Statistical Mechanics2014 HW4英文学习材料

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1、StatisticalMechanics2014HW4ClassicalMechanics1Abeadofmassmslideswithoutfrictionalongawirewhichhastheshapeofaparabolay=Ax2withaxisverticalintheearth'sgravitationalfieldg.a)FindtheLagrangian,takingasgeneralizedcoordinatethehorizontaldisplacementx.b)WritedownLagrange'sequationofmotion.2Adoubleplanepen

2、dulumconsistsoftwosimplependulums,withonependulumsuspendedfromtheboboftheother.The“upper”pendulumhasmassm1andlengthl1,the“lower”pendulumhasmassm2andlengthl2,andbothpendulumsmoveinthesameverticalplan.a)FindtheLagrangian,usingasgeneralizedcoordinatestheanglesh1andh2thependulumsmakewiththevertical.b)W

3、ritedownLagrange’sequationsofmotion.3ThemotionofaparticleofmassmisgivenbyLangrange’sequationswithLagrangianLexpαt/mTV1222whereaisconstant,Tmxyzisthekineticenergy,andV=V(x,y,z)is2thepotentialenergy.Writedowntheequationsofmotionandinterpret.2p1224TheHamiltonianforasimpleharmonicoscilla

4、torisHmωx.2m2Introducethecomplexquantitiesmωipmωipaxanda*x.2mω2mωa)ExpressHintermsofaanda*.b)EvaluatethePoissonbrackets[a,a*],[a,H]and[a*,H].c)Writedownandsolvetheequationsofmotionforaanda*.5ConsidermotionofaparticleofmassminagravitationalpotentialV=-k/randtaketheorbitalplanetobet

5、hex-yplane.TheHamiltonianisthen122kHppxy2mr22Wherenowrxy.Theangularmomentumvectorpointsinthez-directionandhas(z-)componentLxpyypx,andtheLaplace-Runge-Lenzvectorliesinthex-yplaneandhascomponentsKpLmkx/r,KpLmky/r.xyyxa)ShowthatL,H0,Kx,H0,Ky,H0,soL,Kx,andKyareconstantsofthemotio

6、n.b)ShowthatKx,L-Ky,Ky,L-Kx,Kx,Ky-(2mH)L.QuantumMechanics6Supposethatthewavefunctionofa(spinless)particleofmassmisαrβree(r,θ,φ)ArwhereA,andareconstantssuchthat0<<.FindthepotentialV(r,,)andtheenergyEoftheparticle.7TheeigenvalueSchr̈odingerequationforaparticleinonedimensionis2"

7、(x)V(x)(x)E(x).EEE2mItisconvenienttocasttheequationindimensionlessform:ifaisalengthintrinsictotheproblem–i.e.constructedfromtheconstantsthatappearintheequation(,mandthoseappearingintheexpressionforV(x)