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Crib_Prob_Set__4

# Crib_Prob_Set__4 - 4-45 In this chapter we learned that if...

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Unformatted text preview: 4-45. In this chapter, we learned that if 1;!" is an eigenfunctien of the time-independent Sehrb‘dinger equation, then w; (-x, t) = 1,11” (.x)e'iEn”" Show that if ﬁrm and 1!!” are both stationary states of 13', then the state ﬁlth-T) 2 memfx)gwifmtfh + Cn' 1);?” (x)e—r'£nrm satisﬁes the time—dependent Schrt‘j‘ding‘et egua’tidn. Postu'iate 5 gives. the time-dependent Se’hrﬁ'din'ger equation as am; I) H"? r::'f (1‘)},1‘3? We will substitute 111 into each side of this equation separate-1y to Show that the equivalence holds. The left side becomes H‘IJ: H [cmvme—I'Em If”: +6" "(j-[”6 [‘Eh'ﬂ'] _ ' —: 5'5 Uk ‘ —t'E {ﬂu _ Emit"; 149mg, FEIICHVE’H e and the right side is . T1 It i _. ,, ' WI'E. If}: v . 7H5 1;}; I w cm'bmwrne " +- Cn En-Ipne n —iE tE, iﬁ— z ik< mcmtif'rme _‘£M'TI _ gt 8% ‘,_1E‘ '71) [3—1; Den'Ve Equation 13.2 from [3.1. Equations 13.1 are xz'rsineeosei yzrsiné‘sinqﬁ z=rcosﬂ (DJ) We use these equations to wn'te tan qb-as sin’ib rsin6(sinqb)= Z x (1) E311 : . m .45 (:05ng rsi'n'ﬂ 005d: Likewise, we can write (using 'ttigen'emet'rie identities) r2 = rﬁsin2 6- + cos.2 15‘){1=.in2 ¢ + 00.5;2 ¢) 2 :r2 sin2 6 sin2 :35 + r2 c0333 sin2 {33: + r2 sin2 9 ces2 ¢ + r1 cos:2 9 (:031 qb : (r sine sin'qﬁjl + ('r Sine sin (6)2 + (r-cos' 3)1(siri2¢ + cos2 qb) =-x2'+ yz + zz r = (33+ 3}: +22)”: (2) and 'z. = 'r Cos 9 cost? = -—z—---— (.3). (x2+y-2+ZZ)E}2. Equations 11. 2,_ and 3.3.1'6 Equations D2. 1 D—Z. Express the following points-given in Cartesian-coordinates interms of spherical coordinates; iwe‘wr'm 1»: he -. . . . I . I51 HIS 0f the heun§pherc b6 (I, A hﬂlﬂl'sphef'v Li.i‘[email protected](ln1!s If} H {. 6 4. T a”: H q) .. (I [Ctl ‘d 1 5-.) 'J 1 ‘f. (ILS Kawasaki-sharing JV 2 ,3 Sin 8drd6a'qb V2] r tiff sinﬂdﬂf 61¢: 23m i ' .______. . = ,7 k We wiil learn in Chaptcr 6 that 3 2p). hydrogen-atom orbital is-gi’mn by i \ i I = 1’5".»- ME new! sinJB sin-95 Show that 1&2” is normalized. [Don‘t forget to Siqua're will" ﬁrst.) __.__.._____—-—-—~— We want to show that > i i i i i i 1 i i i i i i 00 .n' _ 2::- 1’: f f 116,, 1%,, r sinedrd9d¢=1 0 [J ' 0 First, We square VIZ“: 1’ _ 1 r‘e" sin3 6 sin2 :1) U29, win 327: Then a. = _1,.. 'drrqeﬂf d9 sinmf Sinz¢d¢ 327! ”c.” 0 ‘3 (3—3;)‘(4‘3 (2) (35-) = and we haw: shown that V’zpy is normalized, go 5" (ts OWL (7gp): We WMWL’ 5—20. Show that R ask)” for a harmonic oscillator. Note that (3:2)”2 is the square root of the mean (hf-the square of displacement (the root-mearz-sguaredisplacement} 'of the oscillator. (x2) = f ¢2<x>xzw2mdx =§ . . or H4 1 From Table 5335533) a: (a...) (2‘1ng U643 /2_ So TE (x2) = f In cx>x2w2(x)dx . 50‘ 1/2 00 1 = 2 (—) f dr‘(2ax2 ~— -1)2x2e‘“ 43! 0 . (3) f dx('4;a2x6 .— 4M4 -+ x?)e--“’ H, _ a (§)'”[(*)(.—8—:a)‘+(£g)] s_52=. (2)1” 5—34. In thefar infrared spectrum of H79-Br,_ there is a series of lines 'separated- by 16.72 cm" . C'aIculate the values of the moment o'f'inertia and the intemuciear separatien in H79Br. 0) Assuming that H7931. can be treated as a rigid rotator, ﬁ=2étJ+ 1) 1:0, 1,2,... {5.63) h. E' = 'az'r'zcr (5-543- The lines in the spectrum are separated by 16.72 cm”, so 211 821%] game-x 10-34 "Jr-s) 833(2998 X 1010 ems")! I = 3.35 x 1'3"” kg‘m2 A5223: I632 CID—l We can ﬁnd pt [mil-179131": _ (78.9):(1 .01) 7,9. 