Physics 284 Syllabus

- Special Relativity
- review of the nature of light
- Michelson-Morley experiment
- Einstein's postulates
- Lorentz transformations for space and time
- time dilation and length contraction
- simultaneity and speeding cows
- formalism of Lorentz and relations between gamma and beta
- matrix formalism for Lorentz transformations
- spacetime intervals
- velocity transformations
- relativistic Doppler effect
- laser cooling
- the twin paradox
- Minkowski spacetime diagrams
- relativistic energy and momentum
- energy momentum 4-vectors
- relativistic scattering and decay processes
- electric and magnetic fields
- General Relativity
- principle of equivalence
- curved space
- deflection of light
- time delay
- gravitational lenses and black holes
- Minkowski again
- gravitational redshift
- the twin paradox revisited
- Mach's principle
- Pound-Rebka experiment
- gravity waves and gyroscopes (more tests of GR)
- LIGO
- gravity probe B
- the binary pulsar
- Quantum Mechanics
- particle characteristics of light
- photoelectric effect
- Bremsstrahlung
- Compton scattering
- pair production
- black body radiation
- wave characteristics of matter
- Bragg scattering
- Davisson-Germer experiment
- a model for particle/waves
- de Broglie hypotheses p=h/lambda and E=h nu
- wave packets and superposition
- superposition of two cosine waves
- Gaussian wave packet
- uncertainty
- complementarity of position and momentum
- the Heisenberg principle for energy and time etc.
- uncertainty in measurement
- examples
- the double slit
- radioactive decay
- the Dirac delta function
- interpretation of the wave function
- analogy to light
- probability density
- rules for adding amplitudes and probabilities
- the double slit (again)
- and again with bra-kets
- the Schroedinger equation
- the free particle wave function
- momentum and energy operators
- 1D Schroedinger equation
- normalization
- free particle wave functions (again)
- expectation values
- time independent Schr. equation and E quantization
- infinite square well
- boundary conditions
- atoms and baseballs
- expectation values
- finite square well
- general features of Psi
- concavity and node counting
- wave function sketches
- more examples of the Scroedinger equation
- the 3D infinite square well
- harmonic oscillator potential
- finite barrier potential and quantum tunneling
- Special Topics
- the three Polaroid problem
- the Copenhagen interpretation of QM
- the measurement problem
- Schroedinger’s cat and Wigner’s friend
- Wheeler's delayed choice experiments
- the EPR paradox and Bell’s inequality
- quantum erasers
- quantum tunneling times
- interaction-free measurement
- quantum teleportation
- quantum cryptography
- quantum computing
- the Hydrogen atom
- historical background
- Schroedinger equation in spherical coordinates
- R
_{nl}(r) (Laguerre) and Y_{lm}(spherical harmonics) - boundary conditions and quantization
- relations between E and n, |L| and l, L
_{z}and m_{l} - energy degeneracy
- atoms in magnetic fields
- intrinsic spin
- anomalous Zeeman effect
- the Stern-Gerlach experiment
- spin and statistics

Special Relativity is a theory to describe the ultra fast. It is a generalization of Galilean and Newtonian physics to encompass objects moving at a significant fraction of the speed of light. It is valid for any reference system moving at a constant velocity.

Review of the wave nature of light (interference and diffraction, Huygen's principle), emphasizing Youn'gs double slit experiment. Review of Maxwell's equations. Note that E and B satisfy the wave equation (nice 'proof' of this in Fishbane, Gasiorowitz and Thornton section 35-3), leading to the postulate of EM waves. A review of the origin of the wave equation can be found in Halliday and Resnick appendix III. That EM waves were derived to travel at the speed of light led to the conclusion that light is an EM wave.

Two problems: 1) When light waves, what is waving? Nineteenth century sciencists struggled with the concept of ether. 2) Maxwell's equs. (which say that light always travels at velocity c) are not invariant under a Galilean transformation. One or the other of Maxwell and Galileo are wrong.

If light travels through ether then the measured speed of light should depend on the velocity of the ether. Michelson and Morley set out to measure this speed precisely using a very clever device called an interferometer. They confirm that light always travels at velocity c, just like Maxwell's equs. imply. You should understand the details of their apparatus and the derivation in section 2.2.

