practice midterm 2 solutions

# practice midterm 2 solutions - 5 SP[0 Mtbnm STOLLWONS‘...

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Unformatted text preview: 5!? SP [0 Mtbnm STOLLWONS‘ 1. [25 pts] Identical particles. Three identical particles are conﬁned by a 3-dimensional simple harmonic oscillator potential. Interactions between the particles can be ignored. (3.) [15 pts] If the particles have spin-1, and the system is in its lomt energy eigen- state, what are the allowed values of the total angular momentum, 3'. Explain. SPIN j- =7 boSth => Samm_wavegc“. 3 Loweﬁf energy 97> a“ H7: SHO amends/fault (EMU/1“”) =7 ‘Pm‘ml’ mwef’m SﬂMM‘ :7 SW“ WWW 53mm. Each Parhdc has m: H, 0 m- —I :7 3'4")" =_ . ‘ ‘ ~——-———-3'3\ l0 lulu! sfw‘es "Clea/*Uj (M) [+17 l+l7 ~2— IC):3)M:37 Lr‘sthM, —-{OW€/‘lll9 0P5 mmmfm‘n syn/1M. — :. (:8 Id «We! (:FS/affs) —7 em-hm J 3 MK / '*-? Brewciwilmﬁsl lg J22 sink; orﬁagom( (\DJ=3 :ﬁgﬁﬁog (“(51957 (built? Huh-a) (- Remammj 3 W3 hav‘ﬂ J3, . TWO dram are r ”4914»! SgMMymleam (@l: gap/{\$632. H—l) 10>Io7 +9601“; [@QQO’gg [@9— G.) (@O & [email protected]@1 1+7 1+7 I~7 + Pemf‘ So Jzo m AH‘S. on- Marta Jz[lm;[ 3 370‘9J3MWW‘ (scmmsz mfzdwd‘) GACWHM J;[ £3 one (mewwmb is J=3,m=¢ (b) [10 pts] If the particles have spin—1/2, and the system is in its lowest energy eigenstate, what are the allowed values of the total angular momentum, 3‘. Ex- plain. 2. A spherically symmetric potential “trap” has the radial dependence shown below. Units are chosen so that R; = 1 and R2 = 3R1 = 3. The height of the trap is V0 = 5E0, with E0 = nz/(2mR3). 0 R1 R2 1' The Z = 0 phase shift for this potential (divided by 1r) is shown in the following ﬁgure. (Note that k = le since R1 = 1.) There are resonances (where 60 increases through (12. + 1/2)1r for integer n) at [91 = 1.98715 and k2 = 3.70854. (The former is hard to see from the ﬁgure, but 60 / 7r does dip slightly below —1.5.) It will be useful for the following questions to note that, when R2 —> 00 (with V0 and R1 ﬁxed), there is a single bound state with k = 2.02 and E/Eo = k2 = 4.07. 3.0 (a) [3 pts] First we consider the low-energy region, kR2 << 1. Explain qualitatively and brieﬂy why, in this region, S-wave scattering dominates. Use the plot of 60 to determine an approximate value for the total cross section as kR2 —> 0. (You should not attempt an exact calculation using the potential.) How could your result for a have been approximately anticipated from the form of the potential? Finally, what does one learn from the result that 6006) remains linear up to k z 2? o S-wavc suction/:7 dommk; Lemme ‘HIC eeuth boL/‘ngr Jew/w m Vef (r) keg-Pg M wa0c€cm {rum May/ma fubslmﬁ‘al SIR. What W613 mmw; ’ﬂ/MS Mei/cure 2?- >> k)- at- 1591,44 (‘ t? 0'" it. [HT CLZ (015mg #r/vmlot on ELILSMLF) ' If; V0900 ) *he/y‘ CLI=RL\$~3 ii (TelHTlQ: F’imkvo «Mow; some Wmmﬁ/imﬁ) rélQL View at mum; q .. o For V0—70b €o=-kR ﬁrd’l’lk _ ’ (mend- ucacavft, ) Herc 7 {mm/at? MANN! lifﬂl; leakagcatf— 6 Wwefm m+o r4!“ fwéz, (b) [10 pts] Sketch the S-wave wavefunction 110(1‘) for 0 S r < 6 when 0 < k < 1, explaining brieﬂy any features you show. 74 M Exponenﬁ‘a Lu; 0 . I)” (c) [n pts] Sketch the S-wave wavefunction uo(r) for 0 S r < 6 when k = 1:1, i.e. when one is sitting on the ﬁrst resonance. Explain brieﬂy any features of your graph. What does the narrowness of the resonance imply about the wavefunction? ofsmKr’ Asgg‘ﬁ”. K21 =7 magi—ail hdaﬂ Wa/vdeibfh made, r<R,. MCI/TOW Fffonamce =7 [OI/L0 {Ife‘AM-Q, Qﬁ Agnew—b . (d) [5 pts] Sketch the behavior of the S-wave cross section 0“” as a function of k in the region around k = 191. (2:).- 4’2"; Sm 7’ [a J; (E) (e) [15 pts] Calculate W in the Eikonal approximation. For What values of k do you expect the result to be valid? Which features of the phase shift 60(k) does the eikonal result explain, and which does it not explain. Explain brieﬂy. 805m; A“): “.9. V0’ 2,2 M t1 . K m2- Explaw‘u 523m ) bwf' Mi" "@4374“ I! Show ed Weft '15 f) 36 J Clk‘AOJ QFIM’V‘ ...
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