solution_set_2 - 22 January, 2007 Michael F. Brown...

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Unformatted text preview: 22 January, 2007 Michael F. Brown CHEMISTRY 481 (Biophysical Chemistry) Problem Set 02 To be turned in by: Monday, 29 January Worked examples for this course are dealt with primarily in the Discussion Section; whereas new concepts are introduced in the Lectures. You can get help with the homework problems in the Discussion Sections (12:30—13:20 Tuesday, Koffler 216; and 13:00—13:50 Wednesday, Koffler 216). The problem sets will be graded P+, P, or F and will be used to increase or decrease borderline grades. On all computational problems be sure to use SI units, and indicate your answer to the proper number of significant figures. For maximum credit show clearly how you obtained your answer, i.e. what numbers were combined to yield the final result. Background reading: Silbey, Chapter 9 Back of Chapter Problems related to the homework (optional): Problems 9.3—9.5, 9.7—9.8, 9.13, 9.14 Problem 1. The wavefunction corresponding to a hydrogenic 1s orbital is given by l —r/ao 1P(r)=1lE€ where the Bohr radius a0 = 52.9 pm. (Hint: the above wavefunction is not normalized, if you cannot normalize it then proceed to parts (b)—(e) without the normalization constant.) a) Find the normalized wave function. b) Calculate the expectation value of the radius, given by <r> . 0) Calculate the expectation value of the radius squared, given by (r2). d) Calculate the variance of the radius, defined by (72) — (r)2 . e) Calculate the root mean square radius, defined by (rz>"2. Problem 2. The wavefunction corresponding to a hydrogenic 2py orbital is given by z,0(r,6;p)=1li LsinQ cos¢ e'rlz"0 821 (10 where the Bohr radius a0 = 52.9 pm. (Hint: the above wavefunction is not normalized, if you cannot normalize it then proceed to parts (b)—(e) without the normalization constant.) a) Find the normalized wave function. b) Calculate the expectation value of the radius, given by <r> . c) Calculate the expectation value of the radius squared, given by (r2). d) Calculate the variance of the radius, defined by (r2) — (r)2 . e) Calculate the root mean square radius, defined by W)”. Problem 3. You have just landed ajob at “Desert Biotechnology”, a new start—up company in Tucson investigating the electronic properties of chlorophyll in drought tolerant plants. You decide it is cool to investigate the following wave functions: em" , sin kx, cos kx, sinh kx, and cosh kx. For each wavefunction calculate the energy of a free particle (potential V = 0). [Hintz sinh [or E (ek’r — e‘k" ) / 2 and cosh kx E (6“ + e‘” ) / 2.] Problem 4. Consider a classical harmonic oscillator having a force constant k, mass m, and amplitude A. Calculate the total energy E of the harmonic oscillator: Problem 5. One of the most important tools of modern molecular biology is the electron microscope. Electrons are accelerated in a an electrical potential of Ari) = 105 V and are focused on a biological specimen. Assume that the kinetic energy of an electron is given by eA¢, where e is the charge on an electron. What is the smallest possible object that can be resolved assuming that the theoretical resolution is given by the de Broglie wavelength? 01 —22—07 chem48 l\07\prob\002\mw 100 I C M— _, v; H J W “:2: ::m/, w “ c9 _ W /V = C" 4" r"“":z ” / .5 V «2 .,‘f I r -Ay“ 1:10 ’ I A I ,. H i“? d!" f 5w; (:9 7’ J i ‘ 0 l. a ,,_,) V-.. W; 3/ 6‘77" ( a) E>LVQ€ C—f'fah' 3‘.- k/UZ /U.£ of F 3‘ I g 1 . A A A - < N) r” 1”” 7MP) ra‘dr Sm 59(in d? i 1 F E, g 9: 8 [(30/an é) *66 (7'3 $.71“ afiJirf/eg'rx S, .71 3‘ ‘\ 1/ .0 its” M g ’/ ( 340 > 9* <r“ Wu; pm ] .MW*_~_ __. _. .. a.” /> - ~ w My“ ér \. .2 __w I ‘1‘ L )e 13,2) v- “V” saw» ——_.. 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This homework help was uploaded on 04/02/2008 for the course CHEM 481 taught by Professor Brown during the Spring '06 term at University of Arizona- Tucson.

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solution_set_2 - 22 January, 2007 Michael F. Brown...

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