Graduate Course: Quantum Physics

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

Quantum Physics: Problem Set #2, 2008

This problem set is due Monday 25th February 2008 at the beginning of class

Five problems are needed for a complete score, but you can buy 'credit' for future problem sets by doing more. I want everyone to do questions 2 and 3. If in doubt about whether you can do a particular problem, or do not have access to a suitable computer, please consult me in good time. Note: this page needs the symbol and MT Extra fonts enabled on your browser. Latest version of this document: 9th Feb 2008.
1. Do problem * from Gasiorowicz or Liboff, chapter *, to be decided by you, and I will grade it for credit if I think it is suitable. There are several questions which are interesting, different, or good for revision in addition or instead of those given below. Please consult with me if you want to do this.
2. Consider the square well, with well depth V0, and width 2a, symmetrically placed about x = 0. Discuss this problem from two viewpoints, and use this question to improve your notes on this topic.

First, measure the energy of a bound state from the bottom of the well, and then consider this as a problem for 'free particles' in the momentum representation. Show that a) [p,H] = 0, and that the expectation value of the momentum p = ± hk; b) outside the well, when E < V0, k is imaginary, so that u(x) = exp(±kx), with k = ik, in the x- or "real space" representation. In addition, give the reasons why we need to choose just one of the solutions, not a linear combination of the two, outside the well.

Second, consider the more usual starting point for describing a deep well, with V = 0 outside the well, and -V0 inside the well. Show that, if E < 0: c) inside the well, u(x) = Asin(kx) or Bcos(kx), but that it cannot be Asin(kx) + Bcos(kx), with A and B non-zero; d) Derive the equations needed to determine k and (A or B) by considering the boundary conditions on the wave function at either x = a or -a; e) Work out the k-value and the energy of the second odd-parity excited state if V0 is very large.

Make sure you know what the terms in italics mean before starting this problem. A very similar problem was set, including a diagram and a hint, in the Basic Quantum Mechanics exam in January 1994. Note: this year, there are several of you for whom the above problem may be too easy. If that is the case, do the double well problem, as set out in G3, p 92/93 problems 11 and 13, or G2, problems 16/17 on p 112/113.
3. Set up problem 2 above on a computer, with  a = 0.15 nm, V0 = 250 volts. Using the CUPS simulation program or otherwise, find the energy E of a) the second odd-parity excited bound state and compare your answer with the analytical result you obtained in problem 2; b) the lowest energy unbound state which minimizes the probability of a particle reflecting back into the region x < -a, printing out the transmission and reflection coefficients over a suitable energy range.
4. A particle of mass m is subject to an attractive double-delta potential V(x) = -V0[d(x-a) + d(x+a)], where V0 >0. Consider only energies E < 0.

a) Obtain the wave function of the bound states. You do not need to normalize the function.
b) Derive equations from which the energy eigenvalues can be obtained.
c) Estimate the ground state energy for the limits a → 0 and a → ∞. You may assume that the wavefunction is even.

This double well problem was set on the Quantum Mechanics exam in January 2005.
5. Do one of problems 8, 9 or 10 from G3 chapter 3, page 63-64 (G2, chapter 4, problems 5,6 and 8). Do the problem which stretches you, not the one that you know you can do in advance, and if necessary come for advice before starting the problem.
6. Do problem 11 from G3, chapter 3, p64 (G2, chapter 4, problem 7 p71), becoming familiar en route with dealing with the expansion postulate and integrals involving orthonormal functions.
Note: this question encourages you, as I do, to think of complicated trigonometric functions as complex exponentials.
7. A particle of mass m and energy E is incident from the right (i.e. from r>R towards r=0) on a potential given by V(r), see attached paper form and diagram. Find the intensity ratio of the wavefunction inside the potential region to that of the reflected wave.
This reflection problem was set on the Quantum Mechanics exam in August 2004.