See also Shankland "The Michelson-Morley Expt."

Einstein proposes two postulates: 1) the laws of physics are the same in all reference systems and 2) the speed of light (in vacuum) is the same in all (inertial) reference systems. These postulates lead to the Lorentz transformation, replacing the Galilean transformation.

Start with two simple studies of how clocks and meter sticks behave in two different reference systems if Einstein's postulates are correct. Derive the very important and weird principles of time dilation and length contraction. To test your understanding of time dilation and length contraction, try to figure out what the universe looks like to Mr. Photon.

See also Weisskopf "The Visual Apperance of Rapidly Moving Objects"

See also Bronowski "The Clock Paradox"

Another surprise. The word simultaneous means different things to different observers. One of the course's now famous homework problems, the relativistic cow, illustrates this principle.

Derive the general form of the Lorentz equations. Note how special cases yield time dilation and length contraction. You should be able to repeat this derivation on your own. Memorize the convenient algebra of gamma and beta (which is not well-covered in the book).

Not in the book but quite important. A quick introduction to another way of thinking about Lorentz transformations using matrices. Space and time coordinates together comprise a spacetime vector on which the matrices operate. This is used in one of the homework problems.

Another very useful tool for working with special relativity. Certain quantities called spacetime intervals ARE the same in all inertial reference systems. This observation is handy when faced with certain homework problems.

Velocities also appear different if viewed from different reference systems. Derive the transformation laws for velocities.

Review what you know about the classical Doppler effect (e.g. sound) on your own. Any beginning physics text will have it. The shift in frequency depends upon the velocities of the source and the receiver relative to the air. The relativistic Doppler effect is surprisingly different. The effect depends only on relative velocities. There is even an effect when the motion is perpendicular to the line of sight.

The book gives only a brief description of this is in the special topic on page 55. A totally cool application of the relativistic Doppler effect. Also known as optical molasses. Earned the Nobel prize in 1997. Cool a small gas sample to near absolute zero by shining laser light on it. Later became a technique for producing Bose-Einstein condensates.

See also: Chu "Laser Trapping of Neutral Particles"

See also: http://www.nobel.se/announcement-97/physics97.html

A very famous problem which seems very confusing on the surface. One twin leaves her brother behind on the Earth and warps off to places where no one has gone before. When she returns to Earth, which twin is biologically older? We will pose the problem here and start to understand the apparent discrepancy. A fuller explanation is forthcoming after we learn a little about General Relativity.

Yet another way of looking at relativity problems. In particular, this useful diagramatic tool can help us to understand the twin paradox. It has other common applications in General Relativity.

You guessed it. Both energy and momentum values depend on the reference frame of the observer. Demonstrate that the traditional definition of momentum does not work in the relativistic world. Define relativistic momentum, kinetic energy, rest energy, total energy. Recognize what Einstein's equation E = mc^{2} really means. Pay close attention to the book's example shown in Figure 3.3.

Not in the book. A very brief intro on how to use the matrix formalism on energy-momentum vectors.

Explore the dynamics of the subatomic relativistic world. This is my field of research so you can bet there will be no shortage of homework problems on this topic.

We could spend a great deal more time on this than the brief intro we will do. We will see the general form of the transformation of E and B fields, just to scare you, without deriving it. One problem about a charged wire will challenge you to put together many of the ideas you have learned in special relativity.

Encompasses the study of accelarated reference systems as well as inertial systems. Hence the use of the word "General". It is necessary to describe many features of massive objects (e.g. black holes). The book is extremely curt on the subject of general relativity. In class, we will expand on almost all topics. You are expected to know all of the special effects of general relativity and all of the main experiments that were or will be done to verify the predictions of GR. Much of the lecture material is taken from *Will "Was Einstein Right?".*

Another famous postulate used by Einstein. An accelerated reference frame is equivalent to a stationary reference frame in a gravity field.

