QMGreensite_merged
6.3. THEENERGYOFENERGYEIGENSTATES 95 state”). Energyeigenstatesofhigherenergiesareknownas”excitedstates.”The factthatthegroundst ...
96 CHAPTER6. ENERGYANDUNCERTAINTY anditremainsstaticbecausetheforceontheparticlevanishes,i.e. F=− ∂V ∂x = 0 (6.67) Ontheotherhan ...
Chapter 7 Operators and Observations IsthisadaggerIseebeforeme,thehandletowardmyhand? Come,letmeclutch thee!Ihavetheenot,andyetI ...
98 CHAPTER7. OPERATORSANDOBSERVATIONS = ∫ dxψ∗(x,t)H ̃ψ(x,t) (7.1) where ̃xψ(x,t) ≡ xψ(x,t) p ̃ψ(x,t) ≡ −i ̄h ∂ ∂x ψ(x,t) H ̃ψ( ...
7.2. OPERATORSANDOBSERVABLES 99 Intermsofφx 0 , φp 0 , φn,theprobabilitiescanberewrittenasfollows: Pdx(x 0 ) = |<φx 0 |ψ>| ...
100 CHAPTER7. OPERATORSANDOBSERVATIONS or,in”component”notation ψ′(x)=O ̃ψ(x) (7.12) Alinearoperatorhasthepropertythat,foranytwo ...
7.2. OPERATORSANDOBSERVABLES 101 AnimportantsubsetoflinearoperatorsaretheHermitianoperators.AnHermitian operatorisalinearoperato ...
102 CHAPTER7. OPERATORSANDOBSERVATIONS E = <ψ|H|ψ> H ̃ψ(x) = [ − ̄h^2 2 m ∂^2 ∂x^2 +V(x) ] ψ(x) H(x,y) = [ − ̄h^2 2 m ∂^2 ...
7.2. OPERATORSANDOBSERVABLES 103 Thishermiticityrelationisquiterestrictive; forexample,itisnotevensatisfied bythesimpleoperation ...
104 CHAPTER7. OPERATORSANDOBSERVATIONS ThereforethepositionoperatorisHermitian,andisalwaysreal. HermiticityoftheMomentumOperator ...
7.3. EIGENSTATESASSTATESOFZEROUNCERTAINTY 105 wherewe have integrated twice by parts. This establishesthe hermiticity ofthe Hami ...
106 CHAPTER7. OPERATORSANDOBSERVATIONS FromthefactthateveryobservableO=x,p,E,...isassociatedwithanHermitian operatorO ̃ = ̃x,p, ...
7.3. EIGENSTATESASSTATESOFZEROUNCERTAINTY 107 Ameasurementprocessthereforehasthepropertythat,whateverthephysicalstate oftheobjec ...
108 CHAPTER7. OPERATORSANDOBSERVATIONS TheoremH2 If two eigenstates of an Hermitian operator correspond to different eigenvalues ...
7.3. EIGENSTATESASSTATESOFZEROUNCERTAINTY 109 (II)TheprobabilityP(λα)thatanygiveneigenvaluewillbetheresultof themeasurementis P( ...
110 CHAPTER7. OPERATORSANDOBSERVATIONS Finally,comparethisexpressionforagainsttheexpressionforanexpectation valuefromprobability ...
7.3. EIGENSTATESASSTATESOFZEROUNCERTAINTY 111 EigenstatesofMomentum For p ̃=−i ̄h ∂ ∂x (7.80) theeigenvalueequationis −ih ̄ ∂ ∂x ...
112 CHAPTER7. OPERATORSANDOBSERVATIONS whichhassolutions eigenstates φn(x)= √ 2 L sin[ nπx L ], eigenvalues En=n^2 ̄h^2 π^2 ...
7.4. THEGENERALIZEDUNCERTAINTYPRINCIPLE 113 thattheresultofsuchameasurementwillbetoleavethesysteminastateofzero uncertaintyinbot ...
114 CHAPTER7. OPERATORSANDOBSERVATIONS Fromthiswecanconclude thattheoperatorscorrespondingto positionalongthe x-axis,andmomentum ...
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