PHY215B, Summer '99

A. CONTENTS

1. Special Relativity: the Michelson-Morley Experiment

2. Derivation of the Lorentz Transformation

3. Matrix Algebra

4. Relativistic Space-Time: Four-Vectors

5. Topics in Relativity: Doppler Shift and Pair Production

6. Derivation of the Relativistic Energy and Momentum Formulas

7. Quantized Angular Momentum

8. Electron Spin

9. The Pauli Principle and the Periodic Table of the Elements

10. Energy and Boltzmann Distributions

11. The Fermi-Dirac Energy Distribution

12. The Photoelectric Effect

13. The Compton Effect

14. Continuous Spectra: Planck's Law

15. The Spectrum of Sodium

16. The Spectrum of Helium and Calcium

17. The Anomalous Zeeman Effect and the Land\'e G-Factor

18. Small Oscillations

19. Diatomic Molecules: Properties From Rotation-Vibration Spectra

20. X-Ray Spectra

21. Optical Pumping

22. Diffraction Grating and X-Ray Scattering From Crystals

23. The Nature of Atoms and Electrons: the Millikan, Thomson, and Rutherford Experiments

24. The Wave Equation and Its Solutions

25. Electromagnetic Waves From Maxwell's Equations

26. Optical Pumping

27. Laser Devices

28. Optical Circuits

29. Classical to Quantum Transition

30. Wave Functions, Probability, and Mean Values

31. Newton's Second Law From Quantum Physics

32. Evaluating While Learning: a Project

33. Locating All Schr\"odinger Equation Bound States

34. Chemical Bonding

35. The Helium Atom: Intuitive Approach

B. THE SKILLS

Special Relativity: the Michelson-Morley Experiment

Knowledge

• State the two postulates upon which special relativity is based.
• Derive the Michelson-Morley fringe shift in terms of the velocity of the earth with respect to a presumed ether.
• State how the results of the Michelson-Morley experiment help to justify one of the postulates of special relativity.

Derivation of the Lorentz Transformation

Knowledge

• Derive the Lorentz transformation using the two postulates of special relativity.

Matrix Algebra

Knowledge

• Vocabulary: matrix, determinant, inverse of a matrix.

Rule Application

• Add or multiply given matrices.
• Determine whether two given matrices are inverses of each other.
• Evaluate the determinant of a given 2^v2 or 3^v3 matrix.
• Write a given system of linear equations in matrix form.

Relativistic Space-Time: Four-Vectors

Knowledge

• Vocabulary: event, four-vector, world line.
• Write the Lorentz transformation in matrix form.
• Show that relativistic energy and momentum are the components of a four-vector.

Problem Solving

• Transform the coordinates of a given event from one inertial frame to another.
• Transform given energy-momentum four-vectors from one given Lorentz (inertial) frame to another.

Topics in Relativity: Doppler Shift and Pair Production

Knowledge

• Derive the Doppler shift formula using the Lorentz transformation.
• State the minimum photon energy required to produce a pair of particles, each of mass M.
• Write down the energy-momentum 4-vector for a photon.

Derivation of the Relativistic Energy and Momentum Formulas

Knowledge

• Describe a collision experiment that shows how to define a relativistic momentum which is conserved in the absence of external forces and which reduces to the Newtonian momentum for small velocities.
• Define a relativistic kinetic energy in terms of the work done in bringing a particle from rest to a velocity v under the action of relativistic force F.
• Derive a formula for the relativistic kinetic energy which reduces to the non-relativistic expression for kinetic energy, mv²/2, for small velocities.

Quantized Angular Momentum

Knowledge

• Derive the expression which relates the magnetic moment of a circulating charged particle (in a circular orbit) to the angular momentum of that particle.
• Calculate the possible energy change that occurs when a circulating charged particle is placed in an external magnetic field. Compare the classical result with the correct quantum mechanical result.

Rule Application

• Given a system which has two component parts, which have angular momentum quantum numbers ℓ₁ and ℓ₂, determine the possible values of the total angular momentum of the system and the possible values of its projection along any given direction.

Electron Spin

Knowledge

• Vocabulary: anomalous Zeeman effect, doublet, electron spin, fine structure, singlet, spin angular momentum, total angular momentum, spectroscopic notation, spin-orbit interaction.
• Describe electron spin in terms of the angular momentum of an extended object.
• Describe the physics of the spin-orbit interaction.
• Describe the Stern-Gerlach experiment and state its significance.