Einstein had the brilliant intuition that objects in a gravity field could be thought to travel in (locally) straight lines, just as they would in the absence of the gravitational force, except that spacetime in a gravity field is warped. His conclusion is that mass/energy warps space, and is the foundation of his theory of general relativity. We spend some time figuring out what curved space should look like, something not done in the book. Einstein uses his new theory to solve a famous puzzle, the perihelion shift of Mercury (the first experimental test of GR). At that point he knows his theory is correct.

Here we begin discussion of the second of three famous tests of General Relativity. Reproduce Einstein's thought experiment to deduce the astonishing result that light must curve in a gravitational field. Confirmed by Eddington in 1919, making the front page of the New York Times.

A related effect. Light travel times through curved space do not appear constant to external observers.

See also: Shapiro "Radar Ovservations of the Planets"

For entertainment, we will digress a bit here to talk about some terrific examples of light deflection.

See also: Chaffee "The Discovery of a Gravitational Lens"

See also: Turner "Gravitational Lenses"

See also: Susskind "Black Holes and the Information Paradox"

Not in the book. Another way to think of curved space is through its effect on Minkowski spacetime diagrams. You should be able to draw and interpret simple spacetime diagrams.

The third test of GR. Light changes frequency as it travels through a gravitational field. Deduce this result using our knowledge of special relativity and Doppler shifts. You should be able to reproduce this derivation.

See also: Hafele and Keating "Around-the-World Atomic Clocks"

Now that we know about gravitational redshift, we can finish explaining this famous paradox.

A remaining teaser which we will not thoroughly resolve. How do we know which twin feels the force of acceleration? Related to ideas originating with the 19th century philosopher Ernst Mach.

The Nobel prize winning experiment in the Harvard physics building stairwell that confirmed gravitational redshift.

See also: Pound and Rebka "The Apparent Weight of Photons"

Three more effects of GR you should be familiar with are gravity waves, the "geodetic effect" and "frame dragging". We will digress to discuss a couple of neat experiments that may extend the tests of GR in the next few years by searching for these effects.

Laser Interferometer Gravitational wave Observatory. A search for gravity waves. Briefly touched on in the book in the Special Topic on page 90.

See also: "LIGO: The Laser Interferometer Gravitational-Wave Observatory"

See also: http://www.ligo.caltech.edu/LIGO_home.html

A precision experiment using gyroscopes in Earth orbit to look for the geodetic effect and frame draggin. Consider the Discovery article by Taubes as required reading.

See also: Taubes "The Gravity Probe"

See also: http://einstein.stanford.edu/

Probably an entire lecture devoted to this terrific story of the discovery and interpretation of the first binary pulsar. Work done by two local heroes, a professor and grad student at UMass. Led to the first (indirect) proof of the existence of gravity waves. Awarded the 1983 Nobel prize.

See Also: Weisberg, Taylor, Fowler "Gravitational Waves from an Orbitting Pulsar"

Exam I

Wed. Feb 24 6:30-8pm

Hasbrouck Room 113

Our introduction to QM focusses on the problem of describing physical entities that exhibit both particle-like and wave-like properties. Physicists around the turn of the century encountered several instances where light, at that time usually described by a wave model, appeared to come in discrete, particle-like packages (a.k.a. photons). This particle-like behavior of light was apparent in black-body radiation, the photoelectric effect, and several scattering experiments such as Bremsstrahlung, Compton scattering and pair production. Conversely, it was also discovered that objects that we normally think of as particles, such as electrons, sometimes exhibit wave-like characteristics such as interference. After a brief overview of these experiments, we will begin the task of building a new model of the world that incorporates both particle and wave characteristics simultaneously.

The following four phenomena (as well as black-body radiation, which we do not discuss in class in much detail) provide the most decisive evidence that light comes in packages.

The experiment that won Albert E. the Nobel prize in 1921. Lab work performed by Phillip Lenard (Nobel 1905) and Robert Millikan (Nobel 1923). Albert's explanation of the photoelectric effect using a description of light as packages of energy built upon the model by Max Planck (Nobel 1918) of energy quantization in the black body experiment.

Literally, "braking radiation". It refers to the radiation emitted by a charged particle when it is deccelerated by a charged nucleus.

