Chem 113A Exam 2

Chem 113A Exam 2 - Self Evaluation #2 (Chem‘li3A W07,...

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Unformatted text preview: Self Evaluation #2 (Chem‘li3A W07, Fri. 03i02i07 5:00-7:30pm+20mins, 100+5pts. max=100pts) Name: Mini 1‘! By writing down my name, I confirm that I strictly obey the academic ethic code when taking this exam. (i) Five questions + one extra credit question. Please budget your time. You may want to start with the parts you are more familiar with. Formula sheets are attached at the end. Need more formula? Please raise your hand. (ii) It is very important to show calculation or algebraic details. (iii) Theme and coverage of SE#1 is shown below. (iv) Academic ethics need to be strictly obeyed. No exceptions and no kidding. Theme of SE#2. in 1930 Dirac published “The Principles of Quantum Mechanics” and The PFIHCEDlBS for this work he was awarded the Nobel Prize for Physics in 1933. Some reviewers of Quantum comment that this book "... reflects Dirac's very characteristic approach: abstract but Mach anics simple, always selecting the important points and arguing with unbeatable logic. " Questions in the SE#2 are mainly from material in this book. [1] Applicaions of Simple Quantum Models (total 16 pts) [11(3) (4 pts) Octatetraene P5.4 Calculate the energy levels of the w-network in octaleti‘aene. Cal-1'0. using the particle in the box model. To calculate the box length, assume that the molecule is linear and use the values 135 and 154 pm for C=C and C—C bonds. What is the wavelength of light required to induce a transition from the ground state to the first excited state? lt,ttL\clb‘3‘J’m(5‘-‘i‘) ‘4 . 11. fits”. ’ IN thy-Via“ 1. 2 [1](b) (4 pts) Benzene What is the energy of a delooalized n—electron in benzene in the following state? Assume the radius a=0.1nrn. smc: M cm. 5 atng “a 3 E: ] i K _ 4"sz ' Z? of“ limb” “J 537‘ __. HE. (9.: was xlo'3'L3\(o.lxln'qug .2 2%“ + [11 (c) (8 pts) J P114 ‘H35C1 has a force constant of 516 N m" and a moment of inertia of 2.644 x 10"7 kg mz.w W the light corresponding to the owest energy pure vibrational and pure rotational transitions. In what l £3th (-5)‘ ‘(wa- wan w; Ag (359(0) “ay‘u (st. crab) Inn/0 stadif ‘36 7"”5‘ ‘J 3 [2] From CM to QM (total 40pts) [2](a) (3pts) Dirac guantum condition. The Dirac quantum condition in {f, g} -> [f, g] is a mapping between the classical Poisson bracket of dynamical variables f, g and the commutator of the corresponding quantum operators f, 9. First, please calculate commutator [x,p] based on the Dirac guantum condition. (Hint: classical _§£3_8_38§!_ {f,gl—ax 3p - Poisson bracket is defined as 13,91 3 l“ {le} [2](b) (4pts) Commutator & representation. The choice of the representation for quantum operators x, p must ' satisfy the commutator [x,p] that you just calculated in [2](a). Someone chose the representation as p 9 p and x —) ifl a/ap . is this a valid choice? Please prove your answer by algebra. (Hint: calculate the Yes”; ‘5 q wilt) Ckflltc [2](c) (3pts) Commutator & Uncertainty Principle. Dirac was the first one to recognize the connection between the commutator and the Uncertainty Principle: for QM operators A and B, if [AB] = iC, then AAAB >= 1/2 l<c>l , where (AA)2 = <A2> - <A>2 (definition). Based on your results in [2](a), please calculate AxAp . emu. [XJPJ ="‘R L " AKA? ‘- Fr", Pa) Fold“ x [2](d) (4pts) Commutator algebra. Based on [x, p] yovabtained in [2](a) and the commutator algebra formula listed in the formula sheets, one can derive essentially the commutator of any pair of QM operators. For example. please derive [a , a+] in the reduced units (fi=1 ) where operators a and jffle defined as: a= fim—ix). a‘= _‘fi(p+ix). pi old [9, p1?