Phy63S10-Exam2-Solutions

Phy63S10-Exam2-Solutions - Ska/72w EXAM 2—April 9 2010...

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Unformatted text preview: Ska/72w EXAM 2—April 9, 2010 Phy63 Spring 2010 v.A Name: Rec: Section AmMultipIe Choice (5 points each) 1) A proton has a rest mass-energy of mp02 x 940 MeV (approximate), and a radius of 1 fm (Fermi). In order to have a deBroglie wavelength the size of a typical atom (~ 0.1 nm), what is its kinetic energy? .0082'ev , : fl, 3: fig yawn/d 5 x2 P0 to , S. (b) 12,400 eV(12.4 keV) é’c VH6 mu (2 2 is g; :: _..— (c) 0.164 eV 1 (d) 6,200 eV (6.2 keV) My: reflexes; : L (h: lanky/31.13.. (e)10.06eV ,2 MP 2 fl}: 2) Using the uncertainty principle, estimate the level broadening (uncertainty in the energy of the energy level) for an excited state of an atom, which has a lifetime of Ins: . — A if .O.33p.eV At 4i "’ 2 B _ f 1/ ~ (b)0.44n1eV AE /_ L _.__ _./__ .1 C ” Zafi' 2 27? Aft/1 V5; (0) 0.55 eV __ /24’—0 e l/MM (d) W ' 4;; (gem/mfg) (e) lMeV # [2&0 #586 V (Ll) 3) A hydrogen atom is placed in a magnetic field B, pointing in the +z-direction. For an electron in an 1 = 1 orbital, what is the appropriate energy diagram? Note that the magnetic interaction is given by E m - u . B, where p, is the total magnetic moment, including the orbital and spin contributions. (a) -- 3 '. . ' (b). (a ’ (e) ' 4) In the decay process 2125;sz 9 21233Bi + , what is missing on the RHS (right- hand-side)? 6' + v [mswve (fire/age l/- 8 (6.) 5325:83c3'ff-e) V13 (50 (b) 6'" + i? K , (c) e- Mfffrfl 089W WLLM6“W (d) e+ + v (e) e" + v 5) A certain specimen is prepared as a mixture of 2 radioactive elements X and Y. The ratio in number of atoms is 2:1 (X2Y). Each type of atom is transmuted into atoms other than X or Y. In other words, after each decay, the number of X or Y is decreased by one, respectively. The half life of X is 7000 yrs, and for Y it is 700 yrs. After a time t, it is found that the ratio has become 16:1. What is time t? *ffiz (a) 700 yrs N)“: MXO 6 ”MW"; V- 8 (’15) 612 (b)1560yrs Afr: Me €”%fijg MC (14) (c) 1333 yrs A/ figézfizfiwh 752%] (d) 3333 ’ All : M “H: ”i0 a yI'S NY A}??? J—. , «55$? [?0¢993vo] .2333 yrs : 2 e ? fnghwt I 62-: (Z {n 7 £1, at: 3&2 :f42[?dmim] f: W):23$3WQ' 5 Section B——Truth/Fa1se (3 points each) 1) Emission of a single photon from the ZS to 18 level in hydrogen is an allowed dipole radiative transition. T @ flI/waaf PIIéo/e ficmaBCr‘m A Z: i / 2) To make a laser operate, population inversion must be achieved. @ F 3) In nuclear fission of 235U, each fission reaction produces on average 2.5 thermal neutrons directly in the decay process, which can lead to a chain reaction. T (19 @0444? neg/Mags are lb/ncltece/ czf/ew fiefisf’fiwfi WWW; Q/r-Q S‘IZWEJ fly {$52 prawn-e ryfiamfi/flfim, 4) Stable heavy nuclei with larger atomic number Z tend to have excess neutrons in comparison to protons to help bind the nucleons together. (13 F 5) For a positronium, an “atom” made of a positron e+ and an electron e—, the lowest quantum orbit (neglecting relativistic effects), is given by the Bohr radius, a0 = (ll/21c)2/[kemc e2], where m6 is the electron (positron) mass. T (9 mg ta 249 A: rzfifice/f 5;; ”3? ”face “WM” :4“: v.8 7" MC 7* F 7” .. F 7° Section C—Long problems. Please show all work and justify your solutions/answers. (20 points each). 