Graduate Course: Quantum Physics

John Venables, Dept of Physics and Astronomy, Arizona State University, Tempe, Arizona 

Quantum Physics: Problem Set #5

This problem set is due Friday 2nd May at the beginning of the revision class. Three problems are needed for a complete score, but you may obtain extra credit by doing more. Comps questions are on a separate sheet, and you may submit any of these as single questions. If you are doing a project, you may substitute it for three questions on sets 4 and/or 5. This is the initial version of this document: 23rd March 2008.

1. You can now attempt any problem from Gasiorowicz G3 chapter 11, p185-7, or chapter 14, p233-5(G2 chapter 16, p277-8, or chapter 18, p307-8). Alternatives are any problems in Liboff, chapters 12 or 13. Try any problems from these chapters for credit, discussing with me first if you are not sure which to do.
 
2. Pertubation theory and matrix methods: On problem sheet #3 we discussed the energy level splittings in the (molecular) rotation problem caused when the moment of inertia I1 about the x-axis is almost equal to I2 about the y-axis, but is not equal to I3 about the z-axis. Revise this topic by writing short notes in your own words about the procedures used, and explore the separation into block-diagonal sub-matrices, and the relationship to (degenerate) perturbation theory. See how far you can get in applying the same methods to the case when the angular momentum quantum number l = 3.
 
3. Variational principle and the square well: A particle in one dimension is bound in an infinite well -a/2 < x < a/2, with a parabolic bottom (drawing supplied). The total energy V'(x), can be written as a sum of two terms, V'(x) = V(x) + v(x); here V(x) is the same well with a flat bottom, for which the Schroedinger equation can be solved exactly, and v(x) = Cx2 can be treated as a perturbation.

(a) Give the expressions for the eigenenergies and normalized eigenfunction for V(x).

(b) Calculate the lowest odd-parity state in V'(x), the approximate eigenenergy, and the first three terms in the eigenfunction in terms of solutions to V(x).

This problem was set in the Track 2 Basic QM exam in January 2000. 
 
4. Variational principle and the SHO (this problem is also on set#4): 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 lx4). 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.)
   
5. a) Two neutrons are trapped in a spherical potential V(r) = 0 if r < R and V(r) = + infinity if r > R. Ignoring the interactions between the neutrons, find the ground state wave function of system (including the spin wave function).

b) Assume now that the neutrons interact with each other through the potential Vi = As1. s2, where si is the Pauli spin matrix for the i-th neutron, and A is a constant. Treating Vi as a perturbation, find the approximate ground state energy.

This problem was set in the Track 2 Basic QM exam in August 1995.
 
6. Consider a two level system which obeys the Schroedinger equation:
where W1,W2 and V are real numbers.

1) Obtain expressions for the energy levels and normalized eigenstates of this Hamiltonian.

2) Assume now that W1 = W2 and that at t = 0 the system is intially in the state a = 1, b = 0. Find the time t it takes for the system to reach a (relative) maximum probability of being in the state a = 0, b = 1.

(Note: the above Schroedinger equation is an approximation to the following physical situation: two atoms can bind an electron, one with energy W1 and the other with energy W2. The electron initally on the left atom is represented by a = 1, b = 0; on the right atom by a = 0, b = 1. If the atoms are close together, the electron can hop between the atoms.)

This problem was set in the Track 2 Basic QM exam in August 1996. 
 
7. A system that has three unperturbed states can be represented by the perturbed Hamiltonian matrix , where F > E. The quantities a and b are to be regarded as perturbations that are of the same order, and are small compared with F-E.

(a) diagonalize the matrix to find the exact eigenvalues.
(b) Use second order non-degenerate perturbation theory to calculate the perturbed eigenvalues.
(c) Use second order degenerate perturbation theory to calculate the perturbed eigenvalues.
(d) Compare both results with the exact eigenvalues and discuss their relative reliability.

This problem was set in the Track 1 QM exam in January 2000. 
 
8. The absorption spectrum of HCl gas is the subject of this problem.

1) (30%) Explain the meaning of the term absorption spectrum of a gas, and explain how such a spectrum can be measured in the laboratory.

2) (70%) In HCl gas, an number of absorption lines have been observed with the following wavenumbers (in cm-1): 83.03, 103.73, 124.30, 145.03, 165.51, and 185.86. Are these vibrational or rotational transitions? (You may assume that transitions involve quantum numbers that change by only one unit.) Explain your reasoning briefly. 2a) If the transitions are vibrational, estimate the spring constant (in dyne/cm). 2b) If the transitions are rotational, estimate the separation between the H and Cl nuclei. What J values do they correspond to, and what is the moment of inertia of HCl (in gm-cm2)?

This problem was set in the Track 2 Basic QM exam in January 1996.
 
9. If the above questions do not give you enough choice, or too much challenge, feel free to do other problems from Gasiorowicz and/or Liboff. Some further Comps questions have been put onto a separate sheet, and you may submit any such questions as equivalent to single questions.