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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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- Quantum Chemistry