Problem set 4 and 5 - We, Wye Problem Set#4(Chem 1113A...

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Unformatted text preview: We,» Wye. -/ Problem Set #4 (Chem 1113A W'07, due Thu. 02/08/07 at 11:053m, 100pts) Name: ‘\ V Forming study groups is permissible, but you must construct your solutions independently. By writing down my name, i confirm that I 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. As a “punishment” to the class’ impressive performance, no extra credit question this time. (ii) Please write out algebraic details and arguments. if you use math formulas or literature values, please cite your source(s). (iii) Please include these few pages as the cover 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 #11 (due 02/02/07 Fri., deferred to 02/05/07 Mon. due to SE#1) [2] Preview #12 (due 02/05/07 Mon.) [3] Preview #13 (due 02/07/07 Wed.) [4] PlB-1D (35 rats) (a) .5 Consider a particle in a onet’iimcnsional box defined Il’tx) z 0, a > x > 0 and VU‘) r: so. x 2 a. x S 0, Explain why git of the following unnormalizcd functions is or is not an ’. ptabtc wave function based on criteria such as being ‘nsittent with the boundary conditions, and with the u iatton of tiftxkfxtfldx with probabiiity. rm: .AcosM— c. Cx3(x mo) .42... sinUthza) (Hint: check if the given wavefunction satisfies the BCs and the 7 Max Born conditions on Lecture Note #13- 15. (b) P433 The funcfion #10:) m Axli - (x/afl is an acceptable A > : jg) 3;“ L; LL. 5. {,3 .- wave function for the particle in the one-dimensional infinite “' I depth box of length 0 Calculate the normalization constant 4 and the expectation values (A) and 0:3 ) (c) Once we know the wavefunction at a specific time, we can extract valuable infomation about the quantum systems at that time. First example, if the quantum system happens to be In an eigenstate. Please change “ rom 0. 31a to 0. 35a” to “from 0.3a to 0.4a”. :8 Cal culatc the probability that a panicte m a one- iensional box of length a is found between 0.3m and 5a when it is described by the l’oitowing Wave functions: ‘ ,2 . (17.x 3“? ‘ _ r . .1 y ‘ “Sin 2.. 2, , L i: ' 4 ~ , ‘a , a J " “ L ’ ._§ X ';.B(x+ x1} cl. Q 77 .5“? ““’ mex '“ ‘L 53‘. in; 5 ”A x» it? w \fi ! l" \ would you expect for a classicat pmticlc‘? Compare your {malts in the two cases with the classical result. - ' is a more complete and general theory, and can be used to describe both microscopic and . ' acroscopic systems. Let’s use “particle in a 30 box" for example. Please change the edge length from -' '1nn1 to 10nm. P423 Generally, the quantization of translational morion is not significant for atoms because of their mass. However. this conclusion depends on the dimensions of the space to which they are confined. Zeolites are structures with small pores that we describe by a cube with edge length 1 nm. Calculate the energy of a 1-12 molecule with n;r = n, = n.T h= lO.Con1pare this energy to Hal T = 300 K. 15 a classical or a quantum description appropriate? (b) Please change the temperature T from 298K to 398K. P422 This problem explores under what conditions the classical limit is reached for a macroscopic cubic box of edge length a. An argon atom of average translational energy 3/2 H is confined in a cubic box of volume V = 0.500 m3 at 298 K. Use the result from Equation (4.25) for the dependence of the energy levels on a and on the quantum numbers it”, try. and at. a. What is the value of the "reduced quantum number" a=. hf «in; +113 for T = 298 K? b. What is the energy separation between the levels a and a + l? (Hint: Subtract 15'“+1 from El, More plugging in numbers.) c. Calculate the ratio (l9nr+l —- E¢)/kT and use your result to conclude whether a classical or quantum mechanical description is appropriate for the particle. [6] Superposition of eigenfunctions (20 $1 a. free particles. Please prove your answer by algebra. [$44 Is the superposition wave function for the free particle it) = A.e“~"‘-3"'*3”‘3” + A_e“‘-"3”’£;"”‘ an eigenfunction of the (unintentum operator? [5 it an eigenfunction of the total energy 5:} n 1"? Explain your result. (b)"PIB. Please prove your answer by algebra. P4.12 Is the superposition wave function :14 x) = V’fflsinm'rrx/a) + sin(mn-x/a)] an eigenfunction of the total energy operator for the particle in the box? "" B. Advanced Toplcs (Other advanced textbooks, handouts. lecture notes) “a (None) pendent Study D Modern Applications at Reseach Frontiers [7] Quantum sguare (259$) ‘ This is the STM image for Fe atoms on a Cu(111) surface, obtained first by Dr. Eigler and his group at IBM Almaden Research Center in 1992. and is called a “quantum square". Let's