Arthur Compton (Nobel 1927) studied the scattering of light off of carbon atoms and discovered that the change in wavelength of the scattered light was explained by a model of photons scattering off of electrons.

The "positron" was predicted by P. A. M. Dirac (Nobel 1933) in the 1920s and discovered by C. D. Anderson (Nobel 1936) in 1932. Pair production is the process by which a photon is converted into an electron-positron pair.

In class we will see a brief overview of this topic that lead Max Planck to first hypothesize that energy was absorbed and emitted in discrete quanta.

We begin with a discussion of Xray scattering off of crystals, ala Bragg and Son, thereby understanding how interference takes place in this scattering process.

A very similar experiment with electrons done by Davisson and Germer (which accicently lead to the 1937 Nobel prize) yields an interference pattern for electrons. The amazing conclusion is that electrons also behave like waves.

...serve to define how particles will diffract and interfere. A couple of examples of double slit interference using electrons and baseballs will help elucidate the concept.

Try to build up a model of a particle/wave out of a localized superposition of waves of different wavelengths.

This simplest of examples allows us to understand the difference between group velocity (of the envelope) and phase velocity.

This advanced example gives a more realistic notion of a localized wave packet, and allows us to generalize the definitions of group and phase velocities.

The very famous principle proposed by Werner Heisenberg (Nobel 1932). It follows inescapably from the way in which we constructed the wave packet.

The uncertainty in the position of a wave packet is inversely proportional to the uncertainty in the momentum of the wave packet. This leads to Heisenberg's famous inequality.

Other uncertainty principles for energy/time and angular momentum/angle are left as homework.

A precise definition of uncertainty is established in terms of the rms of many measurements.

The size of the interference pattern in the double slit experiment demonstrates the HUP for momentum and position.

Radioactive decay demonstrates the uncertainty relation between energy (mass) and decay time.

A purely mathematical example showing the extreme case of a localized wave packet.

Spring Break

What does the wave function really mean? We learn that QM tells us that we can no longer predict with infinite precision what values of location or momentum we will measure for a particular wave/particle. We can only make statements about the average values and the rms deviations we can expect from an ensemble of such measurements. This leads us to the interpretation that the wave function provides us with statistical information only.

For light, |E| plays the role of Psi, while the probability of locating the photon at a particular point is proportional to |E|^{2}.

Psi^{*}Psi is interpreted as the probability density for locating the particle in a given range.

The rules for combining two possible paths to an experimental outcome is a subtlety that often confuses the beginning quantum mechanic. We explore a few examples in order to better understand the concepts.

Required reading: Feynman Lectures in Physics vol 3 chapters 1-3

We return to this now infamous example to explore the difference between distinguishable paths (for which the probabilities add) and indistinguishable paths (for which the amplitudes add).

We introduce the Dirac "bra-ket" notation, noting that the amplitudes for consecutive events multiply, and summarizing the previous rules for distinguishable and indistinguishable paths.

Having invented the notion of a particle/waves, it is now necessary to invent new equations of motion to describe the behavior of these objects. Analogy to the classical equation for conservation of energy quickly lead us to the Schroedinger equation for wave functions.

We introduce complex exponentials as the general replacements of cosines for a set of basis wave functions.

Using the definition of free particle wave functions, we can define operators that yield energy and momentum values from those wave functions.

Using our definitions of energy and momentum operators we construct the logical equivalent of the classical conservation of energy in one dimension.

In order for |Psi|^{2} to be interpreted as a probability density, it must be normalized so that the integral over all space is equal to unity. We look at the Gaussian wave packet as an example.

We note that these functions satisfy the Scroedinger equation without a potential energy term, but that they have some interesting features as regards normalization.

...also known as "averages". Expectation values tell us what to expect out of an ensemble of measurements. We explore the definition of "expectation value" following appendix 5 in the book. Then we look at the Gaussian wave packet example to calculate the average momentum and the rms uncertainty on the momentum. Not surprisingly, Heisenberg's uncertainty relation is a logical consequence of the equations.

