Chem 113A Exam 3 - ll b Rotationals ectra> bond len th 5...

Info icon This preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon
Image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 2
Image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 4
Image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 6
Image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 8
Image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 10
Image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 12
Image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Image of page 14
This is the end of the preview. Sign up to access the rest of the document.

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\‘(~i-5~ 3-4) a h , Kgéw (OJZBXto‘WJ-fl (I) la a ("M/0 >{ -l-(.t,6xl0‘l"y‘(, l2 | ( l ) 2,0.ya \ W” )(25X‘0 3’) l : l-L‘wlb‘ w * 2 Law l \l [1 |(c) Vibrational spectra-9 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 H-atom 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 H-atom 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... Hit-a 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 canal-OM vi 3‘45 “oh-IF '3 _-. M can u " ‘l I / tr\}\itr)) a W" “WY“ 1 (Nim- ]+ SE 15“" ”'5“ 7‘ Mg was-«WI 3'5}: 2 l“ Sr Linc. _ I i,- . . 1 I z 3 .._© () in“ ‘i “r “- . grur) Hi..." rip IEMM? “85%.. |( “E0" 1' Incr‘ EN" r mini. .22 H: ER”) —i~" 3‘I1}. Uri)" I. 1 out,“ ‘b “1 a-r I" or 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 etc-the 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 Hl-lJ = 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 H-atom’s electron is in the 2p_1 state (n=2, I=1, m= -1). Please calculate the expectation value <LX2> for H-atom’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(<‘n|L‘llmxg~ ) ti saw only ;uhl (inllm5 I JJ .i‘(‘ 7n~\(7—Bl't’ ): bi, |3|(d) Hyperfine interaction (electron spin-nuclear spin coupling) (4 gts) For the H-atom’s electron in the ts state, its orbital angular momentum is described by the quantum number [:0 and its spin angular momentum is described by the quantum number s=1/2. its nucleus, a proton, has spin angular momentum described by quantum number i » =1/2. Consequently, the electron spin 3 will interact with the nuclear spin [through the hyperfine interaction, as described by the Hamiltonian Hs,= c 8-,! where c is called “hyperfine constant”. Please show by algebra that the commutator [J2, Hs,]= 0, where the total angular momentum operator J= S+l. Please show algebraic details. (Remark: This result is very important to the invention of atomic clock) (in le- Lhmvx \)= 5+1 ham: [5.31:0 We burh A”, l I opcr-Jyur 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=Lx-iLy, Please show by commutator algebra that [L, L2] = - L. Please show algebraic details. nhLzlahanh 1,] z LLX,L;]+i[lw.LZ-7 9 ' '.. o-erator. 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,m|I,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’-.l|m> ~ <\.mlL,_ ,n 4|," ‘- ' : (‘Z~hz)"(‘.’m\ : (rmlimn U-n») 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 <Im|Lx|Im>. (Hint: express LX in terms of operators with well-defined 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; Na-nlmni’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. = Elli-15 ’4‘?!|+‘> 7. 3L; mat-1W : (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 physio-I picture can help us “understand” chemistry. instead of “memorizing" chemistry. especially when infonnafionlknowledge is just a mouse-click away. For example, when explaining Stem-Gerlach experiments. we write down |-Sx> = c11+Sz> + c2|-Sz>. 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‘ slézis-z l-Sx I-5z7 —;1 L527 2 1- {upset (11'4'57‘ x > _. (1052“1 7- \1(L+L_)‘“5'L - q<+szllQquir5L> + (-2. \-52 a + [CL-L - 92"$z> A 5 b Stern-Gedach 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 time-dependence 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 time-independent operator. Please prove Ehrenfest’s theorem. (Hint: Postulate #4, time-dependent 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, time-invariant 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 ground-state electron configuration of many-electron 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 lanai-cream. “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 H-atom 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.-. Elli-IIIIIIIIII ”lg-IIIIIIIIIII Observe qt ‘ ‘5 Lmb . (ll ll‘j'gltgfl-l-l-I-I ”“4.er e‘uqH-‘WI' mumzanazsglssl! w 4‘ L x... b w) ammnunnnamm 5, n _ .___ rm Ila-lumuanfinafi » “- ' - w- to' fill-SEIEIIFIIII ., , n W commie. H“ "h '“ W“ pzosapes ingot-a“ 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- i-cm f In“: A: :13»: the “tr-90““ 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(V-E)]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.. H|q> = q|q> . 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 O|q>. including the normalization constant. H 0‘? g-(Oflfth? qlo+olq>= AmLH‘tb" 2 Ioulg‘s- olq? : 14%|“? :dbqu-olq) _._. q, 5 61;” 0‘1) :.c""L .. a“. c (cl/”l “14> : c." =fl=== End of Chen-rt 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 ...
View Full Document

{[ snackBarMessage ]}

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern