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Unformatted text preview: ll b Rotationals ectra > bond len th 5 ts ased on the results in [1](a) and [1](b), we can obtain valuable information about the rotating diatomic molecule from the spectroscopic data. For example, in 1H‘9F, the frequency of light absorbed in a change from the l=3 to the
[:4 rotational state is measured to be 2.5 *1012 5". Please calculate the bond length of 1H19F. AE:‘\‘€
W 1.36“”): ML
'1 321:!" x
“I 1500"”)
r‘= R 1D}: 5. (kéu X it‘5q‘1‘5’lzri\‘(~i5~ 34) a h , Kgéw
(OJZBXto‘WJﬂ (I) la a ("M/0 >{ l(.t,6xl0‘l"y‘(, l2  ( l )
2,0.ya \ W” )(25X‘0 3’) l : lL‘wlb‘ w * 2 Law l
\l [1 (c) Vibrational spectra9 force constant {5 pts) The quantized energy levels for the vibrational motion of a diatomic molecule can be found in the formula sheets.
1H‘QF absorbes a photon with wavenumber 4138.32 cm'1 to go from the vibrational ground state (n=0) to the 1St vibrational excited state (n=1). Please calculate the force constant for the chemical bond of 1H‘QF. (Hint:
wavenumber = 1/A) [“21 kLV 
=*(%V”A&r3 JJ Atom (total 16 pts)
ISE for Hatom 4 '5. The angular momentum o erator, L2, in2 the spherical coordinate system {r.6.¢} can be expressed as: sine aa aa sinzg'a?
Please show by algebra that the above TISE for Hatom can be simpliﬁed to: —h2 £02 3%) +131}! 4 2mr2[V(r) —E]w = 0
Where V(r)=
‘1; t
I}. "L
75‘: 'l‘
713');  l
I. b Radial art of the total ei enfunction 4
sume the total eigenfunction for H—atom's electron ‘P(r.9,¢) = R(r)Y(B,¢) where Y(B.¢) is the eigenfunction of the
angular momentum operator. L2: i.2 Y(e.¢)=i(i+1)h2 Y(6,¢i) where #0,1,2.3... is a quantum number. Please derive
the differential equation govemin the dial rt R r) of the H~atom's eigenfunction: H¢=EY gun Mucus .‘ z, r
 ., 1 . I...
Hita 1.133%. in, ﬂit. thi g c L2 t n'vm (atrium) = E RIM/Mi
Sign 1* {ﬁg} (ximiie.¢ii* 37:3" “W”! 6)) " 2.91;? B!
' Vihd’
5m: gift“! ! J "9* “M \_ 5. spam”
 3' fir: +‘uhl YIN”) is o canalOM vi 3‘45 “ohIF '3
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‘l I / tr\}\itr)) a
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3'5}: 2 l“ Sr Linc. _ I
i, . . 1
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Hi..." rip IEMM? “85%.. ( “E0"
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1nd JJ 5 0 Radial nodes 3 ts The atomic and molecular structures and bonding are mainly determined by the states : the valence electrons (the chemically active electrons in an atom). Therefore, given the wavefunction of the
valence electrons, it is important to obtain the following information: (i) number of radiailangular nodes: (ii) location
yi’these nodes; (iii) maximum of the probability distribution. The eigenfunctions to the Hamiltonian operator of the
Hal—atoms electron can be expressed as Wm(r,9.¢) = Rn.(r)Y,.,(e, )where R...(r) is called the radial part, and Yh(9.¢)
its called the angular part. Radial nodes exist wherever Rul(r)=0, and angular nodes exist wherever Y”..(e.¢) =0.
iPIease locate all the radial nodes for the 2s orbital. Please show algebraic details. .25..) ":2. 1:5 ﬁnd] Angular nodes (5 ate) (0 Please locate all the angular nodes for the 2pc (n=2. l=1. m=0) wavefunction. Please
'show algebraic details. _(ii) If a valence electron is in a state described by the 2pc wavefunction and forms a
chemical bond. pieasepred etcthe orientation of this chemical bond eigenfunctions of the Hamiltonian operator is still an eigenfunction with the same eigenvalue. That's, if (pt. c2. q)"
are degenerate eigenfunctions of operator H with identical eiegnvaiue E: He. = E cpl (i=1 .2.....n), then the new
wavefunction '4’ = 01% 4' 02¢: + ...+ one" is still an eigenfunction of H with the same eigenvalue E. This is called the “Degenerate eigenstates theorem”. (Hint: show HllJ = EW)
, l c” it") #550 . Ht; H [q tint amt”
Slut. ﬁcram can “run...
at; Hm.“ H (an) t... New.)
