QMGreensite_merged
7.4. THEGENERALIZEDUNCERTAINTYPRINCIPLE 115 wherebissomeconstant.Inthatcase,fromthedefinitionofφ′, B ̃φa(x)=bφa(x) (7.107) Thisp ...
116 CHAPTER7. OPERATORSANDOBSERVATIONS whichprovesthe”onlyif”portionoftheorem. Theproofaboveextendstriviallytocontinuousnon-dege ...
7.4. THEGENERALIZEDUNCERTAINTYPRINCIPLE 117 andsquaringbothsides,wehave |<ψ 1 |ψ 1 >||<ψ 2 |ψ 2 >| ≥ |<ψ 1 |ψ 2 & ...
118 CHAPTER7. OPERATORSANDOBSERVATIONS 7.5 The Time-Energy Uncertainty Relation Itwasnoted,attheendofthelastlecture,thatchangeof ...
7.5. THETIME-ENERGYUNCERTAINTYRELATION 119 1 ̄h |<ψ|[Q,H]|ψ>|∆t = ∆Q (7.132) Wenow applythegeneralized uncertaintyprincipl ...
120 CHAPTER7. OPERATORSANDOBSERVATIONS ...
Chapter 8 Rectangular Potentials Mostoftheeffortinsolvingquantum-mechanicalproblemsgoesintosolvingthetime- independentSchrodinge ...
122 CHAPTER8. RECTANGULARPOTENTIALS ticleintheclassicallyforbiddenregions,andthisfactgivesrisetosomeremarkable phenomenawhichare ...
8.1. AQUALITATIVESKETCHOFENERGYEIGENSTATES 123 Nowsupposetheenergy isintherangeE∈[0,Vmax]. Thentherewillbeclas- sicallyforbidden ...
124 CHAPTER8. RECTANGULARPOTENTIALS changeinanyofthoseeigenvaluesresultsinanon-physicalstate,theboundstate energiesalwaysformadi ...
8.2. UNBOUNDSTATESANDQUANTUMSCATTERING 125 Becauseφ(x)= 0 forx> 0 andbecausethewavefunctioniscontinuousatx=0,we musthave φ(x) ...
126 CHAPTER8. RECTANGULARPOTENTIALS Inanyrealscatteringexperiment,ofcourse, thereisalwayssome finiteuncer- taintyinthemomentumof ...
8.3. THESTEPPOTENTIAL 127 andtheTransmissionCoefficient T = Itrans Iinc = |C|^2 |A|^2 = no.ofparticles/sectransmitted no.ofparti ...
128 CHAPTER8. RECTANGULARPOTENTIALS ThechoiceofAandDisachoiceof theinitialstate. IfD= 0 thentheparticle is initially approaching ...
8.3. THESTEPPOTENTIAL 129 whilethereflectioncoefficientis R = v 1 |B|^2 v 1 |A|^2 = (p 2 −p 1 )^2 (p 2 +p 1 )^2 (8.29) Noticetha ...
130 CHAPTER8. RECTANGULARPOTENTIALS Butinthiscase R = |B|^2 |A|^2 = 1 (8.36) whichmeansthat all incidentparticlesarereflected; n ...
8.4. THEFINITESQUAREWELL:BOUNDSTATES 131 BoundStates E< 0 Webeginwithadiscussionoftheboundstates.InRegionsIandIIItheSchroding ...
132 CHAPTER8. RECTANGULARPOTENTIALS Thisproves that φ(x)is alsoa solution. Thecrucial factusedin(8.46) wasthe symmetryofV(x)arou ...
8.4. THEFINITESQUAREWELL:BOUNDSTATES 133 EvenParityBoundStates Therequirementthatφ(x)=φ(−x)alsoimpliesthatA=B. Sowehavealto- get ...
134 CHAPTER8. RECTANGULARPOTENTIALS Thistime, the requirement that φ(−x) = −φ(x)implies that B = −A, and CE=0. Thewavefunctionha ...
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