1. Do any three short problems from Gasiorowicz chapter 1,(G3, p21-22 or G2, p24-25), and/or chapter 2 (G3, p41-43 or G2, p38-40, and/or Liboff chapter 2. Do problems which stretch you but also revise material you feel to need to be sure about, and may not cover later. As part of this exercise, explore the need to be able to deal fluently with whatever units you meet, and make a list of useful constants. |
2. Do problems 14 & 15 from Gasiorowicz (G3) chapter 1, p22 (these are #15 & 16 in G2, p25). The harmonic oscillator in the first problem is a two-dimensional oscillator, i.e. r is effectively in the x,y plane. Compare this with the second problem, where the shape of the potential is different. This leads to the important result that quantization of energy results in energy levels that depend on the shape of the box: n^{-2} for the hydrogen atom potential V(r) ~ r^{-1}, n^{1} for the simple harmonic oscillator potential V(r) ~ r^{2}, and n^{2} for the square well potential V(r) ~ 0 for r < a, and infinite for r > a, i.e. for k becoming very large in the second problem. Absorb these points for the future! |
3. Spectroscopy is extremely accurate, and so models of hydrogen-like atoms are very important for testing quantum ideas. Use the formulae for the Bohr atom to explore both effects of nuclear charge (Z) and of reduced mass m, by considering motion of the electron and the nucleus relative to the center of mass of the atom. Use your analysis to calculate the Rydberg constant that is appropriate to a) deuterium; b) a ^{3}He^{+} ion, c) a ^{12}C^{5+} ion, and d) positronium. Make sure you know what particles these entities contain, before you start this problem. |
4. Do problem 6 from G3, chapter2, p42 (#4 from G2, p39). This problem can be done either approximately using the handwaving version of the uncertainty principle, or better mathematically using a Gaussian wave packet. Explain carefully whether and when you are using relativistically correct expressions. |
5. Use the uncertainty principle to estimate how long a lead pencil can stay balanced on its tip before it falls over. This problem, from Dicke and Wittke chapter 2, is a typical gedanken experiment inspired by Herr Heisenberg. In your answer be clear which version of the uncertainty principle you are using, and practice explaining your answer to the class in not more than 3 minutes. |
This problem was set on the Track I comprehensive examination in August 1998, with the pencil replaced by an ice-pick. (These pencils and ice-picks are sharp on a quantum scale). |
6. Investigate the decay rates for electrons in Bohr orbits using G3, web supplement 1-B (see page 15), and/or G2 problems 17 and 18 (page 25). Find out the lifetimes of some spectroscopic lines you are interested in, e.g. the Balmer line in atomic H, or the Na D-lines, and see if you can make sense of these numbers in relation to this simple calculation. The aim here is to get a feel for what lifetimes are involved, and how well spectroscopic energies can be determined via lifetime broadening (using the uncertainty principle applied to E and t). If this problem feels too difficult, do question 17 first, and submit it as an alternate or for supplementary credit. Note that G2 uses Gaussian units. |
7. Study black body radiation by doing two coupled problems (G3, chapter 1, problems 1 and 2, p21 (#1/2 from G2, p23/24). Solve the first problem, and show that it is analogous to the kinetic theory problem relating the number of atoms in a gas which flow through a hole in a wall to the atomic density of the gas, n, at pressure p. Make a list of the units you use, and keep track of the units for energy density and total energy in the cavity, and power radiated through the hole in the wall. Both problems are good for a) thinking about units and dimensionless ratios; b) for extracting dependencies on physical parameters c) for getting the answer correct by computing (or looking up) a definite integral. |
8. If you have access to Levi's book "Applied Quantum Mechanics" and to the MatLab language, study Levi's chapter 2. Then study his Matlab examples initially, and develop a graphical example which helps with your background knowledge for this course. We may eventually add the information to the course web pages; i.e. submit your answer as a web-page if you like the idea. But don't do any of this without consulting me first! |
9. Revise your knowledge of Fourier series and transforms by consulting the class webpages on Fourier transforms and writing out the Fourier series for one function, and the Fourier transform for another function, which have not been done in class, nor are currently on the web as a result of previous student efforts. If I'm happy with your result, then we'll add the transform to the list of results; i.e. submit your answer as a web-page if you feel so inspired. |
Latest version of this document: 16th January 2008.