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2+3+preview8 - 51:95 "’ Problem Set #2 (Chem 113A...

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Unformatted text preview: 51:95 "’ Problem Set #2 (Chem 113A W’07, due Thu. 01/25/07 at 11:05am, 100+5pts, max=100pts) Name: lzslgc Shiv. : 4.} Forming study groups is permissible, but you must construct your solutions independently. By writing down my name, I confirm that l strictly obey the academic ethic code when doing this problem set. Please write down the names of everyone who you worked with on this problem set under your name. (i) Seven questions, one for each day. Studies show that the learning process can be more effective and enjoyable if one starts reviewing the material soon after class and does the exercises on a daily basis. (ii) One extra credit question. But the total score of this problem set is capped by 100 points. (iii) Please write out algebraic details and arguments. If you use math formulas or literature values, please cite your source(s). (iv) Please include these three pages as the cover sheet for your solutions. (v) if you have difficulties/questions, please find resources and assistance ASAP. (vi) Academic ethics need to be strictly obeyed. No exceptions! Please refer to the syllabus. A. Basic Material (Textbook) [1] Preview #05 (due 01/23/06 Mon.) [2] Preview #06 (due 01/25/06 Wed.) [3] Preview #07 (due 01/26/06 Thu.) mann distribution 15 ts “1 Assume that a system has a very large number of energy levels given by the formula 8 = so]? with 39 = 2.34 x 10"”.1 , where 3. takes on the integral, values 1,2, 3, . . . .Assume further that the degeneracy of a level is given by 31 = 21‘. Calculate the ratios :15 fit, and nu} /n1 for T == 100 K and T = 630 K, respectively. [5| Mathematical properties of quantum wavefunctions (30 pts) (a) To sati the statistical interpretation of the quantum wavefunction (Born Interpretation), quantum wavefu ions need to be normalized. Some simple exercises. 4 in nonnalizing wave functions, the integration is over space in which the wave function is defined. The flowing examples allow you to practice your skills in two~ rec-dimensional integration. "ormalize the wave function sin(’mrx/a)sin(z:my/a) over )8 range 0 S x S a. 0 S y S b. The element ei’ area in two- imcnsional Caitesian coordinates is (12: dy; n and m are ategers and a and l: are constants. ’ormalize the wave function ef"5“3 cos 6 sin d) ever the ntcrval O S r «c: «a, 0 S 9 5 7r, 0 Sc!) S Bar. The voiumc iement in three-dimensional spherical Coordinates is ‘ 3 sin (3dr d6 dd). The eigenfunctions of a Hermitian operator must be orthogonal to each other. Some simple exercises. .25 Show that the foiiowing pairs: of wave functions. are tthogonai over the indicated range. t-1§3§af3 and (lost: ~ l)€‘°“‘f3t“”=“2,-—w S. x < be where o: is a 5mm that is gmater than zero 2- dog) ("aft am] (r/o0 )e""’?“fi cos 6‘ over the interval 090». 059$?s 03¢$21¢ [6| Eigenvalue problems (30 pts) (a) Solving eigenvalue problems is one of the most important math skills in QM. Some simple exercises. 92.13 Detcmtinc in each of the following cases If the function in the first column is an eigenfunction of the operator in the second column. If so! what is the eigenvaiue‘? 82 , (3w8t3.\’+2yj / 6.x: .1“ *5“ “W7; ‘ a ‘ ’2 a )4}? + 3"” filth“ «- A“)? .x d d x ' .