— [u,a*] = [#5 (r- :13, it; “+1101 _ " 0 fl o ‘ [it it“; P]" [J5 Pl “"- =--i(;1).~r-1=1 \/ [21(e) (4pts) Common eigenket theorem. Given [1%. B]=O and that A|LP> = a|l¥> (i.e, [lib is an arbitrary eigenket of A), please prove by algebra that lllb is an arbitrary eigenket of 3. (Remark: this powerful theorem is very helpful with solving Hamiltonian's eigensystems and with understanding the mystery of the Stern-Gerlach experiments in [5]) UJQJ:AB-BA‘O :. 467611- M‘Shrl' will; 3m) Argill>3= IN” 5* = 6 M10- [3] Quantum SHO: Schrodinger’s and Dirac's approach (total = 34 pts) To simplify the algebra. let‘s use reduced units in this question. In reduced units, m=k=h=1, and the 10 quantum SHO Hamiltonian. H = p2I(2m) + kX2/2, can be simplified to: H = (p2 + x2)l2. Similarly, in reduced units, [x, p] = if: can be simplified to [x. plf i. Now define the lowering operator .3 = E (p — ix). [3](a) (4pts) Raising operator it". Please show by algebra that the adjoint operator of a, the raising operator a+, can be expressed as a" = $03 + l'x). (Hint: (A + B)‘ = A” + 3*; (AB )“ = 3394+ ; (CA Y = c*A*) [3](b) (4pts) Commutator. Please use the definitions of a", H above, [x, p] = i , and the commutator formula in formula sheets to show that the commutator [a", H]= -a*. By Jet lPIPT'J‘: Lp,9p3 ~‘-‘ 0 A I i. .3. z :[ié/£PI]+ [Ft—IPJ‘EIZ 1- L3fixlngJ+ Iii“ ‘1‘ :' {:71 [[51114- fiL: L‘Qllc’zj - l : IE([P,XXJ)1 {JEELlePJ} - plxjx +)([P,!J)* léfEleJP-l' PLleJ) =;a(~'x—.~x)+dc:l+ll> q .1 . ’ .1 - : iLfi( “)0 + N2 (up) “K.— i x*ar .- ‘ + _-.-. (with ‘iI—filrmd -‘-' “q 8 we» ’ l~5h' :4] / 6 [3](c) (4pts) Any QM operators are functions of a and a+. Instead of using x and p as two independent operators, Dirac chose to use a and a+ . Consequently, any quantum operator can now be expressed in terms of a and a+. For example, please express operator x and operator p in terms of a and 3+ . Please show algebraic details. (Remark: Consequently, any QM operator can be expressed as a function of a and a+) «l- xrp ~ m “In at “‘ ‘ d='—fi(‘,-{x) *2}. ml 0 “.1” I ‘ _1_ t— m’r = El?‘ mi + MW“) .2 ‘ 2“ 1'35- ~_L\>o-‘)S 2i sl- ‘LP’rl “P 2 ‘7 a Vi r1: {2 ‘ t '2. z [k at =—-~-x )Q , ‘1 :' ‘filp :(ZX ‘2‘. 1 Xtfifutsq 2 j— (0’,_q) ‘Fi [3](d) (4pts) Raising operator. Let’s denote |E> as an eigenket of H with eigenvalue E, i.e., H|E> = E |E>. Please prove by algebra that a+|E> is also an eigenket of H with eigenvalue (E +1) in the reduced units. P . . _ lease show algebraic details H ‘ E 5 z E IE» on 3‘: -1 2 L“'H1:u-\-H‘uqf: a ‘l‘ HqHE>2 G+H*“*|E> :.Hal‘: ql‘Hiq =3 ulE>+ NE> sqrElE>+atle> bk ~q+(E,\)\E> + “I I «HF->15 em Cucuch 04* H with ctgc'lvqlw fi-I - . [3](e) (4pts) Zero-point energy. Please prove by algebra that E 2 1/z in reduced units. (Hint: Set lQ>=a|Ez and use the property that, for an arbitrary ket |Q>, normalized or not, <Q|Q> 2 O) [3](f) (4pts) Normalization. Based on [2](d) and [2](e). we can write down the