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11.2. THEEIGENVALUESOFANGULARMOMENTUM 175 11.2 The Eigenvalues of Angular Momentum TheenergyeigenvaluesofH ̃,foraparticleinacent ...
176 CHAPTER11. ANGULARMOMENTUM wehave L ̃^2 = L ̃+L ̃−− ̄hL ̃z+L ̃^2 z = L ̃−L ̃++[L ̃+,L ̃−]− ̄hL ̃z+L ̃^2 z = L ̃−L ̃++h ̄L ̃z ...
11.2. THEEIGENVALUESOFANGULARMOMENTUM 177 Therefore, L ̃+φab=Cab+φa,b+1 (11.33) whereCab+ isa constant. This establishesthe fact ...
178 CHAPTER11. ANGULARMOMENTUM Now,we mustbeableto reachthehighest stateφabmax byacting successively on φabminwiththeraisingoper ...
11.3. THEANGULARMOMENTUMCONES 179 whichimpliesthat: bav = 0 bmax = n 2 bmin = − n 2 a^2 = n 2 (n 2 + 1 ) (11.49) Itiscustomaryto ...
180 CHAPTER11. ANGULARMOMENTUM Confirmationof theseresultscomesfromNature. Allelementaryparticles, for example, have an intrinsi ...
11.4. EIGENFUNCTIONSOFANGULARMOMENTUM 181 whilevaluesofLxandLyareindefinite;althoughtheirsquaredexpectationvalues mustsatisy < ...
182 CHAPTER11. ANGULARMOMENTUM becauseinsphericalcoordinates,thervariabledropsoutoftheangularmomentum operators: L ̃x = i ̄h ( s ...
11.4. EIGENFUNCTIONSOFANGULARMOMENTUM 183 andwesolvethefirst-orderdifferentialequations L ̃+Yll= 0 L ̃zYll=lhY ̄ lm (11.70) byth ...
184 CHAPTER11. ANGULARMOMENTUM or d dθ A(θ)=lcotθA(θ) (11.81) whichissolvedby A(θ)=const.×sinlθ (11.82) Then Yll(θ,φ)=Nsinlθeilφ ...
11.4. EIGENFUNCTIONSOFANGULARMOMENTUM 185 sothat Cl+m= ̄h √ (l−m)(l+m+1)eiω (11.91) Likewise <φlm|L ̃+L ̃−|φlm> = <φlm| ...
186 CHAPTER11. ANGULARMOMENTUM Asanexampleoftheprocedure,letuscomputethel= 1 multipletofspherical harmonics,i.e.Y 11 , Y 10 , Y ...
11.4. EIGENFUNCTIONSOFANGULARMOMENTUM 187 havejustcomputed,are Y 00 = √ 1 4 π Y 11 = − √ 3 8 π sinθeiφ Y 10 = √ 3 4 π cosθ Y 1 , ...
188 CHAPTER11. ANGULARMOMENTUM andtheenergyeigenstatesareangularmomentumeigenstates φlm(θ,φ)=Ylm(θ,φ) Elm= 1 2 I l(l+1) ̄h^2 (11 ...
11.5. THERADIALEQUATIONFORCENTRALPOTENTIALS 189 Uponquantization,r,%pand%Lbecomeoperators, r ̃F(r,θ,φ) = rF(r,θ,φ) %pF ̃ (r,θ,φ) ...
190 CHAPTER11. ANGULARMOMENTUM andthenjustexpressing∇^2 insphericalcoordinates ∇^2 =( 1 r ∂^2 ∂r^2 r)+ 1 r^2 ( 1 sinθ ∂ ∂θ sinθ ...
11.5. THERADIALEQUATIONFORCENTRALPOTENTIALS 191 TheFreeParticle ThesimplestpossiblecentralpotentialisV(r)=0. Ifwe write E= ̄h^ ...
192 CHAPTER11. ANGULARMOMENTUM Forthefreeparticleinonedimension,wefoundthatE,Pwasacompletesetof observables,wherePisparity,andal ...
Chapter 12 The Hydrogen Atom Historically, thefirstapplicationoftheSchrodingerequation(bySchrodingerhim- self)wastothe Hydrogena ...
194 CHAPTER12. THEHYDROGENATOM whichreduces,asshowninthepreviouslecture,tothe”radialequation”forRkl(r) d^2 Rkl dr^2 + 2 r dRkl d ...
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