CHEMISTRY 333 – Fall 2006 Exam 1, take home (due Oct 14, Sat, 5 pm) Name __________________ INSTRUCTIONS TIME LIMIT & DUE DATE. As announced in class, you may work on this as long as you like up until the due date. The exam, along with any extra pages securely stapled (see below), is due in my mailbox on Oct 14, Sat, 5 pm. RESOURCES. All work must be done independently. You may not ask for or receive intellectual help from other students, faculty, family, clergy, or higher powers. You may not consult any print or online resources with the following exceptions: your textbook, your lecture notes, and the homework solutions posted on the web. SHOW YOUR WORK. You may use Mathematica and the mathematical formulas in your textbook (DO YOU KNOW THE DOOR CODE FOR THE COMPUTER LAB?) However, you should try to write your answers in a way that shows in as much detail as possible, the mathematical ideas you began with and how you worked with them, and not just the final answers you obtained. Also, include units in your answers, where appropriate. EXTRA PAGES. Your work might very well run on to extra pages. If it does, please write your name and the problem number on each page, and indicate the sequence that the pages should be read for each problem. Rule of the Grader – Given any answer (right or wrong), the more I understand about the thought that the produced the answer, the more generous I can be towards the answer. I am grading process, not just a result.2Engel 5.1 (p. 77-8) describes how to obtain total energy eigenfunctions and eigenvalues for an electron in a finite depth box of width 1.00 x 10-9 m with Vo = 1.2 x 10-18 J. Engel also lists the five lowest eigenvalues for this box. 1. What are the five lowest eigenvalues (in J) for an electron in an infinite depth box of the same width? 2. Which system has lower eigenvalues, the finite depth box or the infinite depth box? Why?33. Use the formulas provided in Engel, and any graphing tools at your disposal, to determine the lowest eigenvalue (1% accuracy in E is adequate) for an electron confined to a finite depth box of the same width, but with Vo = 2.4 x 10-18 J. NOTES: i. Do not draw your graphs on this sheet (I don’t need to see them), but do write the formulas that you graphed (with numerical quantities inserted wherever appropriate) in order to find the eigenvalue. ii. It may be possible to obtain the desired value using Mathematica’s Solve function. I strongly prefer that you obtain your answer using Engel’s graphical method. If you use Solve (or something else non-graphical), please be a good sport and let me know what you used and how you used it.4Referring to Engel’s original finite depth box (Vo = 1.2 x 10-18 J, a = 1 x 10-9 m) and the general formulas that he provides for the eigenfunctions, 4a. What are the normalized eigenfunctions for E = 4.61 x 10-20 J and 4.09 x 10-19 J? NOTE: Please provide an algebraic expression (with numerical quantities inserted wherever appropriate) for each eigenfunction in Regions I, II, and III (see Engel for region definitions).54b. What are the probabilities of finding an electron in Regions I, II, and III, respectively, when the electron’s energy E = 4.61 x 10-20 J and 4.09 x 10-19 J, respectively? NOTE: I am looking for six probabilities in all.64c. Use Mathematica to plot the normalized Region III eigenfunctions from problem 4a. NOTE: Plot both eigenfunctions on a single graph so that they can be compared