Problem Solving

• Given the orbital and spin angular momentum for given atomic energy levels, label the levels with spectroscopic notation.
• Given the spectroscopic notation of an atomic energy level, find the orbital and spin angular momentum quantum numbers.

The Pauli Principle and the Periodic Table of the Elements

Knowledge

• State the Pauli exclusion principle.
• Define "electron shell" and "electron sub-shell" in terms of the quantum numbers of the energy states that comprise shells or sub-shells.
• Arrange the elements in the periodic table given the order of filling of electron sub-shells.

Problem Solving

• Determine the number of electrons that can occupy a given shell or a given sub-shell.

Energy and Boltzmann Distributions

Knowledge

• Define the partition of a system consisting of N molecules of a gas.
• Define thermal equilibrium starting from the concept of a very large number of possible partitions for a given many particle system. Identify the partition for an ideal gas in thermal equilibrium.

Problem Solving

• Determine the number of particles that have their energies in a given energy range, given the partition for the many particle system.
• Given the partition, calculate the average values of kinetic energy, speed and speed squared and the total energy of a system of N molecules of a gas.

The Fermi-Dirac Energy Distribution

Knowledge

• Vocabulary: Fermi energy, degenerate gas.
• State the Fermi-Dirac occupation probability and energy distribution function, defining all your symbols.
• State the conditions under which the Fermi-Dirac distribution may be approximated by the Boltzmann distribution and show that the Fermi-Dirac distribution reduces to the Boltzmann distribution under these conditions.
• Derive the energy density as a function of temperature of an almost degenerate gas of non-interacting fermions.

Problem Solving

• Given a system of non-interacting fermions, calculate the Fermi energy.
• Given the Fermi energy as a function of chemical potential, invert the relation to determine the chemical potential.

The Photoelectric Effect

Knowledge

• Vocabulary: cutoff frequency, impeding potential, photoelectric effect, photocurrent, photoelectron, photoemission, stopping potential, work function.
• Describe the three aspects of the photoelectric effect which contradict the predictions of the electromagnetic wave theory of light.
• Explain how the photon theory of light correctly predicts the experimental observations of the photoelectric effect.

Rule Application

• Given the work function, the density, and the spacing between atoms of a material, and the intensity of light incident on the material's surface, determine how long the electromagnetic wave theory would predict it should take before photoemission can occur.

Problem Solving

• Given two of the following find the third using Einstein's equation for the photoelectric effect: frequency or wavelength of incident light, maximum photoelectron kinetic energy or stopping potential, work function of metal or cutoff frequency.

The Compton Effect

Knowledge

• Derive the equation for the wave length shift caused by the Compton effect. Start from fundamental conservation laws. Define all terms and justify each step.
• State how the results of the Compton experiment contradict the classical electromagnetic wave picture and support the photon theory of light.
• Describe how two properties of the photon allow it, at high densities, to give the appearance of an electromagnetic wave.

Problem Solving

• Given the necessary data about a collision between a photon and an electron, use the Compton shift equation and conservation laws to find either the electron's or photon's final momentum, kinetic energy and wavelength.

Continuous Spectra: Planck's Law

Knowledge

• Vocabulary: black body or ideal absorber or emitter; black body spectrum; spectral and total emittance; Planck's Law, optical pyrometer.
• Sketch the black body spectrum for a progression of temperatures, paying particular attention to the peak shift and to the general height shift.
• Describe in detail how one computes the locus of black body points on the Chromaticity Diagram.
• Describe how an optical pyrometer works.

Rule Application

• Given the integrated form of Planck's law, determine temperature from total emittance and vice versa.

The Spectrum of Sodium

Knowledge

• Sketch the energy levels of sodium and similar one-valence-electron atoms.
• Label the levels with the spectroscopic notation.
• Identify the "D" lines of sodium on the energy-level diagram.
• Identify the P series, S series, D series, F series transitions.
• State the Δℓ and Δ J selection rules for electromagnetic transitions.

Problem Solving

• Given the spectroscopic specifications of two levels of a one-valence-electron atom, determine from the selection rules if it is an allowed transition.
• Given the orbital and spin angular momentum for the energy levels of a one-valence electron atom, label the levels in spectroscopic notation (or vice versa).

The Spectrum of Helium and Calcium

Knowledge

• Vocabulary: fine structure, spin singlet state, spin triplet state.
• Sketch the energy level diagram for He (for one excited electron) and draw in the allowed transitions.
• Sketch the energy level diagram for Ca (for one excited electron) and draw in the allowed transitions.
• State the ΔS selection rule for atomic transitions.