‘0 ,' 6-“ {216% tosinéd 2 schm sin ([6 (S! (m) ‘V (b) The operator “dz/dxz” closely relates to the kinetic energy operator and will appear frequently in the future. some simple exercises about identifying its eigenfunctions and eigenvalues. PZJS Which of the foitowing wave functions are eigenfunctions of the operator dzfdxg‘? If they are eigenfunctions, what is the eigenvalue? /qc“3"‘ +1363” cosax /b/ 3'1112 x fl”? /€./€“" x B. Advanced Topics (Other advanced textbooks, handouts, lecture notes) (None) C. Independent Study 7 Co stant coefficient 2"d-order linear differential e uation 25 ts (a lease first read Handout #2 carefully (attached in Leoture Note #02) and then write down the following W’ “This is to confirm that i read carefully Handout #2 during the following time slot(s): Jan. 19, 2-4pm; Jan 8-9am (just for examples)". Please solve the following constant coefficient 2”°'-order linear differential equation (SOLDE): y“ — 5y’ - 6y = 0. (I) There are 3 cases. Which case is this? (ii) Please derive the general solution for y(x). (iii) Please solve for y(x), subject to initial conditions y(O)/‘= -1 and y’(0) ; 0. (c) SOLDE: y” + y’ + 2y = 0. (i) Ther-‘efaregcases. Which case is this? (ii) Please derive the general solution for y(x).. Please show the calculation details. a Modern Applications & Reseach Frontiers one) E. Extra Credit |8| Classical wave: standing Wave (5 gts) 132.6 Does the superposition (MK?!) 2 A Sinikx "‘ w?) + 214 Silfikx + w!) gamma a standing wava? Answer this questian by using triganometric identities in (201331911313 the two terms. Question Number 2% I30 - ma NW. , . know) OVKV'HC ‘/ WQJWW‘CKW Wat “TSM\\ You} RAYny M (CON! yecLYFJI‘wx CDMKK ‘ /V' Eyiidzlfi (A Problem Set #3 (Chem 113A W’or, dUe Thu. 02/01/07 at 11:05am, 100pts) Name: kskc Elm?» Forming study groups is permissible, but you must construct your solutions independently. By writing down my name, I confirm that 'l strictly obey the academic ethic code whendoing this problem set. Please write down the names of everyone who you worked with on this problem set under your name. - (i) Seven questions, one for each day. Studies show that the learning process can be more effective and enjoyable if one starts reviewing the material soon after class and does the exercises on a daily basis. (ii) Please write out algebraic details and arguments. If you use math formulas or literature values, please cite your source(s). . _ 3 . ‘ (iii) Please include these few pages as the obver sheet for your solutions. , (iv) If you have difficulties/questions, please find resources and assistance ASAP. (v) Academic ethics need to be strictly obeyed. No exceptions! Please refer to the syllabus. A. Basic Material (Textbook) [1] Preview #08 (due 01/29/07 Mon.) [2] Preview #09 (due 01/29/07 Mon.) [3] Preview #010 (due 01/31/07 Wed.) [4| H-like atom: Bohr’s model (35 pts) Imagine you were Niels Bohr. You were born in Copenhagen on October 7, 1885 and got your Master‘s degree w r in PhysicsileDB and Doctor's d.99F3,¢_lU,,,Ph¥$i9§ inir1911ifrom the Copenhagen University- in the autumn of , , ' ' M191 1:4-youemade1a'etayLatLamhridgeawheteyoujrgflEQQyfiflm/ingtherexperimentatworkgoingéommhegz-- ; , w _Cavendish_l_abol:atow under Sir J.J. Thomson‘s guidance. In the Rutherford's liaboratoryiini/lanchester, wherejustjn thoseyears suCh a‘n intensiVe Scientific life and activity 7 . prevailed as a consequence of that investigator's fundamental inquiries into the radioactive phenomena. Having there carried out a theoretical piece, of work on the absorption of alpha rays which was published in the Philosophical Magazine, 1913, you passed on to a study of the structure of atoms on the basis of Rutherford's discovery of the atomic nucleus. By introducing conceptions borrowed from the Quantum Theory as established by Planck, which had gradually come to occupy a prominent position