QMGreensite_merged
5.5. GAUSSIANWAVEPACKETS 75 andthetimeittakestoexpandtosomeverymuchlargersizea(t)=A>>a tA= maA ̄h (5.79) Asanexample,choos ...
76 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE 5.6 Group Velocity and Phase Velocity We have seen that the expectation value of position ...
5.6. GROUPVELOCITYANDPHASEVELOCITY 77 Thecorrespondingexponentialscanbeexpandedintosinesandcosines,theproduct oftwocosinesisshow ...
78 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE Inthecaseofafreeparticle,wehave vgroup = ( dω dk ) k=k 0 = ( d(Ep/ ̄h) d(p/ ̄h) ) p=<p ...
5.7. THEPARTICLEINACLOSEDTUBE 79 whichsimplymeansthattheprobabilityoffindingtheparticleoutsidethetubeis zero. Thesolutionofadiff ...
80 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE Itisusefultonormalizeφn(x),sothatthesolutiontothetime-dependentSchrodinger equation ψn(x, ...
5.7. THEPARTICLEINACLOSEDTUBE 81 Integrateoverx ∫L 0 dxφ∗k(x)ψ(x,0)= ∑∞ n=1 an ∫L 0 dxφ∗k(x)φn(x) (5.121) andusetheorthogonality ...
82 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE wherewehavedefined Xmn ≡ ∫L 0 φ∗m(x)xφn(x) = 2 L ∫L 0 dxxsin[m πx L ]sin[n πx L ] = ...
5.7. THEPARTICLEINACLOSEDTUBE 83 Therealpartof ψ(x,0) issketched inFig. [5.4]. Then thecoefficientsanare easilycalculated: an = ...
84 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE ...
Chapter 6 Energy and Uncertainty Wehaveseenthatforagaussianwavepacket φ(x)= 1 (πa^2 )^1 /^4 e−(x−x^0 ) (^2) / 2 a 2 eip^0 x/ ̄h ...
86 CHAPTER6. ENERGYANDUNCERTAINTY 6.1 The Expectation Value of p n AccordingtotheBornInterpretation,theexpectationvalueofanyfunc ...
6.1. THEEXPECTATIONVALUEOFPN 87 Then,usingthedeltafunctiontoeliminateoneofthep-integrations,wefind <p>= ∫ dpp f∗(p)f(p) 2 ...
88 CHAPTER6. ENERGYANDUNCERTAINTY 6.2 The Heisenberg Uncertainty Principle Letususe(6.16)tocomputethemomentumuncertainty∆pforthe ...
6.2. THEHEISENBERGUNCERTAINTYPRINCIPLE 89 TheHeisenbergUncertaintyPrinciple It is impossible, by any measurement process, to sim ...
90 CHAPTER6. ENERGYANDUNCERTAINTY inpositionandmomentumexistindependentofany”disturbance”ofthesystemby observation, andthatthese ...
6.2. THEHEISENBERGUNCERTAINTYPRINCIPLE 91 Thetotalenergyisminimizedford/dR=0,andthisminimumisobtained at R= 3 ̄h^2 4 me^2 (6.32) ...
92 CHAPTER6. ENERGYANDUNCERTAINTY 6.3 The Energy of Energy Eigenstates Wehaveseenthattheexpectationvalueofenergyisgivenby <E& ...
6.3. THEENERGYOFENERGYEIGENSTATES 93 Ontheotherhand,usingtheintegrationbypartsformula(5.7), Hmn = ∫ dx { − ̄h^2 2 m φ∗m ∂^2 φn ∂ ...
94 CHAPTER6. ENERGYANDUNCERTAINTY Nowrecalltheexpression(4.67),fromprobabilitytheory,fortheexpectationvalue ofanyquantityQ = ∑ n ...
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