Rule Application

• Given the electronic configurations for two inequivalent electrons, determine all possible states and write them in spectroscopic notation.
• Sketch the energy level diagram for a given two-valence-electron atom (for one excited electron only) and draw in the allowed transitions.

The Anomalous Zeeman Effect and the Land\'e G-Factor

Knowledge

• Derive the Landé g-factor.
• Compare the anomalous Zeeman effect to the normal Zeeman effect.

Problem Solving

• Given a particular transition in terms of spectroscopic notation, determine the number of Zeeman components and their splittings in a weak magnetic field.

Small Oscillations

Knowledge

• Vocabulary: small oscillation, point of stable equilibrium, linear approximation.

Problem Solving

• Given a force or potential energy function, show the small oscillation solution's: (i) center point; (ii) force constant; and (iii) frequency. The solution presentation should include: intermediate steps, graphs, and answer checks.

Diatomic Molecules: Properties From Rotation-Vibration Spectra

Knowledge

• Define "wave number" in terms of frequency.
• Draw diatomic molecule vibration-rotation energy levels, label them with quantum numbers, and indicate P-Branch and R-Branch allowed transitions.
• Explain the origin of each term in the expression for the vibration-rotation energy levels of diatomic molecules.

Problem Solving

• Given the spectrum of a species of diatomic molecule and values for two of the wave numbers: (i) identify the transitions and sketch them on an energy level diagram; and (ii) determine the equilibrium separation of the atoms, their inter-atomic force constant, and their frequency of radial vibration.

X-Ray Spectra

Knowledge

• Vocabulary: absorption coefficient, absorption limit, absorption edge, Bragg scattering, bremsstrahlung.
• Describe the production of discrete X-ray spectra.
• Derive the absorption law for X-rays passing through matter.
• State the Bragg scattering law for X-rays.
• Explain the nomenclature for the principle X-ray spectral lines.

Problem Solving

• Determine the maximum energy or minimum wavelength of X-rays produced when electrons of a given energy bombard a target.
• Determine the absorption of X-rays of given intensity passing through a material with a given absorption coefficient and thickness.
• Given the atomic number Z of an atom, estimate the wavelength of the K_a line and of the K absorption limit.

Optical Pumping

Knowledge

• Define spontaneous emission.
• Define stimulated emission.
• Define population inversion.
• Explain the basic principle of laser operation (optical pumping).

Diffraction Grating and X-Ray Scattering From Crystals

Knowledge

• Describe the transition from a two-slit interference pattern to one with many slits having the same slit-to-slit separation. Explain why diffraction gratings' maxima are so sharp.
• For a diffraction grating, derive the expressions for the angles at which maxima are detected.
• Discuss scattering of X-rays by a crystal lattice using Bragg's equation, and describe the experimental arrangement for observing Bragg scattering.

Problem Solving

• Given the wavelength of light incident upon a given size grating with N lines, calculate the angles of deviation of the principle maxima.
• For white light incident on a grating with N lines, calculate the angular separation for two given wavelengths of transmitted light.
• Given three of the following quantities for Bragg scattering, calculate the fourth: separation of crystal lattice planes, wavelength of light, angle of incidence, order of spectrum.

The Nature of Atoms and Electrons: the Millikan, Thomson, and Rutherford Experiments

Knowledge

• Describe and derive the equations which relate the quantity to be determined to the quantities which are experi mentally controlled in: (i) the Millikan oil-drop experiment; (ii) Thomson's determination of e/m; and (iii) the Rutherford scattering experiment.

Problem Solving

• Calculate the deflection of charged particles passing through given electric and magnetic fields.
• Calculate the differential cross section for scattering of given charged particles by nuclei of given charge.

The Wave Equation and Its Solutions

Knowledge

• Vocabulary: amplitude, wavelength, wave number, phase, phase constant, wave function, wave speed, wave equation, harmonic function, sinusoidal wave, traveling wave, boundary conditions.
• State the one-dimensional wave equation and its general solution.

Rule Application

• Given a wave function for a one-dimensional traveling wave, verify that it satisfies the wave equation.

Problem Solving

• Given a sufficient number of parameters associated with a sinusoidal wave, write down the mathematical description of the traveling wave.
• Determine the unknown parameters of a one-dimensional sinusoidal wave, given its displacement as a function of either: (i) position at two different times; or (ii) time at two different positions.
• Determine the unknown parameters of a one-dimensional sinusoidal wave, given the wave function and its first derivative with respect to time at x = 0 and t = 0.

