Download Quantum Mechanics Homework Problems - Prof. David Ginger Jr and more Assignments Quantum Chemistry in PDF only on Docsity! Homework 3. Due Monday Oct 17 at 5pm in Prof. Ginger’s mailbox. Page 1/4 CIRCLE YOUR ANSWERS AND KEY INTERMEDIATE RESULTS USE MAPLE WHENEVER POSSIBLE STAPLE YOUR PAPERS TOGETHER INCLUDE ALL COMPUTER PRINTOUTS (with commentary) Levine Problems 3.31 – Probabilities for particles in a 3D box 3.36 – Degeneracy and 3D box levels 4.29 Morse Potential – also: c) Given that De = 7.31E-19 J/molecule, and a=1.82E-9 m-1 for HCl, calculate the force constant k for the HCl bond. Plot the Morse Potential for HCl and the corresponding Harmonic oscillator potential on the same graph. d) for what displacements from equilibrium is the harmonic oscillator (HO) potential a good approximation (say within 90%) of the Morse potential? e) For what physical situations would the HO potential yield a poor approximation to the actual potential? Additional Problems 1) Consider a particle in an infinitely deep two-dimensional square well with sides of length L. a) What are the five lowest allowed energy levels for this system. b) Use the computer to make 3D plots that display the probability of finding the particle as a function of position within the box. c) Compare the wave functions associated with any one pair of degenerate states. 2) Tunneling is important to many chemical processes, for instance, electrons tunnel during many redox reactions and protons can tunnel during acid/base reactions. On a relative scale, how important do you think carbon atom tunneling is to organic reactions, on the whole? Give a qualitative justification of your answer in writing. Next, justify your answer quantitatively by comparing the relative tunneling rates for an electron, proton, and carbon atom across a rectangular potential barrier 1 Angstrom wide, with height of 1 eV. Assume that the kinetic energy of each particle is 0.5 eV. Ultimately the relative importance of different processes depends on the rate of competing processes and it is true that for some reactions (the automerization of 1,3-cyclobutadiene, the ring expansion of 1-methylcyclobutylefluorcarbene) carbon tunneling plays a central role. 3) In lecture we derived the density of states for particle in a 3D box as: 2/1 2/3 22 2 2 )( E mV ED = hπ , starting with an approximation for the number of states with nx2 + ny2 +nz2 ≤ n2 as a function of n. Test this starting approximation by writing a program or routine in Maple (or C or Fortran or Basic or Labview or whatever) to calculate the ACTUAL number of levels below the energy corresponding to a given n
