QMGreensite_merged
13.1. SPINWAVEFUNCTIONS 215 where Ω= eB Mc (13.62) Ωisknownasthe”cyclotronfrequency.”TheeigenstatesofthisHamiltonianarethen simp ...
216 CHAPTER13. ELECTRONSPIN Notethat(i)σyisanHermitianmatrix;and(ii)theeigenvaluesarebothreal. Aswe foundlastsemester,theeigenva ...
13.1. SPINWAVEFUNCTIONS 217 Notethattheinnerproduct αy·βy= 1 2 [1,−i] [ 1 −i ] = 1 +(−i)^2 = 0 (13.78) vanishes,soαyandβyareorth ...
218 CHAPTER13. ELECTRONSPIN LetT = 2 π/ΩbetheperiodcorrespondingtoanangularfrequencyΩ. Then,from thesolution above, weseethat th ...
13.1. SPINWAVEFUNCTIONS 219 However,wereallywanttheinteractionenergyintherestframeofthelaboratory, whichisalso (approximately)th ...
220 CHAPTER13. ELECTRONSPIN Now [J^2 ,L^2 ]=[J^2 ,S^2 ]= 0 (13.94) sinceL^2 andS^2 commutewithallthecomponentsofJ%=L%+S%,andther ...
Chapter 14 The Addition of Angular Momentum ”Considertheliliesofthefield,howtheygrow.Theytoilnot,neitherdotheyspin...” TheBookof ...
222 CHAPTER14. THEADDITIONOFANGULARMOMENTUM SothismeansthatJz=Lz+SzmustcommutewithJ^2. Whatwewanttodoisto constructeigenstatesof ...
223 FromthisoneeigenstateofJ^2 (thehighestweightstate),itissimpletoconstruct alltheothereigenstatesofJ^2 andJz,bysuccessivelyapp ...
224 CHAPTER14. THEADDITIONOFANGULARMOMENTUM andweget Φ (^32) − 12 = √ 2 3 Y 10 χ−+ √ 1 3 Y 1 − 1 χ+ (14.17) Theremainingstatewit ...
225 butthesearenoteigenstatesofJ^2 .Sowelookforalinearcombination Φ 1212 =aY 11 χ−+bY 10 χ+ (14.24) whichhastheorthonormalitypro ...
226 CHAPTER14. THEADDITIONOFANGULARMOMENTUM Theset of allsix eigenstatesof J^2 ,Jz thatcan be formedfromthe sixstates Y 1 mχ±, t ...
227 asmalleffect,thereisapowerfulmethodknownastime-independentperturbation theorywhichallowsoneto”sneakup”ontheeigenstatesandeig ...
228 CHAPTER14. THEADDITIONOFANGULARMOMENTUM splittingisverysmall;wecan seethatthecorrectiontoatomicenergiesisonthe orderofthesqu ...
14.1. THEGENERALMETHOD 229 or,inketnotation {|j 1 ,m 1 ,j 2 ,m 2 >} (14.46) witheigenvalues J 12 ψj^11 m 1 ψj^22 m 2 = j 1 (j ...
230 CHAPTER14. THEADDITIONOFANGULARMOMENTUM whichwenowquicklyverify. Suppose,e.g.,thatj 1 >j 2 ,sothatjmin=j 1 −j 2 .Then N′ ...
14.1. THEGENERALMETHOD 231 andthenormalizationcondition <Φjmax− 2 ,jmax− 2 |Φjmax− 2 ,jmax− 2 >= 1 (14.62) Ladderoperators ...
232 CHAPTER14. THEADDITIONOFANGULARMOMENTUM isdeterminedbytheorthogonalityandnormalizationconditions <Φ 10 |Φ 00 > = 0 < ...
14.1. THEGENERALMETHOD 233 SothecontributiontotheHamiltonianfortheelectron,duetotheinteractionofthe electronmagneticdipole momen ...
234 CHAPTER14. THEADDITIONOFANGULARMOMENTUM Then,usingthefactthatfortheHydrogenatomgroundstate |ψ(r=0)|^2 = 1 πa^30 (14.83) with ...
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