PHY4604 Problem Set 3: Quantum Mechanics - Dumbell, de Broglie Wavelength, Maple, Assignments of Physics

Download PHY4604 Problem Set 3: Quantum Mechanics - Dumbell, de Broglie Wavelength, Maple and more Assignments Physics in PDF only on Docsity! PHY4604–Introduction to Quantum Mechanics Fall 2004 Problem Set 3 Sept. 13, 2004 Due: Sept. 22, 2004 Reading: “Otherevidence” notes, Peebles Ch. 1 1. Quantum Dumbell. A pair of masses m are held a distance d apart by a massless rigid rod, as shown. 1a) Quantize this system using the Bohr-Ehrenfest quantization condition for the angular momentum, and find the form of the energy levels En. 1b) We used the rigid dumbell in our discussion of the specific heat of Hydrogen gas, H2. Compare your answer for n = 1 in part 1a) with the energy of the lowest rotational state of the Hydrogen molecule deduced from specific heat measurements (or alternatively the temperature at which the rotational states begin contributing to the specific heat), taking m ' mp, the mass of the proton, and d ' 1A. [Hint–compare with Bohr’s derivation for the H-atom.] 1c) Why does this calculation work for the H2 atom (in an order of magnitude sense!) even though we neglected the potential energy due to the Coulomb forces (technically, van der Waals forces) holding the molecule together? 2. de Broglie wavelength. Estimate the de Broglie wavelengths of (a) a thermal neutron; assume the neutron is at high temperatures in thermal equilibrium such that its kinetic energy is about 3kBT/2, where T = 300K is room temperature, and the neutron mass can be taken to be the same as that of a proton, (b) an electron with energy E=1 GeV=109 eV, where the momentum in this case is related to the energy by the relativistic expression E2 = p2c2+m2c4. 1