in the science of theoretical physics, you succeeded in working out and presenting a picture of atomic structure that, with later improvements (mainly as a result of Heisenberg's ideas in 1925), stillfitly serves as an elucidation of the physical and chemical properties of the elements. V You started with the simplest atom, H-like atom (i.e., one electron circulating around a nucleus carrying +Z charge). Based on your training in classical mechanics, you realized that the Coulomb force of attraction on the electron, Foam, provides the acceleration a towards the nucleus: Ze '02 = m a! = m. —-—- 4776072 c e r In order to quantize the energy level of the electron to generate discrete spectra to fit the experimental observation, you propose the 2”"l Bohr condition, quantizatio of angular momentum: I an 11armomentum=mvr*n—b- ~ w123v Q V: Mm VL" vii" gvI . ’L’ *- g 3 5". ZTWGP qfilwebp (a) Given the above 2 equations, please show by algebra that the radius of the electron (i.e., thedistance - between the electron andrthe nucleus) is quantized (i.e., only some specific values are allowed): L 7 "-1 We, (hZAL 203'. = 60 Yiz‘ln ? 607/2132 73' .__._. é.” 3"!“- 3/ = = "" ""“"' 7.. V‘ 77222711,, Z a0 / r‘ (421‘ hie/2V a) k Eel—r Me (b) Please calculate the value of the constant, amin the SI units. (This consta t is later called the“ ohr radius”)~i> (0) Since the radius is quantized, following the 2nd Bohr condition, the velocity of the electron must be quantized, too. Please show by algebra Fooul "‘— L 2.-?\ §;o V! S hie r t if. g L “I 2 I: ‘——— — e z in k V’ min-Hui m = m5 _ Zez ' 17" V LII/EEO " qua/"i 317mg}, 26071.5 ' a) V "-2 2:63 1/2 éohl’) (d) From classical mechanics, you know how the energy of the orbiting electron relates to its radius a velocdy (no need to pzrove this equatl n): v L _— g a q T : €13 h 1% 2. E—lmvz— ze ' Ej—‘é— “ I "2 2 2 ‘ 417607' 633$); 11 e we Given the results from (a) and (c), please show by algebra that the energy of the electron is quantized and has the following form: -f _L & La 1 _ 2 e a 7 {2 e awe t Giuofiaai Sf - 2:47», Elm 7' 2, “‘e _ . L E EF—T V elm-"M ' Eu= W a yawn-‘1 lerf as is) in Cue qua-mew ' " a; L ‘ E—T—‘q—“h' nu .Lh #flfl—lfl-E— _.. he, _- .——%e we . (e) Please show by calculation that 8 e. 1 wk __________1 H _F—-—_*. L L / — 24 2 Li :0 m w». fifl‘l = —(9 18 x10"13_])z - -Ic] “ é , 9) gym xugr‘tc *0)" u" 86671sz " n2 e - Wm >’ 10 c b. I ' . his 4 °i'-'O‘lx13’§‘i& in '9 (O'bLP-‘Ug-VH L (f) The validity of your theoryimodel must be confirmed by experime‘flts. In order to relate the energy levels of the electron to the observed spectra, you proposed the 181 Bohr condition (no need to prove): e) wisely) 2 4Wire-mumw--*rd {Wt-l: -r‘ 8 ( .9- 80: in s‘ufpil”:“(rheu-a. wilt -3!+2y +66 _- i L -- “" COMB XlO 0L f as \ £5: 1'. 1 ‘1 \ h wBased on your results in (e) and the 1st Bohr condition, please predict the frequencies and wavelengths of the first—4 spectrum lines in theBalmer series. corresponding to transitions n=3-)n=2. n=4-)n=2. n=5~>n=2. and n=6-)n=2 for the electron in the H—atom. (9) Please compare your answers in (f) with experimental results. Please cite the souroe(s) of your experimental results. If your theoretical prediction matches the experimental results very well, you can be awarded the Nobel prize in Physics in 1922 (~85 years ago). 