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455exam_answers_Exam_1 - Chemistry 455 A Spring 2008...

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Unformatted text preview: Chemistry 455 A Spring 2008 Reinhardt and Stanich Hour exam #1, 50 minutes in class. 100 points. No Calculators or electronic devices of any type may be used on or during this exam! Each student may have one (1) hand-written 5 by 8 inch note card. 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! No work = no credit! (& we mean that...). Show all work on the exam paper....it you need to use “the back of a page” this must be indicated on the front of the same page.... BUT, when explanations are required, often we are looking for just “a few words.” So be as brie as ssible & don’t start writing paragraphs! L/ \< a . Name: W | UW m#: * :3 fl. ' Signature: (Your signature implies that all work shown is your own, and that you have no accessed any information beyond than hand written on your 5 x 8” card, and have received help from no one but your TA or Professsor.) Points Problem #1 Problem #2 Problem #3 Total Cum“, on LQS‘F Problem #1 (40 points) Particle on a RING of circumference “a”, ( a = 2m). Consider the “first excited state, or n = 1” wavefunction for a particle of mass m moving freely on a ring of circumference “a”. 0<= x <= 3 is a measure of distance around the ring. Assuming that the normalized wave-function is: 1p(x) = i/(Z/a) cos(291:X/a) a) (5 points) Sketch the wave-function over the range 0 < x < a. You may imagine that this may be drawn along a straight line, with points “0” and “a” representing the same point. OR you may draw the wave- function around the circle. Your choice! W x ”9’0” 7) v I x ’ f R a \ b) (5 points) Sketch the probability density corresponding to this state over the range 0 < x < a. Use the same graphical representation as in part “a)”, above. . " n DEM? a x «u c) (10 points) From your sketch, above, deduce, and describe in words your thinking, i) What is <x> on the interval from 0 to 3? Why? (be brief!) a R“ . k g 1 ‘97‘7 “4W7 7‘\ K’ZQP’M’V «4 mew/h”. ii) What is the probability of observing the particle at or near this average value? Why? (be brief!) OLC [hrDL 0(1 WEUKV is j I v Q ' \( QA 04" “d fContinued on next page ...... “(3%" iii) What two boundary conditions should the stationary state wave-function satisfy at the point 0 = 3? Just state them! 94¢ch W6.) K7000] 3%9/1/ iv) In your sketch of part “a)”, above are these conditions met? How can you tell? Be briefl ye) A’LL bt/Lfil/Lthx Vl/vfixV «4‘5“ mm/k WLQJLV JMJOWLX 74%“ 560W}. 44““. WLpVI’LL‘ v) 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 11 = 1 energy eigenstate, sketch a “histogram” of the results you might expect. Use the same graphical representation as you used in in part a. What “interpretaion of the waw—fimction are you 72(usingjglzél< make this histogram? «if 91» X d) (10 points) Is the above function is an eigenfimcgion of KEX°p ? Clearly indicate the eigenvalue in terms of m, a, h, and other appropriate constants. Show all work... L {5‘ Are’ (“f/(fix -_: 2km» (“)(7E‘X) I/g) p m if“? 2: 4V1 CO) (’2‘ / ‘2 Eigenvalue= k KZWIA’D e) (10 points) Is the above fimction an engenfuction of Px “P ? Clearly indicate the eigenvalue In terms of m, a, h, and other appropriate constants. Show all work" /\ Ark] 1" 5(7Trg):2 TA'L /L 3M( (”NIX igenvalue= “7%L ‘ ~/l/L/l/ M Qr2wl" %M4Jtu«/ Ra} ”W £[3QWaC/Lvip‘rl: “ Problem #2. 35 points. Operators and commutation relations. Consider the operatrors PX"p =' -ih d/dx; and X0], = x. a) 7 points: what is the result of applying these operators to a difi'erentiable and normalized wave- fucntion, f(x), in the order: Px°pX°pf(x). Show all work. ”1112 éy(x)¢m) 2: «Via A“) Ail/T XABILY/ b) 7 points: what is the result of applying these operators to a difl‘erentiable and normalized wave- fucntion, f(x), in the order: X°PPX°P(X). Show all work. Marta W C as}??? c) 7 points: What, then, follows from applying the operator [Px°p,X°P] (aka the commutator) to this same function f(x)? £?°’J><"léw: W? W »— xy/it’taw 3/: 2 “5+4 3““) ”HM/y“) - (Maps/u) J -: - 4 ‘3 %— C7~ d) 7 points: What then is the “operator value’ of the! commutator [Px°",X°"]? (Hint: remove the fix) fiom part c) above!) jaw]: jig-I m ”Yul/HQ) e) 7 points. Suppose operator B is for YOU(!) to take two steps forward in the direction you are facing, and operator A is to take two steps forward in the direction you are facing and turn around 180 degrees. What is the “value of the commutator” [A,B]? Drawing a graph may help. (Hint, if the motions described by A, B above are though of as operations, what operation corresponds to [A,B] thought of as a combination of the motions AB and BA?) If this is not evident in a few seconds, move on! Use the back of the page for discussion, if needed! But don’t get stuck on this problem: move on if you don’t see what to do! xg‘ifi {AB 33Afl ”(KA /]Lh (I hath/Q” 9A M w, ‘ $71)“; /%M\ I" ‘1 ”£ng :YWavfli‘ Problem #3 (25 points). Experimental foundations of Quantum Theory (this is problem 1-9 from McQ, on list of your suggested review problems, and is also Worked Example 1-2, p 7 of McQ, one of which was promised to be an exam problem.) At last you get to “do an integral” although likely not the one you expected... Integrating the Planck distribution over all frequencies, v, gives the total radiation density as a function of absolute temperature T. The result is T to a power, i.e TN. You problem is to determine the numerical value of this “power N”. ~ 8 h 00 Ema]: 73-]; -—————(e(_k,%’_’;_ 1) aZv. Given the integral: 1: (eff 1) (ix: g make a suitable change of variables determine the power of T which determines the total equilib- rium radiation density of a body at temperature T. (Hint, try, for example, x = hvlkT, and show all work!) , ' " The exponent of T is , a result called the Stephan-Boltzmann Law . ‘ HM ...
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