微扰理论及其应用

渤海大学本科毕业论文（设计）含时微扰理论及其应用Time-dependentperturbationtheoryanditsapplication学院（系）：数理学院物理系专业：物理学（师范）学号：10030009学生姓名：庞涛入学年度：2010指导教师：韩萍完成日期：2014年5月5日渤海大学BohaiUniversity含时微扰理论及其应用-I-摘要在量子力学中，精确求解薛定谔方程是很困难的，一般只能求近似解，应用微扰理论可以求得近似解。学好微扰理论在以后的学习中具有很大帮助。微扰理论分为两类，不含时微扰理论和含时微扰理论。在量子力学中，含时微扰理论研究的是一个量子系统的含时微扰所产生的效应.该理论是由英国物理学家狄拉克首先提出和发展建立起来的。应用含时微扰理论可以近似的计算出有微扰时的波函数，从而计算无微扰体系在微扰作用下由一个量子态跃迁到另一个量子态的跃迁概率。含时微扰包括常微扰和周期微扰，在这两种微扰作用下，得到的结果是不同的，我们分析计算了在常微扰和周期微扰两种微扰作用下的跃迁概率，得到了一些结论。在常微扰作用下时，我们得到了一个重要公式，该公式被称为费米黄金定则。常微扰是只在一段时间内起作用，时间足够长的话，则跃迁概率与时间无关；而通过计算无微扰体系在周期微扰作用下的跃迁概率，得出的结论是周期微扰的频率只有在一定范围内，才会发生跃迁。只有当外界微扰含有频率mk时，才会出现明显跃迁。此外，我们还讨论了光的发射和吸收，给出了偶极跃迁的选择定则。最后对激光的产生和激光的应用进行了介绍。关键词：选择定则；含时微扰；跃迁概率；黄金规则含时微扰理论及其应用-II-Time-dependentperturbationtheoryanditsapplicationAbstractInquantummechanics,theexactsolutionofSchrodingerequationisverydifficult,generallyonlyapproximatesolutions,usingtheperturbationtheorycanbeobtainedtheapproximatesolution.Tolearnagreathelptotheperturbationtheoryoflearninginthefuture.Perturbationtheoryisdividedintotwocategories,notthetime-dependentperturbationtheoryandtime-dependentperturbationtheory.Inquantummechanics,thetime-dependenttheoryofperturbationistheeffectofaquantumsystemwithtime-dependentperturbationgenerated.ThistheorywasfirstproposedanddevelopedbytheBritishphysicistDirac.Calculatedusingtime-dependentperturbationtheorycanbeapproximatedbyawavefunctionperturbation,thuscalculatedwithoutperturbationsystemundertheperturbationinducedbyaquantumstatetransitiontothetransitionprobabilityofanotherquantumstate.Thetime-dependentperturbationincludedregularperturbationandperiodicperturbation,inwhichtwokindsofperturbations,theresultisdifferent,analysisoftransitionprobabilityinconstantperturbationandperiodicperturbationtwoperturbationeffectwasobtainedbyus,someconclusionswereobtained.Intheconstantunderperturbations,weobtainaformula,theformulaiscalledtheFermigoldenrule.Theperturbationisoftenworkonlyinaperiodoftime,timeislongenough,thetransitionprobabilityisindependentoftime;andthroughthecalculationoftransitionprobabilitywithoutperturbationsystemintheperiodunderperturbations,itwasconcludedthattheperiodicperturbationfrequencyonlyinacertainrange,thetransitionwilloccur.Onlywhentheexternalperturbationwithfrequency,willappearobvioustransition.Inaddition,wealsodiscusstheemissionandabsorptionoflight,givesthedipoletransitionselectionrule.Applicationoflaserandlaserproducedfinallyisintroducedinthispaper.KeyWords：Selectionrule；time-dependentperturbation；transitionprobability；Thegoldenrule含时微扰理论及其应用-III-目录摘要................................................................................................................................IAbstract.................................................................................................................................II引言...............................................................................................................................11含时微扰理论的概述.....................................................................................................21.1含时微扰理论下的薛定谔方程............................................................................21.2跃迁概率................................................................................................................32常微扰和周期微扰.........................................................................................................52.1跃迁概率和费米黄金定则..................................................................................52.2周期微扰..............................................................................................................73含时微扰理论的应用...................................................................................................103.1光的发射和吸收..................................................................................................103.1.1爱因斯坦的发射和吸收系数.....................................................................103.1.2用微扰理论计算发射和吸收系数.............................................................113.2选择定则..............................................................................................................143.3典例分析.........................................................................................................164激光简介.......................................................................................................................184.1激光的产生........................................................................................................184.2激光的应用........................................................................................................19结论.............................................................................................................................21参考文献.......................................................................................................................22含时微扰理论及其应用–1–引言在量子力学中，对于具体物理问题的薛定谔方程，可以准确求解的问题是很少的，一般只能求近似解。因此用来求问题的近似解的方法就显得非常重要。而含时微扰理论就是一种用来求解问题的近似解的方法，该理论是由狄拉克提出并发展建立起来的，含时微扰理论研究的是一个量子系统的含时微扰所产生的效应。应用含时微扰理论，可以从简单问题的精确解出发来求较复杂问题的近似解。它用于体系的哈密顿算