: LY. " [ﬂit ”JEN"
: EH” 1 4”. ch}
1 Hi“ VJ
[31(9) Real function orbitals (4 ﬁts} Deﬁne a new 253 orbital: 2:3x forms of 2pIt and show that 2px is a real function. (Remark: a
has the same energy as 2p+1 and 2p.1 ) (1(2)”2 +1 + 2p.1 }. Please derive the functional
the ”degenerate eigenstates theorem”. 2px 1"le \‘t‘. "i=1". lulﬁ‘hph + «LD
“J—{( “1+1 HI...)
lﬁl Ewan J‘H (“Eul'usmtcﬁ)
=i;((;1\"‘smo(s° we) / eﬁ( ﬁlluﬂwﬂuae + same + toss I‘sm&)
= F21: (ﬁfhwoiz use) {If}; Hz
 ﬁ(;}"‘smeco36 = :33 S‘Wco
' 8's // c Ex ectation value 4 ts Assume the Hatom’s electron is in the 2p_1 state (n=2, I=1, m= 1). Please
calculate the expectation value <LX2> for Hatom’s electron in this specific state. (Hint: L2 Y.m(6,¢) = I(I+1)l"i2 Y.m(6,¢); LZ Ylm(9,¢) = mﬁ Y.m(6,¢) where /=0,1,2,... and m=0,i1,:t2,..., ,il. ) 2
4L l.="> 4t lt‘lim» (\MPCLZ'LIHN ‘g(<‘nL‘llmxg~ )
ti saw only ;uhl (inllm5 I
JJ .i‘(‘ 7n~\(7—Bl't’ ): bi, 3(d) Hyperfine interaction (electron spinnuclear spin coupling) (4 gts) For the Hatom’s electron in the ts state, its
orbital angular momentum is described by the quantum number [:0 and its spin angular momentum is described by
the quantum number s=1/2. its nucleus, a proton, has spin angular momentum described by quantum number i »
=1/2. Consequently, the electron spin 3 will interact with the nuclear spin [through the hyperfine interaction, as
described by the Hamiltonian Hs,= c 8,! where c is called “hyperfine constant”. Please show by algebra that the commutator [J2, Hs,]= 0, where the total angular momentum operator J= S+l. Please show algebraic details.
(Remark: This result is very important to the invention of atomic clock) (in le Lhmvx \)= 5+1 ham: [5.31:0 We burh A”, l I opcrJyur ocl an ‘i~ own rupeclwc. numb/c
,3 r (513\ . “can.“ \ WNW) ,rLs,sJ + L1,51
2 0 IJ 13 [5+1 1]
:vj : LSIIJILIIIJ : a f
s t= llJ‘s’1"'3
w 5mm (5 I] ' 1)] *{t
z c [J11301‘5 'I z z ) 2
1 5([J7‘1JLJ’LJ11521‘LJ’1] ’U 8 why.
[4] Objective #3 “Advanced Topics": Orbital Angular Momentum (total 28 pts)
4(a] 80(3) Lie group (4 ate) Why is angular momentum important? Because the angular momentum operator J is
the most famous member of the 80(3) Lie group! For an operator 0 belonging to the 80(3) Lie group of the Lie
algebra, its x, y, 2 components satisfy the following cyclic commutation relations:
[0,. Oyl= cox. [0,, 0,]: cox , and [0,, 0,]: coy,
where c is a constant. For angular momentum operator J, including the orbital angular momentum operator L, c = i in the reduced units (h=1). Based on these cyclic relations, please prove by algebralt}. Ly]=0. (Remark: similarly. [L2,
L,;l=0 and [L2, 1.21:0 ——no need to prove) Angular momentum is extremely important in QM...” (quote from Cj‘rjn’s QM textbook). in Questions [415}, let’s think about I
4
tr
*3
I'l
;.