In the special (but common) case where the potential energy of a system is constant in time, the Schroedinger equation separates into two simpler equations. In this case, the time dependence of the system follows a universal form. The spatial dependence of the wave function is related to the energy of the particle/wave. One of the most important and amazing lessons of quantum mechanics is that if we confine a particle/wave to a localized space, the possible energy states are restricted to a set of discrete values. This explains, among other things, why electrons confined within an atom are only allowed to occupy certain discrete energy levels. Several examples of potential energy functions are explored.

We look at one of the simplest examples of a confining potential (the infinite square well) in order to see how energy quantization follows from the application of the Schroedinger equation.

After writing the general solutions for the wave function inside and outside the well, we see that the requirement of continuity at the boundary imposes restrictions on the possible momentum and energy values, and leads to energy quanitzation.

Specific examples of atom-sized wells and baseball-sized wells, show us why energy quantization is not noticable in the macroscopic world.

We pursue the infinite square well example to calculate average momentum and position, and rms values for each, ultimately verifying the Heisenberg uncertainty relation once again.

This example of a potential function is similar in many ways to the infinite square well. The one profound difference is the nature of the boundary conditions. In this case, both continuity and differentiability are required in order to solve Schroedinger equation. We will set up this problem in class and leave much of the detailed algebra to a homework assignment.

Exam II

Wed. Apr. 7 6:30-8pm

Hasbrouck Room 113

We briefly digress from the textbook in order to study some of the general features of the wave function in the presence of an arbitrary potential function.

See also: Brehm and Mullin "Introduction to the Structure of Matter"

This exercise is designed to help build your intuition about wave function shape, focussing on concavity and the number of nodes. We note the difference between classically allowed and forbidden regions. Once again, we see how energy quantization results in the case of confining potentials.

We draw some wave function sketches in class for an arbitrary potential function. An interesting specific example is left to the homework.

We continue the study of the Schroedinger equation with several examples which give us some exercise in "practical" quantum mechanics.

We will take a brief glance at this example which displays the new feature of "degeneracy". We will return to the idea of degeneracy more fully when we explore the hydrogen atom potential later in the course.

This example is extremely important because it serves as a good approximation around the minimum of any smooth potential funtion. General solutions to the harmonic oscillator potential involve the Hermite polynomials. A key feature to note is that the energy levels of a quantum mechanical harmonic oscillator are uniformly spaced.

This is an example that is similar in many ways to the finite well potential. The boundary conditions are different, however, leading to important consequences. The most interesting of these is the feature of quantum tunnelling, wherein the particle/wave can tunnel through a barrier in a way that is classically forbidden. This feature forms the basis of the scanning tunnelling microscope and of apparent faster-than -light motion, among other things. The main goal of the finite barrier problem is to calculate the transmission and reflection probabilities.

This section of the course represents a departure from traditional topics in introductory modern physics in order to explore some of the more recent and more entertaining issues in QM research.

What is the intensity of light passing through three polaroid filters, each rotated by 45^{o} relative to the preceeding filter? The answer teaches us we cannot sensibly assign a definite polarization to photons travelling through the apparatus.

We use the preceeding example of the triplet of polaroid filters to introduce the idea of "wave function collapse" and elucidate the traditional Copenhagen interpretation of quantum mechanics.

See also: John Horgan "Quantum Philosophy"

See also: David Albert "Bohm's Alternative to Quantum Mechanics"

See also: Abner Shimony "Reality of the Quantum World"

The picture we are left with from the Copenhagen interpretation is that wave functions evolve in time continuously and smoothly according to the Schroedinger equation until a measurement occurs, which triggers a discontinuous collapse of the wave functions. Not surprisingly, this idea does not sit well with many people.

After a brief note on multiparticle wave functions, the measurement problem is elucidated with that most famous of QM paradoxes, Schroedinger's cat. We follow this demonstration to its logical but apparently ridiculous conclusion with the story of Wigner's friend and Dan Rather's evening news.

See also: http://www.sciam.com/explorations/091696explorations.html

If the preceeding was not enough for you, we also discuss the nature of delayed choice experiments, wherein quantum influence might be said to travel backwards in time.

