MASSACHUSETTS INSTITUTE OF TECHNOLOGY

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1 MASSACHUSETTS INSTITUTE OF TECHNOLOGY 5.73 Quantum Mechanic I Fall, 00 Profeor Robert W. Field FINAL EXAMINATION DUE: December 11, 00 at 11:00AM. Thi i an open book, open note, open computer, unlimited time exam. You may talk to each other to clarify what the quetion mean, but not how to anwer them. You may alo contact me and I will attempt to give you explicit help (including ome direct intruction about how to olve a problem). Name: GRADING: I. II. III. TOTAL:

2 5.73 Final Exam Page 1 I. Aufbau for Adult. Look at page 31-9 of the lecture note. Now conider the following problem of pectrum identification and prediction. A. You have an aborption pectrum and you do not know whether it come from atomic C, N, or O. You do not have acce to a table of energy level or aigned pectral line. What are the feature in the pectrum that would enable you to concluively identify the carrier of the pectrum? Here are the rule: a. The atom tart out in it ground electronic tate. You cannot rely on any tate being initially populated other than the lowet L S J tate predicted by Hund rule. b. The election rule for electronic tranition are l = ±1. Thi i conitent with L = 0, ±1, S = 0, J = 0, ±1. An off-diagonal matrix element of H SO between ame-configuration, ame-j, L = 0, ±1, S = 0, ±1 tate could make weakly obervable certain S 0 tranition. The intenity borrowed by a nominally forbidden tranition i directly related to level hift of both the borrowing and lending tate. c. The nuclear pin of 1 C, 14 N, 16 O are I = 0, 1, and 0 repectively. You can ue the preence or abence of hyperfine tructure to make aignment, but you mut make ome prediction about the qualitative nature of the hf. d. You can ue the Zeeman ect in a variety of way to identify the tate involved in a tranition. If you do o, you mut calculate g- value. e. The Landé interval rule hould prove very helpful. If you ue it, you hould calculate the relative intenitie of tranition to different J component of an S 0, L 0 multiplet tate. The tranition intenitie come from the form of the tranition operator a a 1-electron operator, T (1) [r] where r i the electron coordinate. Thi ha l = ±1 election rule. Z-polarized light excite m l = 0 tranition. You hould anwer thi quetion by identifying the unique pectrocopic ignature of each atom. I am looking for quality (very pecific diagnotic) rather than quantity (vague, qualitative tatement).

3 5.73 Final Exam Page B. Ioelectronic and iovalent comparion. The energy level diagram for two atom with the ame number of valence electron will reemble each other, but the cale parameter [ε nl, F k, G k, ζ nl ] will be ytematically and predictably different. (i) How would the pectra of C, N, and O compare. Make plauible argument about how each of the cale parameter would change. (ii) How would the pectra of Si, P, and S compare to the pectra of C, N, and O, repectively? Again, be a pecific (and ingeniou) a you can about how you expect all of the cale parameter to change. C. A tranition from the N atom ground tate to one of the 4 P tate belonging to either the p 4 or the p 3 configuration i excited. (i) (ii) How would you be able to tell which configuration the 4 P tate belong to? The tranition i excited with a 1f light pule. There will be quantum beat in the fluorecence. Be a pecific a you can be about the fine tructure (J = 5/, 3/, 1/) and hyperfine tructure ( F= J I) contribution to the quantum beat pectrum. Be a pecific a you can be about frequency ratio and the relative intenitie of the variou beat note.

4 5.73 Final Exam Page 3 II. Effective Core Potential (Peudopotential) It i often ueful to implify a many-electron problem to a one-electron problem. The active electron move in the emi-empirically defined ective potential of the ioncore. In thi problem you will ue a one-dimenional ordinary differential equation olver (Dave Lahr MATLAB handout) to define Z (r) for the Ca 1 S and Ca 4 S ion uing the experimentally known quantum defect for the n, np, and nd Rydberg erie of Ca and the 4nl 1 L (l=0, 1, ) Rydberg erie of Ca. The radial Schrodinger equation, expreed in atomic unit i du E V() r dr ur () = rrr () ( 1) u = 0 r ll E i the binding energy of the electron (the zero of energy i et at the ionization threhold) Vr () = Z ()/ r r V () r = V() r l ( ) r ll 1 where, for Ca, br Z () r = 18( 1 ar) e, a 0 and b 0 which ha the required limiting behavior Z Z ( 0) = 0 ( ) =. The b parameter determine the overall ize of the Ca ion-core and the a parameter permit Z ( r) to exhibit ome remnant of hell tructure. Initially, you hould et a = 0 and vary b to obtain a V l (r) that give the correct (i.e. empirically determined) quantum defect for the Ca n Rydberg erie. Then you will adjut a and b to match the quantum defect for the Ca np and Ca nd Rydberg erie

