Question

4. Assume that a particle is in an eigenstate (lm)-a common eigenstate of Îand Îz. (a) Evaluate (îz) and (îy) in that state, where (...) = (lm| .- |lm). (Hint: Express În and Î, in terms of Ît and Î_ and use the result of Problem 3b.) (b) Also evaluate (îz) and (iz) in that state. (c) For a given value of l find the minimum value of Al, and AÎy, where A(...) = V((...)2) – ((...))2. Discuss what happens to the general Heisenberg uncer- tainty relation AÎnai, 2 KL, Îy) in that case.

Answer 1

t we have It = bxtily la lucily. so, eu = lxth a ly = lt - 1 Nov (L): 1 + m ) ta | Lọ) - ( 2 + + +/- 2 m) (la) [ <em)ltlem) + <lm le 1em] we have l+ 11 m) = e-m) (e+m+1) k l l, mely l-lem>=r8+m) 11-m+1) t 1 l, m-> sor (1x) = žaf com).re-myc et mary k le, mti) + < hom[18+o) {t=malgk/. m. 1] oth ( 56 – ) (1+0=1)" {ren 10, m+1) + {E=»)(e-m0-1){ cm|1, mna? . 1 ->)Cevnt) Óm, ntr + [Cs)<1-041) Sm, m-) zo Hene delta function Sm, mer it. Bomo os 6 m, may = 0, Sm, me =0 So, (ln) = 0 Here delta m, mt) m = D co man man n = m+1)

Again (ly) = <em) ex-ley 1 lmy [ Glombream) (lama) dile, m+1) - <em / eta Jie-mey) k)mp] the [le+m) ((a+ma) Smymicom - scetu) ( = m+) &nm-Jö © ¢ = (1, 72, J2 = 6ite? tex lut Raket ich lei) = <emy bez teplem Ett rusl ez sem> t<emle?/em+ Floletonfendu lavellons + . t ltl tllt lt

eflem) = lup+Ilm) = (two ) (6+m+1)! h 18. mn+) 10-/++1) hill - m)Ce+m71) + le, m+1+1) = -m) (e+n+1) *2 1 l, m+2) - ☺ Similanly e² Ilm) = a tom) (e-m+1) toll, m.) = {c+m) (p=m+1) t? ll, m-2) - @ Otf-l lim) = pt ret m) (e-m +1} [l, m-1) = Me+m) (d-m) 16-n+1)(e-m-1) *2 1 , m-1+)) = 1 m2 Vl? - (Mt.)? 6²1.m) & (-64/6m) = (-/--) (44 m#)'k 12, m+1) = (2-satni sez-Antinya tid 1 Bom) - - Clemens *[ <=)(e+n +1)<? Kt, mH, m42) +(79) (6-mav) k?zem/1, 0-2 + {2-67822_ n =))2 *?{ae, mlem) + 40,412m) 7] a to o [ «-»> <cmaSamoa + (@=)(e-ma) Sm, m-2 + a fer(? foro) como - 4 [ P-><e+mu) sa o + (c+m><e-mou) mm? + 2 112-m2) (22-6+14). 1] Et 2.212"-ma) (22- fra) 27 : (012) - Het Me 2 (1+(mm)2) man? find the of the value lim Put from Problem - 3b. You will 16x2} & also Lly).

. (2:12 Similanly (62) = <lm / (li-ly? lem] 18)= -*[ semillalt Ilm] + Kem) (-2-1 emn) ~euberg-Ilan? -Kemle-ft1 2017 ] Put the value of can ☺ ☺ ☺ & © . . 14)=(m) Ce+mu) kilml , m+ 2) + Ce+m)(e-mai) <emle, m-2) -2,10+ m2) (P2-pay")" <inlem (C--){c+ma) Smeet+ (6 *» )/ –90) Formation -2,142 m2)(82 magh)" Sme, m? J Megy re->) (-pouse) Pa © sin = 422) - (en)? - Alm=522 m) (1?- mizig? -o - le (p* – m?)(22 – for:92) * similarly, Aly = (1, 2) - (ly)? Alya (CC - my) (p=104172) f Now, Almoly = th 2 12 - m²) ( p² - (M+7)] a ciu, by] = (enly - byla) [[ en z ly]) = <lm / enlyn by en 1 lon) R = 2 lm / luly/em>- <eml fylu (kuy>

re++l- + -b) 1+ + Olt - ll-la 21 en by = (C++4-) (1-4) - tilbake tenkt - Ipl -1-1-) ly live = (49 3:43)(4+4- tit (halt +94 l--8-14-1-1-) .([lasty]) = {lmi allt h4 +8+14 - _-ta)/em> : -<em) will+l+ + p lt - latt-2-1)/em> use ean . 9 . © ¢ © Elm, by]> T(-1) (pa mtu) k ? Kembe,mta) o (+m) (p= m +1) 6*2 Complem-7) + 12-m2] (p? - for as 2 Kempomy-te? m?) (l? (mnt3}<lam lem? ] - (-m) (@=m+1) ki? 4em1l, m+2) + {{+m) (2 - maty koLlan 16, m-2) -122- ] (72-(+1)%)'<em lem) + Cl2-m2) 627-porn) <cullen?]] 20 - Thuy Alu Aly = 2 ( 67 m2) ( 1² - (04) 32) if lam Alu Aly > 0 e Cenly])=0 . Alusly 7 é {(lu, ly]> Thuy