model the “quantum square” as a particle in a 2D box. g 3 (a) Denote the length of the sides ofthe square box by g, (the longer side with 11 Fe atoms) and t, (the shorter side with 11 Fe atoms) respectiveiy. and the effective mass of the electron by m‘. Please solve the eigenfunctions and eigenvalues for the quantum square’s Hamiltonian. If you use the "separation of variables” theorem. please show that this theorem is applicable to your quantum system. (b) What are the :1K and n, quantum numbers of the state imaged below? (Hint: this is an image of M“) (c) Please find or estimate Lg, L,. and the effective mass of the electron m" from the 1993 Science paper on "quantum corrals" (Science 262. 218 1993) dBased on our results In a c what zs the ene -y of the imaged state in eV ? E. Extra Credit (None) 1 Problem Set #5 (Chem 113A W'OT, due Thu. 031507 at 11:05am. 100m) Name: Xia nab mu, ( fig”, Forming study groups is pennlssible. but you must construct your solutions independently. By writing down my name. i confirm that I 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. As a “punishment” to the class’ impressive performance, no extra credit question this time. (ii) Please write out algebraic details and arguments. If you use math formulas or literature values. please cite your source(s). ,t (iii) Please include these few pages as the cover sheet for your solutions. (iv) if you have difficultiesiquestions. please find resources and assistance ASAP. For examples. TA office hours and small-group study sessions. (v) Academic ethics need to be strictly obeyed. No exceptions! Please refer to the syllabus. A. Basic Material (Textbook) [1] Preview #14 (due 02i09i07 Fri.) [2] Preview #15 (due 02i12107 Mon.) [3] Preview #16 (due 021410? Wed.) 4 PIB A ll ion : on u ated it-bond solvated electro a d uantum tunnelin 25 ts Comment I have heard: “Learning science is like eating dark chocolate—bitter at the beginning and sweet at the end“. We haVe gone far and deep in this course; let‘s celebrate our accomplishments by seeing how these principles apply to real-life research. The following are applications of “particle in a box" to modern chemistry. (If you still feel very bitter. please find resources and assistance ASAP. We are here for you) (a) The “particle in a box“ model can be applied to calculate the energy structure and transitions in rt—electron conjugated molecules. This simple model is called the “free electron model”. The stmcture of B-carotene is shown below. (i) Please estimate the length of the 10 box using 138 and 157pm for C=C and 0-0 bonds (ii) For HOMO. n=? (iii) For LU M0. n=? (iv) Please calculate the absorption wavelength (in units of nm). (v) what is the color of carotene ? (b Solvated electrons, application of “particle in 8 SD box“. :When metallic sodium is dissolved in liquid sodium chloride, . electrons are released into the liquid. These dissolved elec- ' trons absorb light with a wavelength near 800 nm. Suppose live treat the positive ions surrounding an electron very - crudely as defining a three-dimensional cubic box of edge L, and we assume that the absorbed light excites the electron ' from its ground state to the first excited state. Calculate the ? edge length L in this simple model. (c) Quantum tunneling. (Hint: Prof. Lin always modifies Example Problem to make sure students can not just copy the answers down. Why didn't he modify this question?) EXAMPLE PROBLEM 5.1 As was found for the finite depth well, the wave function amplitude decays in the barrier according to aidx) = Acxp[-—. i2m{V0 — Eli/ii2 x]. This result will be used to calculate the sensitivity of the scanning tunneling microscope. Assume that the tunneling current through a barrier of width 0 is proportional to |A| 2exp[—-2. )‘2m(Vu -— £3232 a]. a. If VtJ — E is 4.50 tell, how much larger would the current be for a banier width of 0.20 run than for 0.30 nm? '1 b. A friend suggests to you that a proton tunneling microscope would be equally effective as an electron tunneling microscope. For a 0.20-nm barrier width, by what factor is the tunneling current changed if protons are used instead of electrons? B. Advanced Topics (Other advanced textbooks. handouts, lecture notes) 5 PlB-i D: zer oint one and Uncertain Princi Is 20 There exist proposals and books for extracting the zero-point energy (ZPE) from such quantum systems as PlB—‘iD. Are they science or science fiction (no need to answer)? (3) Please calculate Axfor the ground state of FIB-1D quantum system with box length i. and particle mass m. (b) Then please estimate the ZPE by using the uncertainty principle AxAp 2 NZ. (Hint: use the exact