QMGreensite_merged
4.4. EXPECTATION,UNCERTAINTY,ANDTHEQUANTUMSTATE 55 Asanexample,weprovethethirdoftheseidentitiesusingtheformulaforinte- grationby ...
56 CHAPTER4. THEQUANTUMSTATE Aphysicalstatehasthe propertythat,giventhestateat sometimet,wecan determinethestateataslightlylater ...
4.4. EXPECTATION,UNCERTAINTY,ANDTHEQUANTUMSTATE 57 ThedeviationoforderNp−^1 /^2 ineq. (4.61)hasthesameorigin,andispresentinany s ...
58 CHAPTER4. THEQUANTUMSTATE TheBornInterpretationenablesustopredictthevaluesofthesetwonumbers, giventhequantumstate|ψ>.These ...
4.4. EXPECTATION,UNCERTAINTY,ANDTHEQUANTUMSTATE 59 Likewise,theexpectationvaluefortheobservablex^2 is <x^2 >= ∫∞ −∞ x^2 ψ∗ ...
60 CHAPTER4. THEQUANTUMSTATE ...
Chapter 5 Dynamics of the Quantum State Historyisjustonedamnthingafteranother. Anonymous Theclassicalmotionofaparticleisrepres ...
62 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE Ehrenfest’sPrinciple d dt <qa> = < ∂H ∂pa > d dt <pa> = −< ∂H ∂qa &g ...
5.2. THESCHRODINGERWAVEEQUATION 63 Equation(5.8) shouldbeunderstoodas aprediction, giventhequantumstate ψ(x,t),fortheexpectation ...
64 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE ThenrewritingtheSchrodingerequation(anditscomplexconjugate)as ∂ψ ∂t = i ̄h 2 m ∂^2 ψ ∂x^2 ...
5.2. THESCHRODINGERWAVEEQUATION 65 = ∫ dxψ∗ ( − ∂V ∂x ) ψ = <− ∂V ∂x > (5.15) exactlyasrequiredbyEhrenfest’sprinciple. Wes ...
66 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE Finally, weneedtochecktheconsistency oftheSchrodingerequationwiththe BornInterpretation. ...
5.3. THETIME-INDEPENDENTSCHRODINGEREQUATION 67 andonecanthenreadilyverifythe2ndEhrenfestequation ∂t<%p>=<−∇V > (5.27 ...
68 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE or,moreexplicitly [ − ̄h^2 2 m d^2 dx^2 +V(x) ] φ(x)=Eφ(x) (5.35) isknownastheTime-Indepe ...
5.4. THEFREEPARTICLE 69 5.4 The Free Particle WhenthepotentialV(x)vanisheseverywhere,theSchrodingerequationinonedi- mensionreduc ...
70 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE Thesetofeigenfunctionsandcorrespondingeigenvalues { φp(x)=eipx/ ̄h,Ep= p^2 2 m } −∞<p& ...
5.5. GAUSSIANWAVEPACKETS 71 Wenowhaveaprescription,giventhewavefunctionatanyinitialtimet=0,for findingthewavefunctionatanylatert ...
72 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE Webeginbycomputingtheinitialexpectationvalues 0 , 0 andtheinitial uncertainty∆x 0 inposi ...
5.5. GAUSSIANWAVEPACKETS 73 thisbecomes f(p)=(4πa^2 )^1 /^4 e−a (^2) (p−p 0 ) (^2) /2 ̄h 2 (5.67) Then,substitutingf(p)into(5.44 ...
74 CHAPTER5. DYNAMICSOFTHEQUANTUMSTATE = 1 √ πa^2 (t) ∫ dx(x−v 0 t)^2 exp [ − (x−v 0 t)^2 a^2 (t) ] = 1 √ πa^2 (t) ∫ dx′x′^2 exp ...
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