The Einstein, Podolsky, Rosen paradox was proposed as a demonstration of the absurdity of quantum mechanics as an explanation of the world. Much to the surprise of the original authors, John Bell later demonstrated that the apparatus of the EPR paradox could be used to experimentally differentiate between QM and any "local realist" theory of physics. The experiments have since been performed and, counterintuitive as it may seem, quantum mechanics is verified.

See also:the section on the EPR paradox and Bell's theorem in the bibliography

We study the results of several related experiments in quantum erasures as an advanced lesson in the rules for wave function combination and distinguishable paths. An entire class period is devoted to this classtalk-style exercise.

See also: Andrew Watson " Eraser Rubs Out Information To Reveal Light’s Dual Nature"

Recent experiments in quantum tunelling have revealed conditions under which particle/waves appear to traverse a quantum barrier at speeds exceeding the speed of light. We focus on the meaning of a wave packet in the process of trying to understand this strange result.

See also: Raymond Chiao etal. "Faster than Light?"

With the right experimental setup, it is possible to gain some information about an object without any interaction with the object (not even scattering a photon off of it). This very strange result is possible only because of the nature of quantum mechanics and the distinguishability (or not) of different paths.

See also: Paul Kwiat etal. " Quantum Seeing in the Dark"

A technology that would make Scotty proud. It is now possible in the laboratory to make an exact replica of a single photon (i.e. replicating its complete quantum mechanical wave function) at a distant

location, while destroying the quantum state of the original photon through wave function collapse. Unfortunately, this transference does not happen faster than light.

See also: the section of QM teleportation in the bibliography

A short discussion of this topic shows that there are some classic issues in cryptography that are profoundly altered with the advent of quantum mechanical keys.

See also: Charles H. Bennett etal. " Quantum Cryptography"

We go over some of the building blocks of a hypothetical QM computer and discuss the fundamental differences with classical computers, which center on the difference between bits and "qbits".

See also: the section of QM computing in the bibliography

Reviewing the Rydberg equation and the Bohr atom.

Translating the differential equation from a cartesian coordinate system to spherical coordinates (for a spherical potential such as the H atom potential) requires knowing how to apply a Jacobian transformation.

Separation of variables allows us to divide the H atom problem into several parts. Solutions involve the Laguerre functions for the radial part of the wave function and the sperical harmonic functions for the angular part of the wave function.

Just as was true in earlier 1D Schroedinger equation problems, boundary conditions lead to quantization rules. In this case, the 3d boundary conditions give rise to quantization of energy (*n*), angular momentum (*l*) and a component of angular momentum (*m _{l}*).

Further examination between the values of energy and angular momentum and the integration constants *n*, *l *and *m _{l}* demonstrate some failures of the original Bohr model.

As in the case of the 3D infinite square well, many solutions of the 3D H atom potential share the same energy level.

Degeneracy is partially broken by putting the atom in an external magnetic field. In this case we expect degenerate energy levels to split into an odd number several slightly different energy levels. This is known as the normal Zeeman effect. It is an effect used to great advantage in laser cooling experiments.

Two experimental puzzles led to the conclusion that there is intrinsic angular momentum (spin) associated with an individual electron. Modern perspective realizes that intrinsic spin is associated with most particles, including protons, neutrons and photons.

In some cases, atomic energy levels split into an even number of sublevels, a result inconsistent with the normal understanding of the Zeeman effect as resulting from the orbital angular momentum of electrons.

Another famous experiment, and the source of Stern's Nobel prize in 1943, is the Stern-Gerlach experiment, which demonstrates that degenerate neutral atoms can have an even number of component angular momentum states, inderectly indicating that the electron comes in two flavors of spin.

One of the most amazing features of QM spin is how it relates to the behavior of an ensemble of particles. A collection of identical particles in close proximity with half-integer spin (a.k.a. fermions, of which electrons are an example) will refuse to occupy the same QM state. This is the reason why electrons in atoms occupy different energy levels, and do not all fall to the ground state. In contrast, a collection of particles with integer spin (a.k.a. bosons, of which photons and some neutral atoms are examples) all prefer to occupy the same QM state. This feature is the basis for operation of the laser and for the construction of Bose-Einstein condensates.

Final Exam

Thu. May. 20 10:30 am

Hasbrouck Room 124