5 5.73 Final Exam Page 4 Eau (..) 1 / hc= R = ( cm ) 4R ECa ( nl) hc= ± n µ p µ = µ = d µ = ( ) l For the Ca 4nl 1 L Rydberg erie, the ective potential of the Ca 4 S ion-core could be parametrized a where V () r = Z ()/ r r l ( 1) r ll cr Z () r = Z () r ( e 1) Z ( 0) = 0 Z ( ) = 1 Thi form for Z ( r ) treat the ect of the 4 electron a imply additive to the ective potential of Ca. The c parameter will be much maller than the b parameter, becaue the 4 orbital will be much larger than the Ca ion-core. Once you have optimally decribed Z ( r ) by fitting to the Ca nl Rydberg quantum defect, you hould adjut c to fit the Ca 4nl 1 L quantum defect. µ =. 931 µ = p d µ = ( 1 4 l ) = ECa n L hc 1R ( n µ ) l Here i the procedure that I propoe you follow. Ue the MATLAB ODE olver upplied by Dave Lahr (and you MAY collaborate with each other in making thi work). A. Chooe an initial value for the b parameter and et a = 0. You hould chooe b large enough o that Z ( r ) i between and 3 at r = m, which i the official ionic-radiu of Ca. Solve the 1-particle radial Schrödinger equation for Ca 8. The boundary condition are u l (0) = 0 and u l ( ) = 0. You are looking for an l = 0 eigentate

6 5.73 Final Exam Page 5 with even internal node. Once you find the 8 eigentate, you need to adjut b o that the eigenenergy i E 8/ hc = 4R/ 8 µ = cm [ ] = ( ) cm 1. 1 Thi i an iterative proce. Once you have found a atifactory value for b, repeat the iterative proce tarting with a = 1. When you are done you will have two pair of a,b value that give the correct energy for the Ca 8 tate. Then ue the two pair of a,b value to compute the energy of the Ca 8p tate (ix internal node). Both of the calculated energie are likely to be lightly incorrect. Devie (and explain) an iterative trategy o that you can find a pair of a,b value that give the correct energy for both Ca 8 and Ca 8p. [ ] E 8p = 4R8 µ p. Now your Z ( r) function hould be pretty cloe to perfect. Tet thi by computing the energy of the Ca 8d tate (5 internal node). [ ] E 8d = 4R8 µ d. B. (Optional). Now that you have determined an empirically optimized Z () r for Ca, determine for Ca 4nl (l = 0,1, and ). Optimize the c parameter in ( ) cr Z () r = Z () r e 1 to obtain the correct energy for the Ca 48 1 S tate, ( ) = R µ 1 ECa 48 S hc ± 8 µ = [ ] Once you have done thi, check thi Z ( r ) to ee how well the Ca 48p 1P and 48d 1 D tate are predicted. If the reult for 48p and 48d are unatifactory, ugget a plauible reaon for the dicrepancy.

7 5.73 Final Exam Page 6 Cloing comment: Thi ective core potential method could be ued to find the energie of the outide-core electronic tate of a charged metal olid phere, a charged metal hollow phere, or a charged quantum dot.