Ax that you calculated from (a) and AxAp 2hi2 to obtain 4H5 2 G. Since ground state is an eigenstate of H, measuring energy on this state will give ZPE with 100% probability. and thus <H> == ZPE. Consequently. C can serve as an estimate of the ZPE originated from the Uncertainty Principle) (c) imagine you serve as a proposal reviewer for the National Science Foundation (NSF). Would you recommend the NSF to financially support these research projects based on their scientific validity? (Hint: compare the ZPE you estimate in (b) based on the uncertainty principle with the exact ZPE) 6 Stern- erl ch ex riments an ition Princi le 25 ts This is the Stem-Geriach experiment #3 we discussed in lectures. (a) Please change the second 86 device from 36, to SG, and then predict the outcome (including probability) after the third 86 device. Please clearly show your calculations. Note: 862 means Stem-Gerlach device measuring 32. (Hint: |+Sz> = C1I+Sy> + Cal-Sp. where |C1|=|02|=1isqrt(2)) (b) Please change the first 56 device from $6, to SG, and then predict the outcome (including probability) after the third 86 device. Please clearly show your calculations. (c)The following setup is the Wigner device we discussed in lecture: two Stem-Geriach devices. 8G2 and 86x, connect sequentially. Note that the detector is only installed after SGx. After a Ag atom passes 86:. a well- devised magnetic field can refocus its trajectory. Assume a single Ag atom in the eigenstate of |+Sx> leaves the oven and enters 862, where [+89 denotes the eigenstate of 8,, with eigenvalue +hI’2. What will be the states and probability detected by the detector if there is no observer (so-called "Wigner's friend") present? Please clearly show your calculations at each time step. (d) Same as (0), but assume the observer (“Wigner’s friend") is present immediately after $62 to observe the single Ag atom's trajectory? Please clearly show your calculations at each time step. (e) Continued from (d). What if "Wigner’s friend" is present but unconscious or totally drunken (say, due to a dozen cans of beer)? Please clearly show your calculations at each time step. u mu. un. uuuwulgm cqualtull tUl llllh‘KlUUlCIIl SII'ICB [1'18 atom IHICTaCIS Miners {two )éf‘x x z! C. Independent Study A )1 (None) D. Modern Applications 8. Research Frontiers u 0 uantum corral Hand nut #3: Science 282 This is the STM image for Fe atoms on a Cu(111) surface, obtained first by Dr. Eigler and his group at IBM Almaden Research Center in 1992. and is called a "quantum corral“. Let's model the ”quantum corral” as a particle in a circular box. The details of “quantum corrals" can be found in Handout #3: Science 262, 218 (1993). (a) For two~dimensional quantum systems with angular symmetry, polar coordinate system is a good choice: 1: = r c050, y = r sine. Please show by algebra that alraxz + aha} = aziar2 + (Ur) aiar+ (lirz) 62:69 [hint d f(r,e)ldx = d f(r,e)idr'(dri‘dx) + d f(r,e)lde*(deldx)]. (b) Please derive the corresponding Hamiltonian, the time-independent Schrodinger equation. and the Born (boundary) conditions in the polar coordinate system (no need to solve yet). (0) Please re-write the TISE you derived In (b) In terms of dimensionless variables ofor radius and 52 for energy, where the dimensionless variables are defined as: c: rte and s = 2ma2El'h2. where a is the radius of the circular box. (d) Using the “separation of variables" method in solving differential equations, we can "divide and conquer”. Assume the eigenfunctions of the Hamiltonian have the form Rafi) = R(o )603. Please derive the two differential equations governing R(o ) and 6(3), respectively. (e) The differential equation and the boundary (Born) condition governing 6(6) is exactly the same as that In "particle on a ring" (PS#5) and you have it solved already. Now let‘s focus on solving R(o). Denote F as the "separation of variable" constant. Show that the differential equation governing R(o) can be rewritten as arcs—‘4 so“; fiazmafi+mmafi+w43m=o where ,8= 05, t=0,l,2,. .. . (0 Show thatRUS) = O at the. This is the boundary (Born) condition for R(o). (g) The radial equation in (d) is the Bessel equation whose solutions are called Bessel functions MB). Please identify how the allowed values of the energy E are related to the nth zero crossing an; ofJKfi). {To be continued in PS#6 -— you may want to save a copy of your solutions to [7]) 11 m mull-IIIIIIIII Igl-llllllllll lqfiflgflllllllll uncannggggs!!! aamnnngngsnnnn Ila-mumzanfinafi.» -02 . ._., __ _ .534 IIIE'Afi-lllllll a 2 4 6‘ a no .12 a E. Extra Credit (None) Question Number 7 ...
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