91 (1__.661.jx 1.0-27 kg) = 1.653 ”x 10-27 kg .11 Now we can use the rel-at'iensh'ip'r- = 0/er to ﬁnd r. 3.35 x 10"? leg-m1 [/2 r— ) :1.42X10'1Om=142pm 1.653 x 10“”kg . (,2) Madam, “the Co‘r‘reI’PmM/ﬁ Cm: swam/x w the waxed]: 5ng 1k ‘3. "’ " “r m : WMM ,. 3252.. War _, be... .— —- W «33"- -— lmar [1.4.le b WW DY Wicker Q-AQD— [to "" ﬂex :- “jg/T " w PEI/“l Pg/Jn c 4 5/ 1+6 an" H I .Zom :: '- ~\ New :- %AVHt} :2 we M67 _ L1H 5—36. Figure 5. l l Compares the probability distribution associated with. the harmonic oscillator wave function \$109?) to the classical distribution. This problem 1llust1 ates what 13 meant by the classical distribution. Consider x-(t) z A si‘n(wt + (15} which can be written as ' Now m“d;x _ A2 __ x2 12.. 11 II (1) This equation gives the time that the oscillator spends between x and x + dx. We can convert _ Equation 1 to a probability distribution 1n 1: by dividing by the time that it takes fer the oscillator to .go from— A to A. Show that this time is idea and that the- probability distribution 1n x is or p(x)dx = -—-_-—-“ (2) rw‘ A2 7 it2 Show that p{x_} is normalized. Why does p(x) achieve its maximum value at x : :IzA? Now use the fact that; = omx, where e: : (It-1.1 #12)” 2, to show that Cir? Pléld’i = —'—."'— fix/01142 —-§2 Show. that the limits of 1? are :l:(o:A2)”2 =-:l:(_2-1)"2 and compare this result to the vertical lines shown in Figure 5.11 [Hint You "need to use the fact that 1041/2: E") (11 == 10)] Finally. plot Equation 3 and compare your result with the curve in Figure 5 1 l. (3) F I G U R E 5.11 The probability distribution function of a harmonic oscillator in the v = 10 state. The dashed line is that for a classical harmonic oscillator with thetsam'e energyi-The'verticai linesat if m :l:4.6 represents the extreme liinits‘ of. the classical harmonic motion; The variable a) is the angular velocity of the oscillator, deﬁned as to = law where v is in cycles per second. In going from —wA to A the function xtr) goes through 5 cycle, so .211" (% cycle} If w.= __________.____ .3: .... t w- 5 . ,,V....‘-.‘, .muucib Substitute w" = 12:" into Equation l and divide by t: (It _ dx RCA-2' h It? Interpreting dr/r as a prdbabiiity distribution inux, we ﬁnd :1- pixja'x :: —~———x The maximum values of p(:) an: at c r :izA because these we the points at which the classical ha: manic osciiiator has zero velocity Suhsti lutingE -: amx and dé‘ : amdx d‘é‘ Hm pCéMé‘ : Since the limits ofx an: iA, the limits of é“ arc ia'b'A— = fwz. But kA /2~ —..-- (42— w 215mm”— A150 at = (“kW-m. so (out )'/?._.—~(21)'f2 given by the dashed cur-v3 in Figme 5. J 1.- 5:10 3,1,hw 30 ~— 4. 58 The plot of Equation 3 IS QMWWL Praia (ms . L 'Fvam 171mg macheat 533W; We am See , ct) A4; zero 1701M , Jake pmbmhﬁ 95f cuss/EM Mitt/Mar Va has maximum WW , «uh/ate the Wit-Bum (ﬁlm-2W has if; max/imam WW6 . 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Dame ~ L is ‘ (CR) @3230: [<E1>- <9] , ¢ 1 .L I @th 24:53-5“? :0 Dane . ...
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Crib_Prob_Set__4 - 4-45 In this chapter we learned that if...

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