’1' H
9? [41m Ladder operators [4 gts) Now deﬁne the raising operator L. and the lowering operator L_for orbital angular
momentum: \/ J L.=l.,+iL,,. L=LxiLy,
Please show by commutator algebra that [L, L2] =  L. Please show algebraic details. nhLzlahanh 1,]
z LLX,L;]+i[lw.LZ7 9
' '.. oerator. That is when L+ acting on I,m>, it results in a new eigenstate with ' number. (Hint: resuts from [4](b)) . ’f ]= ~L+
> =lt\_ * 1+) Mr” M l l
L; “Hm #1  > Lth‘Lth" l
: l\,m>l L+ "" L : L*Lz‘*l
LXt3lh>tLtnm> ' L”
‘L‘ h‘ : Cm¥‘\ L? h, J\/ 4(d) Normalization (5 gts). We can translate the statement in [4](d) into the following equation:
L+l,m> = c,mI,m+1> where c,m is a constant and the eigenstates I,m> and ll,m+1> are normalized. Please prove by algebra that c,,,, = 1[(l — m)(l + m +1) . l5
hm»: c. mm a
Li l m ( +u) (C.M\.m+l$)+= “millcm‘k )
T 4"”!an
(trim) : .
2 <‘H‘I'L
Mm lulu. \‘X 7. and nulhply "uchY‘Vs kej’
/
+ IIMH» '
\mr’lc l
. llm> 2 < “n M 1
(\M ll, L} ‘ Cm *L (W (l my] I ,ml~> _ Cm.
I " 1W!»
‘ — \m{ L1 L1z H >
4,1Liinb <1.mlL’.lm> ~ <\.mlL,_ ,n
4," ‘ ' : (‘Z~hz)"(‘.’m\
: (rmlimn Un») 2..
2 (i~n\(m+") = "m ‘2‘ ‘Cmi vth) (It n+4) JJ 10 4 e Ex ectation values 4 ts Similarly, one can also prove that L_l,m> = dimll,m—1> where d,m = 1((l + m)(l — m +1) (no need to prove this). Please calculate <ImLxIm>. (Hint: express LX in terms of operators with
welldefined operations on Im> ) I} i
L,‘ s 1x * ”V m {Lh‘L—3‘h) A‘n\\_x\\m§ ; (\M\ L  2 LX;\_\1 ‘2)
m L Hm>J
Z {[(‘ni L*l ‘H> + <‘ ‘ 0).} ‘1‘
2 L; 1H... \_ ~ .. .,———— 0m limH> . L x
1; Nanlmni’l lu,’ + L)
"'i\S(hn)il~"*1‘)(\M’h‘>] :' {(0)
:0 .T J\/ 14m Zeeman effect (6 pts) A rotating charged particle (e.g., electron) with angular momentum L in the presence of an external magnetic field B along the z direction has the following energy E = LZ/(2I) 11sz where I is the moment of inertia and the magnetic dipole momentum ,uZ:—yLZ . This is called the Zeeman effect. The corresponding QM
Hamiltonian can be expressed as: H = L2/(2I) 7 «[8; L2. (i) Please derive the eigenvalues of this new Hamiltonian. "'4 (ii) What is the energy difference between these two states 2,2>, l1,+1> in the presence of an external magnetic
field 82 along the z direction? 37:7 i) H‘lHE‘f ii) Ali. = Elli15 ’4‘?!+‘>
7.
3L; mat1W : (2mg) 4431(4)]
: E: k} ' 3’31 1’2" 2.1 [‘411L1)Hzcm] 11 [5] Objective #3 “True Understanding”: Angular Momentum Theory (total 16 pts)
[ﬁlls] Supemosition principle (5 pts). A transparent physioI picture can help us “understand” chemistry. instead of “memorizing" chemistry. especially when infonnaﬁonlknowledge is just a mouseclick away. For example, when
explaining StemGerlach experiments. we write down Sx> = c11+Sz> + c2Sz>. Please determine the coefﬁcients 1d2.Pl h Ibi ta'l.
