1. Degeneracy- solid state application: Calculate and draw the
energy spectrum (density of states) for free electrons in: a) a rectangular box with sides (a,L,L) with a = 1nm < L = 1 micron. This is a first approximation to a quantum well. b) a rectangular box with sides (a,a,L) with a = 1nm < L = 1 micron; a quantum wire. c) a rectangular box with sides (a,a,a) with a = 1nm; a quantum dot. In order to think about this problem it may be good to consult Liboff, section 8.8, (L4 p336-342 or L3 p349-355), and also to look at the notes for lecture 2 and lecture 3 of my Sussex QMMS course, where I also set this question. |
2. Degeneracy- astrophysical application: Do problems 1 and 8 from G2 p167, as one linked problem (This is G3 problem 12, p212, plus a problem on a neutron star, ask for a copy). Write notes, and provide reality checks, on the validity of this problem using astrophysics textbooks as appropriate, for the education of fellow students, including myself. |
3. Degeneracy- nuclear application: Do G3 problem 17, p 214 (G2, problem 7, p167). This problem mentions N = 126 and Z = 82, which are 'magic' nuclei, as indicated in G2 p181-2, where it is explained that the simplest model doesn't work in detail. Find out about nuclear interaction potentials and explain how the discrepancy is resolved. (This problem requires access to a Nuclear Physics textbook, so please ask for suggestions and handouts). |
4. Consider an electron bound to an infinitely massive, spinless proton,
forming a hydrogen atom. The electron is in the n = 2 state, with an energy
E = - R/4, where R = 13.6 eV. a) (30%) When the electron spin is included, the n = 2 manifold of states is 8-fold degenerate. Why? Enumerate the states. b) (70%) To first order in l (which is a small positive constant), determine the perturbed energies of the 8 states in the n = 2 manifold, under the perturbation V = lL.S, where L is the orbital angular momentum operator of the electron and S is the electron's spin operator. |
This problem was set in the Track 2 Basic QM exam in August 1996. |
5. Consider a spin 1/2 free particle moving along the
x-axis with a projection of (1/2)h/2p
along the z-axis. The particle originates at large negative x and moves
to the right. At x= 0, this particle experiences a spin-dependent
force of the form V(x,s)
= -Ad(x)
[s_{+}
+ s_{-}],
where s_{+}
and s_{-}
are the raising and lowering operators for spin, and A > 0. The particle
has mass m, energy E, such that k^{2} =
8p^{2}mE/h^{2},
and the incident particle has unit amplitude.
What is the probability of finding a particle with projection -(1/2)h/2p (along the z-axis) moving to the left (along the -x axis)? |
Note: the above situation is an approximation to the use of a thin ferromagnetic foil as a switch for ultra-cold neutrons, based on polarizing the neutron spin initially, and being able to reverse it using the internal field of a ferromagnet. Technically it is an example of the use of product wavefunctions, in this case as a function of space and spin. This problem was on the (Track 2) Basic Quantum Mechanics exam in January 1999. |
6. Get some more practice with energies involving S.S and/or L.S by doing G2, chapter 15, problems 4 and 7 together (p264-5). An alternative from Liboff is chapter 12, problems 12.29 and 12.30 (L4, p622 or L3, p640). Don't try all 4 problems for credit- enough is enough! |
7. Consider the handout of the periodic table, and the elements Z = 1 to 11
(G3, p227 or G2, p319 or equivalent). a) Write a small computer program to calculate Z_{eff} from the Ionization Energy, I, for all these elements, using the Hydrogen-like expression I = 13.6Z_{eff}^{2}/n^{2} (eV). Comment on any patterns that you see in the results, being clear about what value of n you are using. b) For Z = 11, what would you expect the L, S and J values to be for the first excited state? What are the possible values of these quantum numbers? Estimate the excitation energy, using estimates for the centrifugal barrier, and Z_{eff} from part a) above. c) Using the 'Terms' given for elements 5-10, show that the J-values are consistent with Hund's rules regarding the combination of L and S needed to form J. See what you can find out about the physical origin of L.S coupling. |
8. Compare the accuracy of the variational principle with first order perturbation theory, in the case of a 1D simple harmonic oscillator with quartic anharmonicity (i.e. a perturbation of the form lx^{4}). Use the operator formalism, so that you do not need to work out integrals. How does your approach to this problem change if the restoring frequency of the SHO tends to zero. Is there an exact solution in this latter case, and can you derive it? (In case this problem sounds daunting, it is basically G3 problem 6 p186 (G2 problem 3, p277), coupled with G3 problem 11 p235 (G2, problem 11, p307.) |
9. If the above questions do not give you enough choice or challenge, feel free to do other problems from Gasiorowicz (G3 chapters 8, 10, or 12, G2 chapters 10, 12 or 15) and/or Liboff, chapters 10 and 12. Some further Comps questions have been put onto a separate sheet, and you may submit any of these as equivalent to single questions. All Comps questions since 1998 are available on the department website (ask for password). |