energy diagram (Hamiltonian A eigenvalues) En = (n+1/2) where n=0,‘l ,2... Since the energy levels are discrete, we can label them by the quantum number n, i.e.. H|n> = 5,. |n>. Consequently, we can translate the statement in [3](d) into the following equation: a"|n> = fin|n+1> where [3,, is a constant to be determined and the eigenkets |n> and |n+1> are normalized. Please prove by algebra that ,8" =an+1 . Please show algebraic details. (awn->11 = (-01] ‘1) ‘h “114‘; :2 [9.1 [11143) l BnLMI b“! ‘ ("‘1‘3" _..' “I” if 147/ LMH qfln‘? = 4 tot-1H (50* finflnv (E + ill-1|...) : [gal-I” in“ in t P- 1.” («+19ch :80 +4 PM x mm = Mai I; V 8 [3](g) (5pts) Expectation values. Given aln>=JZ |n-1>, a+ln>= x/n +1 |n+1>, and your result in [3](c), we can now calculate effortlessly the expectation value of any quantum operator for an arbitrary state ket of quantum SHO. For example, please calculate <n|x2|n>. ‘Z. anlen>= mime-w] ln> * +q+q H33] iVl> :[“\1i(qq+aq —’ (h) Mun» r J, (n + ln> + Jamar» 1’ {<n'\qnln>+ l X” q z (Dhaka): “(hluln.l> '1 R MANN? ‘- O l + 4n‘fl*q+‘M>°\I;-;l<flloln I imihrxén\n+ L) :8 °- (nlx‘er: §<n lq+qln> t11<nlqafln> = Ml; (ninth-l») +~l (m (“mm-D» L a l > -‘;(fi<n|n>)*z(mml(nln 6 ‘\' _l A: In 3mm =§m+ll= N z \/ [3](h) (5pts) Uncertainty in x. in this course, your capability in using the conventional Schrodinger integral approach is also expected. Please calculate <n|x|n> by Postulate #4. Based on your results of <n|x|n> and <n|x°*|n>, please calculate the uncertainty Ax for an arbitrary eigeket of quantum SHO, |n>. Recall that by definition (Ax)2=<x2>—<X>2. N N = .1. liq wk: n and)”; 2 , 63* <an‘">= IVY/\an “IX V W x . a»: =ij (Nn\1(”n‘1 "X‘ewkx — XO‘): w‘ BL: 00 2 = [1° mm.» me 4 / i3) mtg)":ij an odd {Mitten over- at a‘hfificfiltlfiknqj,§kn an“ 40 .{L' l l L (m‘=\<x >— q» z 2 <n|x"'ln>~[<n|xm>) hawk-c5 O / 9 [4] Dirac dual space and bracket notation (total 16pts) [4](a) (4pts) Adioint operator. Please use Dirac dual space, dual correspondence (D.C.), and the definition of the adjoint of an operator to prove that (cA )+ = c*A+ , where c is a constant and A is an operator. kc? \an \/ [4](b) (3pts) Hermitian o ator. Please use the 5_QM postulates to explain why x and p operators are Hermitian. ; f 2‘ M fluff: (K‘5’. q ‘0 'ICSPonJIKg opcrq+gr. «SWIM-c 5 ‘- measure“ 1- Smcc Were oust q cowcSPwJ‘g QM OWNHMJ any r j. ‘ m GILL “M: eljcanulocs HcW‘VC/i SM“ “1‘ clgchv'dvg 0“ I“, {0, Elm 5'“ “WI sprinters “(JPCCHWIY. 1L9 OP‘WIMVS "10$" Hun lac Hc’WlH-“h_ «r3 :3," p05 Mink. . 10 [4](c) (4pts) Hermitian operator ls angular momentum operator Lz= xpy— ypx a Hermitian operator? Please prove your answer by algebra. (Hint: if O=O+, then 0 is a Hermitian operator. You may find formula sheets h I f I. epU) By J14 0+ q‘L + K2A4192p + (L2 3* = (XPy “fle ’+\ 2 (Xpflt (Wk) ‘ :‘Lz ? “WM” KCX‘Pg1777. .L \s no we J [4](d) (5pts) Real eigenvalues theorem. One of the reasons that all QM operators correspond to experimental observables must be Hermitian is that the eigenvalues of a Hermitian operator are all real. That is: for eigenvalue equation O|LP> = A|LP>, if O+=O then A is real. Please prove this theorem by algebra (Hint: Dirac dual space and dual correspondence) % -: (Anoi- L Olll>~‘(<vlol + /r ,,.. = «til/9* :: is 0W5 or O Mfims by match» = «limit» I ‘4’ 0+:O Mn 7.. (Mo-INVHM 5M cc {MI \5 a‘WNS “All HHAC operating M“$lL< "mean is halal m .r _ t ‘ g ' Cl“ “"44 gol‘ old/Ermi- Ialvc, 11 > The Stern-Gerlach experiments are used in our course to highlight the concepts of superposition principle, compatible observables, commutator, and uncertainty principle. \/ [5] Stern-Gerlach experiments (total = 16 pts) [5](a) (4pts) Stern-Gerlach experiment. Following is the Stern-Gerlach experiment discussed in class meetings. The Stern-Gerlach device SGz resolves the z-component of the spin angular momentum, 8;. Please predict the possible outcome(s) (including probability) after the 3rd SG device. Please clear state your reasonin or rove our answer b al ebra. source (oven) K l s T l‘Sx >‘z Cl biz) '3' Cl “51> f l l I l mm in), up:- r l .-- 55°16 3.2 = + ‘55 503,6 5L 2 “E t l v [5](b) (4pts) Compatibiliy. Common eigenlet theorem: if [A,B]=O, then the eigenkets of A must also be eigenkets of B (to be proved in [2 (e); no need to prove here). Please predict the possible outcome(s) (including probability) after the 3r SG device, where the new kind of SG device, 865, measures the length of the Ag atoms’ spin angular momentum, 82. Please explain your reasoning. Please clear state your reasoning or prove your answer by algebra. (Hint: [$2, SZ]=O) +Ks kc sum: wth §: sum. $37 me.-R’ my Mensa“. y 3 l L \SESZ (xlfl’5z) C ) ‘ . when. “23 = I i l l I [5](c) (4pts) Wigner device WITHOUT observer (“Wigner’s friend”). A silver atom in state |+Sx> enters a / 12 Wigner device in the z direction (the area enclosed by the dotted line) and then enters a Stern-Gerlach device in the x direction (SGx). Please predict the possible outcome(s) (including probability) after the SGx u no observcr', *L" ‘ -'.w°%) 5x: 1-:t J [5](d) (4pts) Wigner device WITH observer (“Wigner’s friend”). Same as [5](c) above, but now there is an observer (called “Wigner’s friend”) to observe Ag’s trajectory inside the Wigner device. Please clear state our reasonin or rove our answer b al ebra. (Hint: such observation of trajectory provides information on SZ of the Ag atom and collapses the wavefucntion to the corresponding eigenkets of S2 operator) 0‘ Th" observw‘ cartels out any preview , obs: Hutlm, L 1M3 cox.) “t. rush-dun 0“- Hrsx>‘ bccong ‘3‘ 5"? nut) " 3X) ‘C‘l’ = ""2 Avoh “kc. €wa+ emulon . 13 [Extra Credit] Dirac picture (also called “interaction picture”) (5 pts) The Dirac picture, also called the “interaction picture", is useful to describe quantum phenomena with Hamiltonian operators that depend explicitly on time t. Of course, this content is far beyond most Chem113A level, where only time—independent Hamiltonian operators are considered. Assume we have a Hamiltonian H(t) = Ho + H1(t), where Ho does NOT dependent on time, but H1(t) depends on time. In the Dirac picture, both state kets and operators evolve in time, as defined by: |W(t)>D = exp(iHot/h) |LP(t)> and H1D(t) = exp(iHot/h) H1(t) exp(—iHot/h), where the superscript “D” means in the Dirac picture. Please derive the master equation used in the Dirac picture: d Iw(t)>D/dt = (-Vh) HPm Iw<t)>° 0 y. Question Number Points Earned V . I16 ‘ it) 1/34 0/16 Ho. I16 » Total - 3 )_I100 .— ...
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Chem 113A Exam 2 - Self Evaluation #2 (Chem‘li3A W07,...

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