Physics 284 Syllabus

1. Special Relativity
2. 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.

1. review of the nature of light
2. 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.

3. Michelson-Morley experiment
4. 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.

5. Einstein's postulates
6. 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.

7. Lorentz transformations for space and time
1. time dilation and length contraction
2. 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.

3. simultaneity and speeding cows
4. 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.

5. formalism of Lorentz and relations between gamma and beta
6. 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).

7. matrix formalism for Lorentz transformations
8. 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.

9. spacetime intervals
10. 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.

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

10. relativistic Doppler effect
11. 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.

1. laser cooling
2. 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.

13. 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.

1. Minkowski spacetime diagrams
2. 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.

14. relativistic energy and momentum
15. 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 = mc2 really means. Pay close attention to the book's example shown in Figure 3.3.

1. energy momentum 4-vectors
2. Not in the book. A very brief intro on how to use the matrix formalism on energy-momentum vectors.

3. relativistic scattering and decay processes
4. 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.

16. electric and magnetic fields
17. 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.

3. General Relativity
4. 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?".

1. principle of equivalence
2. Another famous postulate used by Einstein. An accelerated reference frame is equivalent to a stationary reference frame in a gravity field.

3. curved space
4. 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.

5. deflection of light
6. 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.

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

3. gravitational lenses and black holes
4. For entertainment, we will digress a bit here to talk about some terrific examples of light deflection.

7. Minkowski again
8. 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.

9. gravitational redshift
10. 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.

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

3. Mach's principle
4. 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.

5. Pound-Rebka experiment
6. The Nobel prize winning experiment in the Harvard physics building stairwell that confirmed gravitational redshift.

11. gravity waves and gyroscopes (more tests of GR)
12. 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.

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

3. gravity probe B
4. 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.

13. the binary pulsar
14. 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.

Exam I
Wed. Feb 24 6:30-8pm
Hasbrouck Room 113

5. Quantum Mechanics
6. 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.

1. particle characteristics of light
2. 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.

1. photoelectric effect
2. 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.

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

5. Compton scattering
6. 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.

7. pair production
8. 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.

10. 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.

3. wave characteristics of matter
1. Bragg scattering
2. 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.

3. Davisson-Germer experiment
4. 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.

4. a model for particle/waves
1. de Broglie hypotheses p=h/lambda and E=h nu
2. ...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.

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

1. superposition of two cosine waves
2. This simplest of examples allows us to understand the difference between group velocity (of the envelope) and phase velocity.

3. Gaussian wave packet
4. 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.

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

1. complementarity of position and momentum
2. 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.

3. the Heisenberg principle for energy and time etc.
4. Other uncertainty principles for energy/time and angular momentum/angle are left as homework.

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

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

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

5. the Dirac delta function
6. A purely mathematical example showing the extreme case of a localized wave packet.

Spring Break

7. interpretation of the wave function
8. 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.

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

3. probability density
4. Psi*Psi is interpreted as the probability density for locating the particle in a given range.

9. rules for adding amplitudes and probabilities
10. 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

1. the double slit (again)
2. 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).

3. and again with bra-kets
4. 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.

11. the Schroedinger equation
12. 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.

1. the free particle wave function
2. We introduce complex exponentials as the general replacements of cosines for a set of basis wave functions.

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

5. 1D Schroedinger equation
6. Using our definitions of energy and momentum operators we construct the logical equivalent of the classical conservation of energy in one dimension.

1. normalization
2. 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.

3. free particle wave functions (again)
4. We note that these functions satisfy the Scroedinger equation without a potential energy term, but that they have some interesting features as regards normalization.

5. expectation values
6. ...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.

13. time independent Schr. equation and E quantization
14. 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.

1. infinite square well
2. 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.

1. boundary conditions
2. 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.

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

5. expectation values
6. 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.

3. finite square well
4. 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

15. general features of Psi
16. 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.

1. concavity and node counting
2. 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.

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

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

1. the 3D infinite square well
2. 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.

3. harmonic oscillator potential
4. 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.

5. finite barrier potential and quantum tunneling
6. 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.

19. Special Topics
20. 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.

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

3. the Copenhagen interpretation of QM
4. 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.

5. the measurement problem
6. 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.

1. Schroedinger’s cat and Wigner’s friend
2. 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.

3. Wheeler's delayed choice experiments
4. 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.

7. the EPR paradox and Bell’s inequality
8. 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.

9. quantum erasers
10. 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"

11. quantum tunneling times
12. 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.

13. interaction-free measurement
14. 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.

15. quantum teleportation
16. 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.

17. quantum cryptography
18. 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.

19. quantum computing
20. 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".

21. the Hydrogen atom
1. historical background
2. Reviewing the Rydberg equation and the Bohr atom.

3. Schroedinger equation in spherical coordinates
4. 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.

5. Rnl(r) (Laguerre) and Ylm (spherical harmonics)
6. 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.

7. boundary conditions and quantization
8. 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 (ml).

9. relations between E and n, |L| and l, Lz and ml
10. Further examination between the values of energy and angular momentum and the integration constants n, l and ml demonstrate some failures of the original Bohr model.

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

1. atoms in magnetic fields
2. 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.

24. intrinsic spin
25. 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.

1. anomalous Zeeman effect
2. 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.

3. the Stern-Gerlach experiment
4. 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.

5. spin and statistics
6. 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