1) For the harmonic oscillator in 1-dirnension, the Schrodinger equation is: (p2/2m + l/2rno)2x )X‘P mEfl’n, where the momentum operator is p =(\hbar/i)d/dx. The normalized ground state wavefunction is given by LPo i Boexp[—ax2/2]. Note that _ +03 2 2 1/2 . 3 W 3—0;: Exp[-ax ]dx (at/a) . 62 - if (a) If the second excited state wavefunction has the form: ‘Fz = B2(1 — sz)exp[-ax2/2], determine the constant C based on the requirement that: . +133 3 T2* T0 dx = 0 , i.e. the ground state and 2"‘1 excited state -co wavefimctions are orthogonal. Also, determine 130, but do not determine B2 (8 + 2 points) [Hintwuse integration by parts. ] ‘ 0" Q32 7% 2 I 7' 59 f fl/Wncedga%m. f3): 4 abc 2/: 602 we ”DC :[gfj Be “fi/BUSéé) 7 2. max” C3 :E’gt4lZ30428221Lm [/“CZ 6 #6:; 2. 606 am .1 —I‘2 .06 2,4152 C ‘ - ”a z... ’jea A”, :3 fificécf :6 99625229 fi‘gzsz’” ”(M)m’z€ $22—$47 m $ :53 W /C=26L. (9 2 (b) Compute <x> and <(AX)2 > : <x2> - <x> for the ground state (1 + 3 points). , or) , n {M .2 <1): £6qu c/«a/x ref” Xe—de {fog (A? awash?) <@X)2> :' 05%»??? 05> #554 from flea/L4 (ea) (2(2):???) [177/251, ._ (c) Compute <p> and <(Ap)2 >= <p2 >- <p>2 for the ground state (2 + 4 points): z sax <p>: I: of fifiggmxfi’gw 1/77/4921); we fig; twee) <z¢i> : «61> " <79“ '3 “52> ”7770037?ng 4’" 9/ 41:35me “3‘3: ,tg/fa (LIV/5772i: it emigflmMJi/fly 7 MM“. 2&3 {we W m ’7 fiififi 2. 2) An atom with atomic number Z = 100 is stripped of all electrons except of one last deep-core e". (a) What is the radius rn for principal quantum number n (3 points)? (c) Next, the K (n=1) and L (n=2) shells are filled with electrons, and one more electron is added What is the energy of the lowest energy level that the extra electron can occupy, and what 18 its radius (4 + 4 points)? X” 55226 BB: 2 efae’m . 2.323212? 2312M: 5%c/aéwg. 3% awe Wfiéfizy C3 f5: a): I}: «do . 2:5‘zf'y/am (d) 4 more electrons are now added What is the lowest energy electronic configuration (4 points)? Weéafi‘ 2 1». (P _;_ Z 7137;: /§‘- /92 2322p¢3523£ Mt [aytchafis’ «Ma- 3) Polonium 21234P0 undergoes a-decay to Pb. (a) at is the disi ' of the decay process (5 points)? ' 2 x 2 . 203 9L —-—--2 We) lflfyM: Cd :: (121/.953695’58 - 45002503 ”20;. 9715535)“- X 95/ 494%? W W (b) What is the available kinetic of the a—partiele (10 points)? [’0 21.96am @3473; ( Macias): 2‘ — "if mflc Viki 1" {MFA Vi?) ‘" Egg? Hey ” ~ h .— _, ”Maggi/e (“mg/M Mme“m' ”Ma: ”fieflufl ”hr Rama? '— 0 r> 3“” 13/9" '5; ._._ 1 / kill/2 2 ”7 fl_> : Magi/{1k +5 (”3% 7:13;? : Engage [ff %]1&?J?fl€y _ — i 7": 1%., , :(SMoHeL/Z khfflg‘m%%% %%*%%x§%fiflfi¢/ (0) Assuming that barrier to be a square barrier, what is the transmission probability if the barrier height is 30 MeV, and the width is 10 fm (5 points) ? ,2cL (7 jgzggiiigifiZ%V #, ; _*fl_fiwrlfl,da /'::6 77 .— ‘ [S— ‘ , ; 2'@/g%X‘f0/% (249/5004) Effln -2fkaX(é'/§4)'C / e a , , wa' WWM#WMfiHi: “if j; :5 figxix-Fxm/ 1 V, 3 ' 2‘9 .2 6 C. : [if—f/Y'mmf Chemical and -Nuc_le'ar_ Informatum for Selected Isotopes .3 '- .12992835; I: 050115-55 1000'1371 =5999fi986% -:.-~°41 22991221 .- '3: 11.2 ’1 -- .. “ [1'00] .P0}_.O__I_i_i2n:|_1'_- . .- 211988863 _ “1.21uonuaoag-g; ;_-,.n-,; ‘HSGGHQbu-Q'L-I ' ' 212222;” ...
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