CHEM 342. Spring 2002.  Next PS   Previous PS  Answers  PS#2.doc (Word 97)     Problem Set #2   Note: Relevant Chapters in Mortimer are Chapters 14 and 15. Orthogonality 1. Prove that the functions and are orthogonal . Hint: 2. Give a mathematical definition for the Kronnecker delta . What is the numerical value of the Kronnecker delta when the two eigenfunctions are orthogonal? What is the numerical value of the Kronnecker delta when n and m are the same eigenfunction (i.e. n = m)? In addition to these two values, can the Kronnecker delta be equal to any other numerical values?   Operators 3. Find the result of operating with and on the function . Is f(y) an eigenfunction of or of ? 4. Find the following commutators for any function f(x). (a) (b) Hint: , , , and . 5. Find the result of operating with the operator on the function . What values must the constants have for to be an eigenfunction of ? 6. Find the result of operating with the operator on the function . Is it an eigenfunction? 7. The function is a well-behaved wave function in the interval . Calculate the normalization constant (A), and the average value of a series of measurements of x (i.e find the expectation value: ). Expectation Values 8. For the wave function and the operator , give an expression that could be used to calculate the average value obtained from repeated measurements (i.e. show an expression for ).   Particle In a Box 9. Calculate the value of A so that is normalized in the region . Hint: 10. For a particle in a one-dimensional box , we used eigenfunctions of the form . Explain why we could not use 11. The ground-state wave function for a particle confined to a one-dimensional box of length L is The box is 10.0 nm long. Calculate the probability that the particle is between 4.95 nm and 5.05 nm. Hint: 12. What is the ground state energy (i.e. n = 1) for an electron that is confined to a box which is 0.2 nm wide. [Hint: Planck's constant, h,is J s; the mass of an electron, me, is kg]   Uncertainty 13. The speed of a certain proton is 4.5 ´ 105 m/s along the x-axis. If the uncertainty in its momentum along the x-axis is 0.010 %, what is the maximum uncertainty in its location along the x-axis (i.e. )?   Tunneling 14. The wave function inside an infinitely long barrier of height V is . Calculate (a) the probability that the particle is inside the barrier; and (b) the average penetration depth of the particle into the barrier (i.e. the expectation value ). Because the barrier is infinitely long, this wave function is valid for . Hint: . PLEASE NOTE The work you hand in should be neat and well organized, and it should show the strategy and steps you used in solving the problems, as well as the bottom-line answers (or solutions). In grading the problems, both your work-up and your final answers/solutions will be examined and evaluated. The work handed in for grading must carry a pledge that the work is entirely yours and was done without any collaboration with other persons (except for the course instructor and TA's). You are encouraged to work with others in doing the exercises and problems found in the textbook, but all work handed in for grading should be done independently.