8 5.73 Final Exam Page 7 III. Wavepacket Dynamic in Atomic Rydberg-Land. I am going to lead you through a implified verion of the experiment decribed in the attached paper, Nonexponential Decay of Autoionizing Shock Wave Packet by Thoma and Jone [Phy. Rev. Let. 83, 516 (1999)]. The purpoe of thi problem i to develop the dual kill of decribing the evolution of Ψ(t) a a pecific linear combination of eigentate and of decribing the time dependent picture of wavepacket in motion. The relevant energy level tructure of Ca and Ca i ummarized in the level diagram: 4dm d 7470 cm cm Ca 4p P 1 1/,3/ 318nm [Ca 4d 4p] (femtoecond excitation) 4pmd cm cm 1 Ca 3d D 3/,5/ 393nm [Ca 4p 4] (femtoecond excitation) cm cm 1 Ca 4 S 4 nd 1 D (4 < n < 33) 393nm (nanoecond excitation) 365 cm 1 4 4p 1 P 1 43nm (nanoecond excitation) 0 cm 1 Two nanoecond laer are ued to populate one of everal 4nd 1 D eigentate, where 4 < n < 33. A femtoecond laer pule (393nm) excite the 4 electron to the 4p orbital. The frequency of thi tranition i expected to be very cloe to that of the Ca 4p P 3/ 4 S excitation ( cm 1 or 393nm). The pectral width of the 00f pule i ufficiently narrow that 4p P 1/ S 1/ i not excited. The excitation probability (becaue of n 3 -caling) for the Rydberg electron i negligibly mall. The

9 5.73 Final Exam Page 8 hort pule excitation tranfer the nd eigentate from Ca 4nd to Ca 4pnd where it i not an eigentate. Thu Ψ(0) i a uperpoition of quai-eigentate Ψ( 0) = cm 4p md. m Thee are quai-eigentate becaue they can decay by autoionization into the continua of Ca 4 εl (l = odd) and Ca 3d ε l (l = odd) via matrix element of 1/r 1. The energie of thee 4pmd quai-eigentate can be taken to be given by a implified Rydberg formula (neglecting quantum defect): E 4pmd = E( Ca 4p P3 / ) hc R/ m. For pecificity, let 3 m 37 (ix tate) and let C m be 6 1/ for all ix m value. The 00f 393 nm PUMP pule launche a wavepacket that, at t = 0, i identical to the nd orbital in the 4nd initial eigentate. The outer lobe of thi wavefunction i located near (but not at) the outer turning point of the 4pmd quai-eigentate. The wavepacket will evolve in a way that you will need to figure out. The ocillation period i called the Kepler period. Note that if you chooe wavefunction phae o that the innermot lobe i alway poitive, the outermot lobe will exhibit a phae that ocillate with principal quantum number, in thi cae ( 1) m. A. For the 3 m 37 wavepacket, what i the Kepler period, T k? At different time during a Kepler period the way in which the wavepacket created by the 393nm 00f PUMP pule i affected by the 318nm 00f PROBE pule change. The PROBE pule act on the inner (4p) electron. The 4d 4p tranition amplitude from all of the quai-eigentate component are in phae when the wavepacket ha returned to it original form at t = 0, T k, T k and motly out-of-phae at t = T k (, 3T k /, ). B. For the 3 m 37 wavepacket at t = T k /, decribe the time-dependent amplitude of each of the quai-eigentate m-component in Ψ(T k /). Be explicit about the phae of the innermot and outermot lobe of each m-component. When the initially created wavepacket i at it t = 0, T k, T k form, it rate of autoionization into the Ca 4 εl (l = odd) and Ca 3d ε l (l = odd) continua will be different from when it i at it t = T k /, form. Figure of the attached paper ugget that the autoionization rate i minimized at t = 0, T k, T k and reache one or two maxima during each Kepler period. The autoionization i due to matrix element of 1/r 1. The inner part of the md Rydberg orbital i mot important in determining the magnitude (caling a m 3/ ) of 4pmd1/ r 3dε l and 4pmd1/ r 4εl matrix element. 1 1

10 5.73 Final Exam Page 9 C. Explain the ocillation in the autoionization rate of the 4pmd wavepacket. I will be very impreed by an explanation that account for the two maxima in the autoionization rate during each Kepler period. The 318nm 00f PROBE pule excite 4d m d 4p md [via Ca 4d 4p]. [The quantum defect for the 4dm d and 4pmd quai-eigentate are lightly different.] Thi new wavepacket alo autoionize. The autoionization ignal due to the PUMP alone and the PUMP PROBE are ditinguihed via the kinetic energy of the ejected electron. So the ytematically time-delayed PROBE pule i capable of ampling the population in the PUMP-produced Ψ(t), with approximately 400f time reolution. If there were no autoionization from the PUMP-produced wavepacket, there might till be a delay-time dependence of the PROBE-induced excitation probability. D. Derive an expreion for the delay-dependent PROBE excitation rate. Explain your aumption in deciding whether the excitation rate i maximal at t = 0, T k, T k or at t = T k /, 3T k /, or independent of t. Thi i related to your anwer to part B.