PHY215B, Summer '99

A. CONTENTS

1. Relative Linear Motion and Frames~of~Reference

2. Special Relativity: the Lorentz Transformation, the Velocity Addition Law

3. The Length Contraction and Time Dilation Effects of Special Relativity

4. Relativistic Energy: Thresholds for Particle Reactions, Binding Energies

5. Relativistic Momentum: Particle Decays

6. Appearances At Relativistic Speeds

7. The Equivalence Principle: Introduction to Relativistic Gravitation

8. Characteristics of Photons

9. Discrete Spectra

10. The Bohr-Sommerfeld Model of the Atom

11. Transitions and Spectral Analysis

12. De Broglie Waves

13. The Uncertainty Relations: Description, Applications

14. Quantum Tunnelling

15. Numerical Demonstration of Energy Quantization for Atomic Hydrogen

16. Wave Particle Duality: Light

17. Exponential Decay: Observation, Derivation

18. Some Uses of Radioactivity

19. Fundamental Forces and Elementary Particle Classification

20. Conservation Laws for Elementary Particle Reactions

21. Fundamental Forces: Ranges, Interaction Times, and Cross Sections

B. THE SKILLS

Relative Linear Motion and Frames~of~Reference

Knowledge

• Vocabulary: frame of reference, observer.

Problem Solving

• Given the position vectors of both an object and an observer as seen by a second observer, find the position, velocity and acceleration vectors of the object as seen by the first observer. Use suitable notation for labelling observers and observed. Sketch any of the vectors with respect to any frame of reference, as requested.
• Given a kinematical equation containing both symbols and numbers plus a word description of the units involved, properly insert the units into the equation and carry out any appropriate units algebra.
• In any given kinematical problem, determine whether a = dv/dt is valid.

Special Relativity: the Lorentz Transformation, the Velocity Addition Law

Knowledge

• Vocabulary: Galilean transformation, Lorentz transformation, relativistic velocity addition law.
• Derive the Galilean transformation from the non-relativistic velocity addition law and show that it is invariant under interchange of the observer labels. Explain why one intuitively believes that this should be so.
• Explain clearly how the observed constancy of the speed of light with respect to all observers shows that the Galilean transformation must be erroneous.
• Given the Lorentz transformation, derive the relativistic velocity addition law from it.
• Show that the relativistic velocity addition law is in agreement with the fact that the speed of light is the same for all observers.
• Show that, for low enough relative speeds, the difference between the relativistic and non-relativistic velocity addition laws becomes unobservable.

The Length Contraction and Time Dilation Effects of Special Relativity

Knowledge

• Vocabulary: laboratory frame, rest frame, length contraction, time dilation, twin paradox.
• Given the Lorentz transformation, derive the relativistic length contraction and time dilation factors.
• Show that the Lorentz transformation is independent of which frame of reference is considered to be moving and explain how this leads to the twin paradox.

Problem Solving

• Given length and time intervals measured in one frame of reference, find the corresponding length and time intervals measured in a different frame of reference.

Relativistic Energy: Thresholds for Particle Reactions, Binding Energies

Knowledge

• Reduce the expression for relativistic kinetic energy to its non-relativistic form, starting from the general expression for Taylor's series.

Problem Solving

• Calculate CM-frame thresholds for given 2→2 particle reactions using Conservation of Energy and the masses of the four particles involved, and vice versa.
• Calculate binding energies using Conservation of Energy and the masses of the separated and conglomerate given objects, and vice versa.

Relativistic Momentum: Particle Decays

Knowledge

• Reduce the expression for relativistic momentum to its non-relativistic form, using the general expression for Taylor's Series for the expansion of a function about a point.
• Show that F=ma is generally valid only for v²<<c².

Problem Solving

• Given a particle's rest mass and velocity, calculate its relativistic momentum and energy.
• Use conservation of energy and momentum to work decay problems, 1 particle → 2 particles, in the center of mass frame.

Appearances At Relativistic Speeds

Knowledge

• Derive the correct appearance of a moving object, taking into account the apparent Lorentz contraction of the object and also the finite speed of the particles of light by which one observes it.
• Show that, for ordinary every-day speeds, the apparent angle of rotation, produced by retardation/contraction, is too small to be seen.
• Describe the appearance of a moving cube as its v/c → 1.
• Show that, under that assumption of the Lorentz contraction alone, a moving object would appear distorted with respect to its rest-frame shape.
• Show that a distant straight line traveling at speed v will appear rotated through an angle θ = cos^-1√(1 - v²/c²).

The Equivalence Principle: Introduction to Relativistic Gravitation

Knowledge

• Vocabulary: gravitational mass, inertial mass, active and passive gravitational mass, (generalized) tidal force.
• Explain how Galileo's experiment and Newton's second law establish the proportionality of gravitational force to mass.
• Outline the methods, actual and idealized, used to measure gravitational and inertial mass.
• Give three examples where inertial forces come into play on you.
• State: why Einstein suspected that inertial and gravitational forces were essentially the same; and to what level of precision they are known to be equivalent.
• State one way you could tell that the Earth's pull was present if you were an astronaut orbiting the earth in a sealed capsule of finite size, and another way if you had a window.
• State the objective of the Dicke-Eötvös experiment. Describe what is looked for in the behavior of the torsional penelulum and why.
• Design a simple experiment that could be used to measure the local acceleration of gravity, g. Show that the velocity of the article being used is immaterial to the measurement.
• Compare Einstein's Equivalence Principle (ignoring any limitations) with Newton's explanation, assuming the earth's gravity field is present.
• State whether you believe Newton's third law is absolute and justify your belief.

Characteristics of Photons

Knowledge

• Name the seven major regions of the electromagnetic radiation spectrum on a chart that shows the frequency spread of each region.
• Name a characteristic photon source for each of the seven major frequency regions of the electromagnetic spectrum.
• State a very approximate relationship often found between photons, wavelengths and the dimensions of their sources.
• Locate the basic colors of the visible spectrum on a wavelength chart.
• Sketch a graph of normal eye sensitivity vs. wavelength.

Rule Application

• Given information relevant to one of the four quantities associated with a photon - energy, momentum, wavelength, and frequency - calculate the other three.

Discrete Spectra

Knowledge

• Vocabulary: quantization, energy level, energy level diagram, ground state, excited state, ionization, hydrogen-like ion, quantum jump, principle quantum number, atomic one-electron system, atomic number, spectrometer.
• State the energy level formula for atomic one-electron systems.

Rule Application

• Given the atomic number Z for an atom, state the symbol for its corresponding hydrogen-like ion.

Problem Solving

• Construct an energy-level diagram for any atomic one-electron system, showing energy and quantum number values for each level.
• Calculate the wavelength and frequency of the photon emitted when any given atomic one-electron system makes a transition from one specified energy level to another.
• Given a distribution of emitted energy over a discrete set of wavelengths, use linear interpolation on a table of tri-stimulus values, plus a Chromaticity Diagram, to determine the light's chromaticity coordinates and perceived color.

The Bohr-Sommerfeld Model of the Atom

Knowledge

• Vocabulary: Bohr radius, Rydberg constant.
• Derive the allowed radii, allowed speeds, and allowed energies of electrons in the Bohr model.
• State Sommerfeld's three refinements of the Bohr model.
• State the deficiencies and limitations of the Bohr-Sommerfeld model.
• State the differences between the quantum mechanical picture of the atom and the Bohr-Sommerfeld model.

Problem Solving

• Given the masses and charges of a two-particle atomic system, calculate the allowed energies of the system and draw an energy level diagram to scale.

Transitions and Spectral Analysis

Knowledge

• Vocabulary: band spectra, continuous spectra, line spectra, fluorescence, phosphorescence, resolving power, dispersion, spectroscope.
• State the three general types of information spectroscopy yields and cite an area of application for each type.
• Compare and contrast emission spectroscopy with absorption spectroscopy, including the types of light sources used.
• Briefly outline a procedure for spectral analysis, including: (a) the mechanism for exciting the sample atoms or molecules; (b) how the various wavelengths are separated; (c) why the spectra usually appear as lines; (d) how the wavelengths of the spectral lines are measured; and (e) how the sample atoms or molecules are then identified.
• Describe the three mechanisms by which the atoms and molecules make transitions between energy levels.

