Download 6 Problems on Quantum Mechanics of Physics - Homework 5 | PHYS 560 and more Assignments Quantum Physics in PDF only on Docsity! Homework Five Ch 5: due Monday October 8, 2007 PHYS 560 1. (Shankar 5.2.2) 2. (From Shankar 5.2.3) Consider V (x) = −aV0δ(x). Show that is admits (allows) a bound state of energy E = −ma2V 20 /2/hbar2. Are there any other bound states. (Use hint in Shankar, and be very careful about how to handle continuity or discontinuities). 3. Consider an electron in a ‘quantum well with width L = 1 nm and depth Vo = 3 eV. (Requires the results of Exercise 5.2.6, i.e., particle in a well. These solutions are worked out completely in many quantum texts. So this is a sneaky way to get students to go carefully through them). (a) How many bound states are there? Do not forget even and odd bound states. (b) What are the energies, relative to the bottom of the well, of the bound states? (c) Suppose that a photon can be emitted when an electron makes a transition from either odd to even bound states, or from an even to odd bound states. What energies of photons can be emitted from this quantum well? (d) What are the conditions for this system to have only one bound state? (give nu- merical quantities related to physical parameters of well, with units). For Extra credit (or at least amusement) discuss of what these parameters would mean to constructing a quantum well (a real one out of real materials) that had only one bound state. 4. (based on Shankar 5.2.5) Consider a particle of mass m in the state |n〉 of a box of length L. Now, sudden expansion of box Asymmetrically from total size L to 2L without disturbing the wave function. For example, the wall at −L/2 stayed in place, but the right hand wall is now at +3L/2 (The point here is that since it is not done slowly, it is equivalent to having a particle prepared in wave function 1, now put in Hamiltonian appropriate for box of size 2L but shifted over. What is the probability of finding the particle in the ground state of the new box? 5. (Shankar 5.3.1 6. (Shankar 5.4.3) Consider a particle subject to a constant force f in one dimension. Solve for the propagaor in momentum space U(p, t; p′, 0). (answer given in Shankar). Transform back to coordinate space and obtain U(x, t, x′, 0). 1 Appendix: normalizing the ψ̃(p) of Shankar 5.4.3 Normalizing the wavefunction, ψ̃E(p) = A exp [ i ~ (p3 − 6mEp 6fm )] requires a bit more thought than the usual blindly plugging it into the equationA2 ∫ |ψ(p)|2dp = 1. Please please please, Review the section on Operators in infinite dimensions in Chapter 1, in particular, consider the comments regarding the normalization of |k〉 eigenkets. This brings revision printed: October 1, 2007 Professor Thompson cthompson@niu.edu
