket
- explain what is meant by a dispersion relation
- know
that the relation between frequency and wavelength for a specific kind
of wave determines the relation between wavelength and group velocity
- explain
why the group velocity for electron wavepackets should be inversely
proportional to wavelength and (qualitatively) how this leads to the
relation between frequency and energy for electron wavefunctions
- write
down the time-dependent wavefunction for a
momentum eigenstate and relate its frequency and wavelength to the momentum
- explain
why knowing the time dependence of momentum eigenstate wavefunctions
allows us to determine the time-dependence for general wavefunctions
(of a free particle)
- explain why the Schrodinger equation determines the time-dependence of a wavefunction
- explain why the Schrodinger equation can be interpreted as a relationship between wavelength and frequency for wavefunctions
- write
down the potential function for simple physical
systems including electrons in wires or electrons near other charges
Bound states and
atomic spectra
- describe what is meant by an energy
eigenstate, and state important properties that all energy eigenstates satisfy
- know what is
meant by the statement that energy eigenstates
are “stationary”
- Describe
how we would check using the Schrodinger equation whether a certain
initial wavefunction is an energy eigenstate with energy E, and
how this leads to the time-independent Schrodinger equation
- describe what is meant by a bound state (either in classical physics or in quantum mechanics)
- explain
the crucial difference between the allowed energies for bound states in
quantum mechanics as compared to classical mechanics
- describe
how we would go about finding the possible energies for bound states
for a quantum system described by some specific potential energy
function
- explain why properties of quantum bound states can explain why atoms are stable and why atomic spectra have discrete frequencies
- be able to calculate the emission spectrum of an atom or molcule given its bound state energies (an vice versa)
- be
able to predict the wavelengths of light for which a gas of atoms will
be opaque or transparent given its bound state energies
- explain what is meant by zero point energy and why this energy is a consequence of the Heisenberg Uncertainty Principle
- state the
possible energies that might be
obtained in a measurement performed on a simple superposition of energy
eigenstates
- explain
qualitatively how the probability density for a superposition of energy
eigenstates behaves differently from that of an energy eigenstate
- predict
probabilities for measurements of energy
or position for simple superpositions of the energy eigenstates
Tunneling
- describe the behavior of the
wavefunction in regions where
the particle is classically forbidden
- describe
the qualitative evolution of a wavefunction
initially localized in one half of a double well
- describe
the evolution of a wavefunction for a particle that is travelling
towards a potential energy barrier that it does not have enough energy
to pass through classically
- explain
how tunneling can be used to understand radioactivity and to produce
atomic scale "images" with scanning-tunneling electron microscopes