De Broglie Waves

Knowledge

• Describe the Davisson-Germer electron diffraction experiment. State the relevant equations and indicate the appropriate variables with a sketch.
• State the Principle of Complementarity and tell why the wave and particle theories of matter are incompatible and mutually exclusive.
• Explain the significance of the wave function associated with a particle and use the diffraction of electrons by a single slit as an illustration.

Rule Application

• Calculate the deBroglie wavelength of a particle, given its momentum or kinetic energy.

The Uncertainty Relations: Description, Applications

Knowledge

• Vocabulary: level width, lifetime, uncertainty, radioactive, wave packet.
• State Heisenberg's uncertainty principle and illustrate it with at least two examples.
• State the energy-time uncertainty relation.

Problem Solving

• Given a situation in which position and momentum (or lifetime and energy) are to be simultaneously measured, determine the possible precision of measurement of one variable given constraints on the precision of measurement of the other.

Quantum Tunnelling

Knowledge

• Correlate a tunnelling wave packet's evolving behavior with general properties of the Schr\"odinger equation and with periods of exponential and non-exponential decay.
• Show that quantum tunnelling violates expectations from classical mechanics.
• Discuss your reaction to the probabilistic basis of quantum mechanics.

Numerical Demonstration of Energy Quantization for Atomic Hydrogen

Knowledge

• Vocabulary: normalization, normalization factor, normalized wave function, radial Schrödinger equation, radial wave function, 3-point recurrence relation, net-point notation.
• Given the general radial Schrödinger equation for zero angular momentum, derive a 3-point recurrence relation for the radial wave function.
• State the boundary conditions on the radial wave function at the origin and at infinity and give the reasons for them.

Wave Particle Duality: Light

Knowledge

• Vocabulary: quantum field theory, wave-particle dualism.
• Contrast the quantum field theory description of light with that of the classical particle and wave models. Describe the predictive power of each of these.

Rule Application

• Give reasons why any given light phenomenon can or cannot be satisfactorily described as each of these:
  • (a) classical particle
  • (b) classical EM wave
  • (c) quantum field

Exponential Decay: Observation, Derivation

Knowledge

• Describe the relationship between the exponential decay law and typical finite-number data.
• Derive the exponential decay law, for the number of undecayed systems, starting from the "no aging" (constancy of decay constant) assumption.
• Derive the exponential decay law, for the probability that a single system is undecayed, starting from the "no aging" (constancy of decay constant) assumption.

Rule Application

• Determine whether or not a given set of decay data is consistent with an exponential description.
• Determine the mean life for a given set of exponential decay data.

Some Uses of Radioactivity

Knowledge

• Starting from the exponential decay law, derive the relationship between the "disintegration constant" (also called "decay constant") and the "half-life."
• Solve these problems in Physics, Alonso and Finn: 22.14c (including a numerical check), 22.16, and 22.17. Closed book, no answers provided.

Fundamental Forces and Elementary Particle Classification

Knowledge

• Vocabulary: antiparticle, baryon, electromagnetic interaction, elementary particle, fundamental forces, hadron, lepton, muon, meson, neutrino, nucleon, pair annihilation, pair production, strong interaction, weak interaction.
• State the four fundamental forces of nature.
• List seven known elementary particles that do not take part in the strong interaction.

Problem Solving

• Determine, from the nature of the particles involved in a given reaction, whether the reaction goes via the strong, electromagnetic, or weak interaction.

Conservation Laws for Elementary Particle Reactions

Problem Solving

• Determine whether a proposed elementary particle reaction is allowed or forbidden by each of the absolute conservation laws (conservation of energy, momentum, charge, baryon number, electron number,and muon number).
• Determine whether a proposed elementary particle reaction is allowed or forbidden by conservation of isotopic spin or strangeness.
• Given a proposed elementary particle reaction, determine which interactions allow it, which forbid it.

Fundamental Forces: Ranges, Interaction Times, and Cross Sections

Knowledge

• Vocabulary: cross section, flux, total cross section, intermediate vector boson.
• State the characteristic times associated with the strong, weak and electromagnetic interactions among elementary particles.
• State the characteristic ranges associated with the strong and weak interactions.

Problem Solving

• Determine the fractional number of strong (or weak) interactions that can occur in an interaction in which the target particle density is given.
• Given the target particle density and the fractional loss of projectile beam flux through the targets, calculate the total cross section for all reactions.