QMGreensite_merged
8.4. THEFINITESQUAREWELL:BOUNDSTATES 135 Theenergiesarenon-degenerate; Eachenergyeigenstateiseitherevenparityoroddparity,andthi ...
136 CHAPTER8. RECTANGULARPOTENTIALS where Aei √ 2 mEx/ ̄h representstheincomingwavepacket Be−i √ 2 mEx/ ̄h representsthereflecte ...
8.4. THEFINITESQUAREWELL:BOUNDSTATES 137 substituteinto(8.74) Ae−ika+Beika = [ 1 2 (1+ k q )ei(k−^2 q)a+ 1 2 (1− k q )ei(k+2q)a ...
138 CHAPTER8. RECTANGULARPOTENTIALS Inthiscase T= 1 and R= 0 (8.83) i.e.thereisnoreflectionatall.Aparticleofthisenergyiscertaint ...
8.5. TUNNELLING 139 cansimplytaketheformulasforthescatteringstateoftheattractivewell,replace V 0 everywhereby−V 0 ,andthesewillb ...
140 CHAPTER8. RECTANGULARPOTENTIALS andthetransmissioncoefficientisapproximately T≈e−^4 Qa=exp[− 4 √ 2 m(V 0 −E)a/h ̄] (8.94) Th ...
Chapter 9 The Harmonic Oscillator ThereareonlyaveryfewpotentialsforwhichtheSchrodingerequationcanbesolved analytically. Themosti ...
142 CHAPTER9. THEHARMONICOSCILLATOR isalinearequationofmotion. Thisequationmightdescribeatransverse waveon astring,orasoundwavei ...
9.1. RAISINGANDLOWERINGOPERATORS 143 Thenthereisasimpleidentity a^2 +b^2 =c∗c (9.10) Letsfindthecorrespondingidentityforoperator ...
144 CHAPTER9. THEHARMONICOSCILLATOR oscillator,however,thiscommutatorisverysimple: [A,B] = √ 1 2 m √ 1 2 kx^2 , ̃p = √ k ...
9.1. RAISINGANDLOWERINGOPERATORS 145 Intermsoftheraisingandloweringoperators,wehave H ̃ = ̄hω(a†a+^1 2 ) x = √ ̄h 2 mω (a+a†) p ...
146 CHAPTER9. THEHARMONICOSCILLATOR Thecommutationrelation(9.26)alsosaysthat aa†=a†a+ 1 (9.35) so H ̃φ” = ̄hω[a†(a†a+1)+^1 2 a†] ...
9.1. RAISINGANDLOWERINGOPERATORS 147 whichhastheuniquesolution φ 0 =Ne−mωx (^2) /2 ̄h (9.42) TheconstantN isdeterminedfromthenor ...
148 CHAPTER9. THEHARMONICOSCILLATOR Proceedinginthisway,wefindaninfinitesetofeigenstatesoftheform φn = cn(a†)nφ 0 n= 0 , 1 , 2 , ...
9.1. RAISINGANDLOWERINGOPERATORS 149 = (a†a+1)(a†)n−^1 = a†a(a†)n−^1 +(a†)n−^1 = a†(a†a+1)(a†)n−^2 +(a†)n−^1 = (a†)^2 (a†a+1)(a† ...
150 CHAPTER9. THEHARMONICOSCILLATOR Rescalingx y= √ mω ̄h x (9.62) Thegeneralsolutionbecomes φ(x) = 1 √ n! (mω π ̄h ) 1 / 4 ( 1 ...
9.2. ALGEBRAANDEXPECTATIONVALUES 151 andwecanalsoderiveasimilarequationforaφn: a|φn> = 1 √ n! a(a†)n|φ 0 > = 1 √ n! [(a†)n ...
152 CHAPTER9. THEHARMONICOSCILLATOR + √ n<φn|a†|φn− 1 >+ √ n+ 1 <φn|a†|φn+1> = √ n(n−1)<φn|φn− 2 >+(n+1)<φn ...
Chapter 10 Symmetry and Degeneracy Weliveinaworldwhichisalmost,butnotquite,symmetric. TheEarthisround, nearly,andmovesinanorbita ...
154 CHAPTER10. SYMMETRYANDDEGENERACY figure. Ontheotherhand,supposeeverypoint(x′,y′)ontheoriginal sketchwere movedtoanewpoint(x′ ...
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