东北大学《大学物理》课件-第25章Quantum mechanics

wavefunction波函数probabilityamplitude几率幅probabilitydensity几率密度waveofprobability几率波matrix矩阵time-independentstate定态

potentialwell势阱KeywordsAndTerms东北大学《大学物理》Chapter25Quantummechanic(Tbchap.34)25.1TheBirthofQuantumMechanics

25.2WavefunctionandBorn’sinterpretation25.3TheHeisenberguncertaintyprinciple25.4TheSchrödingerequation34-134-234-3,434-5,625.5Particlesinaninfinitelydeepsquarewellpotential34-825.1TheBirthofQuantumMechanicsOnestream34-1Planck(Planck’squantumhypothesis,1900)Einstein(Photontheoryoflight,1905)deBroglie(matterwave,1923)ErwinSchrödinger(wavequantummechanics,1926)TheotherstreamPlanck(Planck’squantumhypothesis,1900)Bohr(Bohr’smodelabouthydrogenatom,1913)WernerHeisenberg(matrixquantummechanics,1925)1

.wavefunction

一个沿x轴正向传播的频率为f

的平面简谐波:Wavefunctionforafreeparticleinonedimension用指数形式表示：波的强度取复数实部Realpart微观粒子的运动状态波函数薛定谔方程描述微观粒子运动基本方程25.2WavefunctionandBorn’sinterpretation34-2

对于动量为P、能量为E的一维自由微观粒子，根据德布罗意假设，其物质波的波函数相当于单色平面波，类比可写成：ThreedimensionsOnedimension2Born’sinterpretationforwavefunction亮波强电子到达多暗波弱电子到达少Double-slitinterferenceofelectronsTheintensityIofanywaveisproportionaltothesquareofthedisplacementamplitude.Bornproposedthatthewavefunctionamplitudeisrelatedtotheprobabilityoffindingtheparticleatthecoordinatesandtheexactvalueoftheprobabilityisdeterminedby2

-waveofprobabilityProbabilitydensity波函数本身没有意义，但是其模的平方有意义，在某处德布罗意波的强度与粒子在该处附近出现的概率成正比–几率波ProbabilitydensityMeaning:atsomeinstanttheprobabilityoffindingtheparticleatacertainpoint表示：t时刻粒子出现在空间某点r处的概率密度。Meaning:Atsomeinstant,theprobabilityoffindingtheparticlewithinthevolumedVaboutthegiveposition表示:某点r附近体积元dV

中的概率单个粒子在哪一处出现是偶然事件；大量粒子的分布有确定的统计规律。电子数N=7电子数N=100电子数N=3000电子数N=20000电子数

N=70000出现概率小出现概率大电子双缝干涉图样px0Electronbeam△xx

one-slitdiffracitonofelectrons25.3TheHeisenbergUncertaintyprinciple(34-3,4)PPxelectronbeamslitxJustforthefirstorderdiffraction

ispositionuncertaintyasitpassingthroughtheslitInvolvingotherordersMorecarefulcalculationHeisenberguncertaintyprincipleWecannotmeasureboththepositionandmomentumofanobjectpreciselyatthesametime.Themoreaccuratelywetrytomeasuretheposition,thegreaterwillbetheuncertaintyinmomentum.2.Uncertantaintyprinciplerelatesenergyandtime反映了原子能级宽度△E和原子在该能级的平均寿命△t之间的关系。

基态辐射光谱线固有宽度激发态

E基态寿命△t光辐射能级宽度平均寿命

△

t~10-8s平均寿命

△t∞能级宽度△E0Discussion：1.Theuncertaintyisinherentinnatureanditistheresultoftwofactors:thewave-particledualityandtheunavoid-ableinteractionbetweenthethingobservedandtheobservinginstrument.2.Anymovingparticlehasnocertainobits.Whatwecangetistheprobabilityofitsposition,momentumatnextinstant.3.Formacroobjects,hissosmallandtheprincipleareunusuallynegligible.Foratinydust,macroscopicparticleitspositionuncertaintycouldbeDetermineSolution:Example:Known:Determine:itspositionuncertaintySolution:Example(34-1)Positionuncertaintyofelectron例

波长=500nm的光波，沿x轴正向传播。如果测定其波长的不准确度为，

求同时测定光子位置坐标的不确定量。解1.Thetime-dependentSchrödingerequation(34-6)SchrödingerprincipalcontributionwastoformulateadifferentialequationthatwassatisfiedbytheamplitudeofthedeBrogliewave.(1)Afreeparticlem(withnoforceonit)

inonedimension(movingalongthexaxis)25.4TheSchrödingerequation34-5,6ErwinSchrodinger1933“Noble”Wave-particledualityWavefunctionTakethefirstderivativewithrespecttotafreeparticle

One–dimensionaltime-dependentSchrödingerequationforafreeparticleTakethesecondderivativewithrespecttoxbaseontheenergyconservation(2)One-dimensionaltime-dependentSchrödingerequationforaparticlewithpotentialenergyU(x,t)一维运动粒子含时薛定谔方程拉普拉斯算符：Generalform

薛定谔方程是非相对论量子力学的基本动力学方程，其地位与经典力学中的牛顿方程相同。(3)

three–dimensionaltime-dependentSchrödingerequationforaparticlewithpotentialenergyU(x,y,z,t)

Thefunctionofpotentialenergyisnotchangeablewithtimet,-time-independentstate

2.Thetime-independentSchrödingerequation(34-6)SubstituteΨintoSequation：time-independentSchrödingerinonedimensionEquation34-53.Somerequirementsimposedonwavefunction(P801)(1)ItshouldbeacontinuousfunctionSincetheprobabilityshouldbecontinuousfrompointtopointandnottotakediscontinuousjumps.(2)Itshouldbenormalized（归一化）Forasingleparticle,theprobabilityoffindingtheparticlesummedoverallspacemustbeexactly1.波函数：单值、连续、有限00<x<L0LxWhen

Electronscannotjumpoutofthewell25.5Particleinaninfinitelydeepsquarewellpotential34-8PotentialenergyU(x)asDenote:generalsolution：Quantumnumber0sin,===kLALxyQDetermineA,B,k(Ψiscontinuous)Boundaryconditions0LxNormalization:0Lx

probabilitydensity

energy

wavefunction

waveequation0Lx1.EnergyisquantizedDiscussion：groundenergyEnergy

excitedenergy0LxCorrespondenceprincipleQuantummechanicsclassicmechanicsn=1n=2n=30x0x2probabilityfunctionandprobabilitydensityMinimum:(n+1)个Maximum:n个4E1E116E19E13thepropertyofstandingwaveWavefunctionx0a例

设质量为m

的微观粒子处在宽为a的一维无限深方势阱中,

求(1)粒子在0<x<a/4区间中出现的概率,并对n=1和n=的情况算出概率值。

(2)在那些量子态上,a/4处的概率密度最大?解(1)概率密度粒子在0<x<a/4

区间中出现的概率（2）a/4处的概率密度极大值对应n=2，6，10，···等量子态。例

求在一维无限深势阱中粒子概率密度的最大值的位置.解:一维无限深势阱中粒子的波函数为将上式对x进行一次求导，并令其等于零即可得最大值得位置为例如最大值位置最大值位置