c an o eases owage racde IS ‘1\\(C‘\‘5z"+ 9‘61») “N bgtlcc 4155; _ 5;1.§¢7
(”2‘ —sx\l5; I‘dﬁw: 33,241; L9; \‘Sz7 i ClCz‘ 51‘
1 ' H5 2
:' (“>2‘ slézisz lSx I5z7 —;1 L527
2 1 {upset (11'4'57‘ x >
_. (1052“1 7 \1(L+L_)‘“5'L
 q<+szllQquir5L> + (2. \52 a +
[CLL  92"$z> A 5 b SternGedach e n'ments 3 ts Based on your results in [51(a). please predict the possible outcome(s) (including probability) after the 3 SG device. Please clearly show your reasoning/algebra. J/ 5 c Ehrenfest’s theorem 5 ts The timedependence of the expectation value <O> is governed by the
Ehrenfest’s theorem: d/dt <O> = (i/h) <[H, 01>, where H is the Hamiltonian operator and O is a timeindependent
operator. Please prove Ehrenfest’s theorem. (Hint: Postulate #4, timedependent Schrodinger equation, and Dirac’s dual space correspondence) (Remark: for most 2D/3D quantum systems, [H, .4 = 0, therefore the expectation value
<J> is a conserved, timeinvariant quantity.) 9,”9‘ <J>IS consenc‘l, 6 ‘[
“° l“ ¢ atuk‘i ' l‘l' 4.1515 ‘MSW‘J w Hum i<d730 Cl)~
3+ 12 15nd) The physical origins for Hund’s rule (3 pts) We can now understand the physical origins of the Hund’s rule, which is used in determining the groundstate electron configuration of manyelectron atoms. Please explain
either (i) or (ii). (i) Hund’s rule: #1 “When electrons are added to Orbitals of equal energy, they will first half—ﬁll every orbital,...” Why
this configuration can lower down the energy? (ii) Hund’s rule: #2 “...with the spins remaining parallel, befOre pairing in any orbital". Why this configuration can
lower down the energy? 0) Bump: al Aojchcmx mm, iloc Sflhs an Mu. J‘sonstwehvc lanaicream. “I‘m it. u, m. chJ +0 orLlM‘S, The" <Kcr§q \uo\ dtcnafud. 1" / +0 13
[6] Objective #5 "Research Frontiers" (total 10 pts) x/ [61(a1 Radial part Rm in Hatom and guantum canals (5 pts)
Please use the following equation expalini therbehavior of R0) at r > 0 for 3 (i=0). p (i=1). and 0' (i=2) states. t» “in: .; 1 u].! r.. ElliIIIIIIIIII ”lgIIIIIIIIIII Observe qt ‘ ‘5 Lmb . (ll ll‘j'gltgﬂllII ”“4.er e‘uqH‘WI' mumzanazsglssl! w 4‘ L x... b w)
ammnunnnamm 5, n _ .___ rm
Ilalumuanﬁnaﬁ » “ '  w to' ﬁllSEIEIIFIIII ., , n W commie. H“ "h '“ W“
pzosapes ingota“ l‘ daft/“(‘1 tPU”I 5'9"th c. +° *. noe‘cﬂs c .
9“ “m” “ .44 mac, and»: “We “"6 11m.“ Marc mu, aim indeed“ "" (”4):ng Lurks “A
u! for blows an cm. and
. cl 5 ist icm f
In“: A: :13»: the “tr90““ pair—ram] kHP‘ m.r h, ado Sma'1;1n H‘w‘ 9'" "M '5 ‘ ham“ ‘3
[sub] Quantum tunneling in mglecular cagcadg (5 gts) Please show by algebra that the meBllE for a particle with mass m at position x=0 to tunnel to position x0 in region II is F?(x°)=exp(2k x0) where qurt[2m(VE)]ih. E is
the particle’s total energy, and V is the height of the potential barrier. (Do you like my drawing?) E“ "I; \/ M» we 30 14 An important goal of this course is to extend the essential QM principles to new quantum systems. Having learned the Dirac
algebra approach to solving the simple harmonic oscillator, we will now apply the Dirac algebra to investigate the properties of a
new quantum system. The following exercise demonstrates again the elegance and power “of the Dirac algebra. [Extra Credit] Application of Dirac algebra to new quantum systems (total = 10 p13) The Hamiltonian of a new quantum system can be expressed as: H = 0‘0. where 0+ is the adjoint of 0. Let ]q> be
the normalized eigenkets of H labeled by their eigenvalues. i.e.. Hq> = qq> . The anticommutator is deﬁned as {A.
B} = A5 + BA. Instead of the ual commutation relations. let the O's have anticommutation relations: {0” .0} =
0’0 + 00* = ‘l. . lﬁlla} Anticommutator (4 pts) Please show by algebra that. for the Hamiltonian H deﬁned above. {H,O} = O. [Ellbl Lowering ogrator (6 pts) Please evaluate Oq>. including the normalization constant. H 0‘? g(Oﬂfth? qlo+olq>= AmLH‘tb"
2 Ioulg‘s olq? : 14%“?
:dbquolq) _._. q, 5 61;” 0‘1) :.c""L .. a“. c (cl/”l “14>
: c."
=ﬂ=== End of Chenrt 13A W197. Wish you a great spring break— you much deserve it. Thank you for doing your best all the
time in this class. We, the teaching staﬁ, wish to see you succeed! ====== '—
L'— , I28 ...
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 Winter '07
 Lin
 Quantum Chemistry

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