2 = l {3“ 54 _ F— _ (I .II. [5] Practice SE#1 (25 pts) £393) *9 a — WV * “7— L‘ I One uestion from Practice SE#1. Guess which one? . ‘ I —' -i . i “ (m were = (e-wwL-W [6] Practice SE#1 {25 ate) 9 k1 -"- f ' s c. . One question from Practice SE#1. Guess whiCh one? i if = Li .5 ¥ HQ ____ 9_ _J' j: 1;. H— g, ‘W‘I‘b Viiii‘jgl 1 IE; 3— “if B. Advanced Topics (Other advanced textbooks, handouts, lecture notes) ' a y '," -_sa#,»————w . ,J L; \ (None) (1“ l )u :1 u (i, r _y/ 1%;— uirw Li "J: —-—:‘_f—(.’R¢(l Yro_ _ C__,' -il M (C'iiJESependent Study I E f: (3 2 WW? 6764'“?an i . "'1- _ .- . . 1,6 .'._ ‘ .‘- f” -I ,.__1 twin.) m (w / .r —'.i. -_ -; l ,. . a h “if: U'q"..3<-—l—‘-;-T‘1$.'Ll“' # LI .H‘J in?» "a 3'! Li . 8L}: .l' J in '~ ' _ fifties r'..»'-_;:.,._ U we r» .- ‘ . a I - J ;, .1 -. x Ir ' "L- I: 1 II , .. " ~' Ii .1_ . . (L rid) _ .. \- _ [i ' .- _ I 3 J. Modern Applications 8: Reseach Frontiers .]Z| Working Princigle of the Stargate (total 15 gts) z’jx‘" The following figures highlight the history for the invention of the Stargate device: Fig. 1: artificial molecules—i"t Fig. 2: artificial molecules—2"” Fig. 3: 1st attempt. April 1,2010. generation. Science 262. 218 (1993) generation, Nature. 403. 489 (2000) failed, as shae is not uite riht) Fig. 4: 1“ successful attempt. Fig. 5: “Supersize it 1" Jan. 1. 2037 Fig. 6: "I’m Loving it 1”.Apri[ ‘l. 2038 Jan.16.2035 ' A "ii—MG 1 11: _l.(_'_)l._‘_l.5 i- The critical moment in the invention of the Stargate is the construction of a special artificial molecule. called “quantummirage” (Fig. 2). It is first predicted based on the Schrodinger equationinqthgwfifixulsohn's interpretation that. if one puts an atom (say. an Fe atom) at one focus of the “quantum mirage". then there is probability that one can observe this atom appearing at the other focus. This prediction was later confirmed experimentally by Dr. Donald M. Eigler in the IBM Almaden Research Center (published in Nature. 403. 489, Feb. 03. 2000) and was finally transformed into the 1st Stargate device by someone in our ChemZOA—1 Fall 2006 class. The figure below shows the propogation of the sound wave in. say. the Disney Hall in downtown LA, when the orchestra. the singer. or the band is located at one focus. Please use this figure to explain the working principle of the "quantum mirage" within one page. (Hint: 2 steps in solving a quantum chemistry problem: i) how is the time evolution and spatial distribution of the wavefunction ?; ii) based on Born‘s interpretation. what is the physical meaning of the wavefunction from (i) ?) Preview #08 (Chem 113A W’07, due Mon. 01/29/07 at 11:05am in class) Assigned reading: Engel 4.1 /Attendance: By writing down my name, I confirm that I was physically present during the entire lecture. Name: fig? 5 Academic ethics need to be strictly obeyed. Forming study groups is permissible, but you must construct your solutions independently. Please write down the names of everyone who you worked with on this preview under your name. ‘ Back: private communication (optional: any questions, comments, Ieaming difficulties, etc., that you want to tell me or the TAs privately) [1] Free 9' article. Please show algebraic details. 1 P43 Are the total energy eigenfunctions for the face particle is: one digjcggtign, Wu} : A+e*‘\i“2’”5l”2)‘ and ip"(‘x): ’ = Ace“”\5“2””55”3“, eigenfunctions of the one-dimensional iiimar momentum operator? if so, what are the eigenvalues? it'k: IA, ell LJOME A71) X : A+ etth 1: Pvl'bp} ‘(hear Mow-'Wtum} (3" " "3K2— , Ci +x . ~ 1‘, N“ (ix " ,. .W @i *‘~> WW ...
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This note was uploaded on 03/06/2008 for the course CHEM 113A taught by Professor Lin during the Winter '07 term at UCLA.

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2+3+preview8 - 51:95 "’ Problem Set #2 (Chem 113A...

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