QMGreensite_merged
10.1. THEFREEPARTICLE,ANDMOMENTUMCONSERVATION 155 Ifwerelabelthecoordinatesx→x′, H ̃[−i ̄h ∂ ∂x′ ,x′]φα(x′)=Eαφα(x′) (10.9) andt ...
156 CHAPTER10. SYMMETRYANDDEGENERACY Then TH ̃[ ∂ ∂x ,x]ψ(x) = H ̃[ ∂ ∂x′ ,x′]ψ(x′) = H ̃[ ∂ ∂x ,x]ψ(x′) = H ̃[ ∂ ∂x ,x]Tψ(x) (1 ...
10.2. PARITY 157 Wehavealreadynotedthattheenergiesofthefreeparticleare2-folddegenerate; i.e.foreachenergyEtherearetwolinearlyind ...
158 CHAPTER10. SYMMETRYANDDEGENERACY thereforetheHamiltoniansofthefreeparticle,theHarmonicoscillator,andthefinite squarewellarei ...
10.2. PARITY 159 underx→−x;itisthereforeaneven-parityeigenstate.Theraisingoperator,onthe otherhandtransformsas a†(−x,−∂x)=−a†(x, ...
160 CHAPTER10. SYMMETRYANDDEGENERACY reverseorder,firstx′=−xandthenx′′=x′+a,theparticleendsupatx′′=−x+a, whichisadifferentpoint. ...
10.3. THEPARTICLEINASQUARE 161 where V(x,y) = v(x)+v(y) v(x) = { 0 0 <x<L ∞ otherwise (10.54) ThevariablesintheHamiltonian ...
162 CHAPTER10. SYMMETRYANDDEGENERACY Acompletesetofenergyeigenstatesisthen { φnm(x,y)= 2 L sin[ nπx L ]sin[ mπy L ], Enm=(n^2 +m ...
10.4. THEQUANTUMCORRAL 163 Therefore β^2 = 1 =⇒ β=± 1 (10.71) Becauseφnm(x,y)andφmn(x,y)havethesameenergy,sodoesanylinearcombi- ...
164 CHAPTER10. SYMMETRYANDDEGENERACY andtheLaplacianoperator∇^2 ,intwodimensions,isgiveninpolarcoordinatesby ∇^2 ≡ ∂^2 ∂x^2 + ∂^ ...
10.4. THEQUANTUMCORRAL 165 Inthreedimensions,angularmomentumaroundthez-axisisgivenby Lz=xpy−ypx (10.86) Intwodimensions,thisisth ...
166 CHAPTER10. SYMMETRYANDDEGENERACY andthesearetheonlyvalueswhichcouldresultfromanaccuratemeasurement. In short,thevaluesofangu ...
10.4. THEQUANTUMCORRAL 167 Thisisaboundstatesolution,sincetheparticleisboundbythepotentialinsidea circle,andweknowthattheenergie ...
168 CHAPTER10. SYMMETRYANDDEGENERACY 10.5 Complete Sets of Observables Tosummarizethelessonsofthislecture: Iftwo(ormore)hermiti ...
10.5. COMPLETESETSOFOBSERVABLES 169 Example:Itwillbeseeninthenextlecturethattheenergydegeneracyofaspherically symmetricHamiltoni ...
170 CHAPTER10. SYMMETRYANDDEGENERACY ...
Chapter 11 Angular Momentum Whenthepotentialenergyinthreedimensionsdependsonlyontheradialcoordinate, thepotentialissaidtobe”sphe ...
172 CHAPTER11. ANGULARMOMENTUM 11.1 The Angular MomentumCommutators Angularmomentuminclassicalphysicsisdefinedas L%=%r×%p (11.3) ...
11.1. THEANGULARMOMENTUMCOMMUTATORS 173 where [L ̃z,V(r)] = x[py,V(r)]−y[px,V(r)] = −i ̄h ( x ∂V ∂y −y ∂V ∂x ) = −i ̄h ( x ∂V ∂r ...
174 CHAPTER11. ANGULARMOMENTUM eigenstatesofangularmomentum,byanalgebraicmethodswhichcloselyresembles thetreatmentoftheharmonico ...
«
13
14
15
16
17
18
19
20
21
22
»
Free download pdf