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This 2 page Class Notes was uploaded by Carmela Kilback on Wednesday September 9, 2015. The Class Notes belongs to CHEM 475 at University of Washington taught by Staff in Fall. Since its upload, it has received 50 views. For similar materials see /class/192611/chem-475-university-of-washington in Chemistry at University of Washington.
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Date Created: 09/09/15
Chemistry 475 Review for Exam 1 Autumn 2002 Reinhardt Heatwole Caplow First rework all homework problem Then work the exam below The problems have been taken from actual prior Chem 475 and 455 exams Do expect at least one homework problem to appear on you exam Note that all answers must be supported by work clearly shown or a an explicit explanation in words of how the result was arrived at Possiblyusefulformulae A gfli39JvA A39D 1 Kg muq 7f 739L 39 217 J 7314 ZFz 7 C quotrquot lzr gtX FPlt Fri9L4 u nned aka A el 1m mgr 1 I ffw39 40d7z S g zgt 47 Wntgrgfx Problem 1 Particle in a Box of length a Consider the second excited state or n 3 wavefunction for a particle in a box of length a Note that this function has 2 nodes In this state the normalized wavefunction is wx IQa sin3nxa a Sketch the wavefunction over the range 0lt x lt a What is it s value outside of this range b Sketch the probability density corresponding to this state over the range 0 lt x lt a c From your sketch above deduce and describe in words your thinking i what is ltxgt397 ii what is the probalibty that on making a single measurement the particle will be found in the interval 0 lt x lt aZ39 iii what is the probability that on making a single measurement that the particle will be found in the interval 2a3 lt x lt a iv If you were to make a large number of measurements of the position of the particle where each time the system was initially prepared in the above n 3 energy eigenstate sketch a histogram of the results you might expect v Give a simple argument that we would expect ltpop 0 in this state vi Using the inde nite integral on the front of the exam show that this wave mction is normalized as claimed Problem 2 Operators and eigen mctionseigenvalues i 3 xii a Consider the function lX 3 e p where p IS a posmve real number i is this wx an eigenfunction of KEop 7 If yes what is the eigenvalue Show all work ii is this WX an eigenfunction of pxquot the momentum operator for motion in the x direction If yes what is its eigenvalue Show all work iii is this WX an eigenfunction of xup the operator for the position in the x direction If yes what is its eigenvalue Show all work iv sketch the probability density for this function and comment on where you would best look to have maximal probability of nding the particle on a making single measurement of its position x vWhat is the wavelength of this wave function vi In what direction is the particle moving Towards x or x Problem 3 A new operator The action of R the refection operator on any function of x is to change the sign of x Thus R x fx for any suitable wave function fx A Show that R is a linear operator That is show that R afx a Rfx and that R fx gx for any functions f and g and any possibly complex constant a B Is x3 an eigenfunction of R show work if yes what is the eigenvalue C is sinx an eigenfunction of R show work if yes what is the eigenvalue D is x4 x an eigenfunction of R show work if yes what is the eigenvalue Problem 4 Another new operator Consider the operator D133 differentiate plus 3 Dp3 acts on functions as follows Dp3 fx dfxdx 3 A what is Dp3 sinocx B What is Dp3 x3eXpctx C Is Dp3 a linear operator a Is it the case that Dp3a x a Dp3 x for a constant a Show work b Is it the case that Dp3 fx gx Dp3 x Dp3fx for any two functions fg Show work 0 Is Dp3 a linear operator Problem 5 The Bohr atom This problem is from your homework The energy of an electron in the 11 Bohr orbit for a nucleus of atomic number Z is E z2 m e4 8 a h2 n2 We are approximating u by m electron mass Using the formula above derive the general Rydberg Balrner formula 17 ZZR1n2 1m2 and arrive at an explicit expression for the Rydberg constant R in terms of the fundamental constants hecm1tx0 State in words how and why you have used the BohrEinstein frequency relation in your derivation Hint it might be helpful to draw an energy level diagram Problem 6 More on the Bohr atom amp the particle on a ring a Consider a de Broglie standing wave with an integral number of wavelengths around the ring Derive the Bohr quantization rule I va nil where R is the radius of an orbit Interpret the requirement that we have a whole number of wavelengths around the ring in terms of your knowledge that a Schrodinger wave function must be single valued c In Bohr s picture he assumed that n l234How does this differ om the quantum picture of a particle on a ring of radius R and circumference 211R In particular what sort of deBroglie wave would correspond to n 0 And why didn t Bohr consider both positive and negative n Would this make a difference in the Bohr formula for the energy levels of an electron in the H atom Would it affect their degeneracy How so Is this correct from your prior lmowledge of the quantum states of the H atom d Can you derive the correct formula for the quantum energy levels for a particle on a ring by simply tting deBroglie waves around the ring If yes do so and show all work b v
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