Electromagnetic Waves From Maxwell's Equations

Knowledge

• Vocabulary: propagation (of a wave), polarization (direction of), plane-polarized (wave), monochromatic (wave).
• Given Maxwell's Equations, the "curl-curl" vector identity, and the definitions of the gradient, divergence, and curl operators, derive the wave equations for electric and magnetic field vectors at chargeless currentless space-points.

Rule Application

• Given the definitions of the gradient, divergence, and curl operators, verify that a given electromagnetic wave, consisting of coupled electric and magnetic waves, satisfies Maxwell's Equations.
• Given the direction of polarization, direction of propagation, frequency and amplitude of a monochromatic plane-polarized electromagnetic wave, write down the electric and magnetic fields in vector form. Sketch the situation.

Optical Pumping

Knowledge

• Define spontaneous emission.
• Define stimulated emission.
• Define population inversion.
• Explain the basic principle of laser operation (optical pumping).

Laser Devices

Optical Circuits

Classical to Quantum Transition

Knowledge

• Starting with Newton's Second Law and the Coulombic interaction, derive the classically-predicted relation between the hydrogen atom's energy and radius. Use the ground state energy of the atom to find the Bohr radius.
• Describe the failure(s) of the classical orbital model of the atom.
• Demonstrate the classical particle mechanics would not be expected to be correct for the hydrogen atom.
• Recite the rules for construction of the full Schrodinger Equation. Illustrate the rules by writing down the Schrodinger Equation for a given potential energy function.
• Draw the wave function for the first excited state of a free particle between impenetrable walls and show that it satisfies the Schrodinger Equation and appropriate boundary conditions. Determine its energy.
• Describe in words and diagram how energy quantization is produced by the Schrodinger Equation plus boundary conditions.
• Explain why the zeros of a rising-exponential-coefficient plot show a system's allowed energies.

Wave Functions, Probability, and Mean Values

Knowledge

• Define quantum mechanical uncertainty in position and momentum in terms of RMS deviation from the mean.
• Show how a system's quantum mechanical uncertainties in position and momentum are obtained from its wave function.

Rule Application

• Given a system's wave function, find the position and momentum uncertainty product for the system.

Newton's Second Law From Quantum Physics

Knowledge

• Discuss the reason(s) why any wave packet in a constant force field must obey classical particle physics.
• Given the Time-Dependent Schrodinger Equation, derive Newton's Second Law and the first correction term to it.
• Show and discuss the condition(s) under which Newton's Second Law is a good approximation, including the details of how the condition(s) would be evaluated in practice if you were given a wave packet and force law.

Evaluating While Learning: a Project

Knowledge

• Vocabulary: topic sentence, closure.

Locating All Schr\"odinger Equation Bound States

Knowledge

• Show that zero-angular-momentum bound state solutions to the Schr\"odinger equation have the form: u₀(r,K) = A(K)e^Kr + B(K)e^-Kr in all regions where the potential energy is constant, but not elsewhere.
• Show that bound states can only occur where A(K) = 0.
• Describe a method of locating all Schr\"odinger equation bound states in a given energy range for a given potential energy function.

Chemical Bonding

The Helium Atom: Intuitive Approach

Knowledge

• Write the classical expression for the energy of an n-electron atom.
• Write the Schroedinger Equation for an n-electron atom.
• Explain how the average coulomb potential energy of an arbitrary Hydrogen wave function depends on the diameter of the region which contains the wave function.
• Similarly for the average kinetic energy.
• Explain how the dependence on size of (3) and (4) above determines the size of the region containing the ground state wave function.
• Explain why the wave functions of Helium should be symmetric upon interchange of the coordinates of electron~1 and electron~2.
• Explain the meaning, in terms of probability, of the expression |ψ|²dV₁dV₂.
• Describe the simplification in the Schroedinger equation that results from setting V₁₂ = 0. Show that the Schrödinger equation reduces to two separate hydrogen-like equations whose solutions can be combined to give an approximate solution to the Helium problem.
• Describe the "full shielding" approximation and show results for the case V₁₂ = k_e e²/r₁ for the electron~1 and V₁₂ = k_e e²/r₂ for the electron~2, similar to (8) above.
• Describe why the "best shielding" approximation is better than those of (8) and (9) above.