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页数:16页
时间:2018-07-23
《量子力学之狄拉克符号系统与表象》由会员上传分享,免费在线阅读,更多相关内容在教育资源-天天文库。
1、Dirac符号系统与表象一、Dirac符号1.引言我们知道任一力学量在不同表象中有不同形式,它们都是取定了某一具体的力学量空间,即某一具体的力学量表象。量子描述除了使用具体表象外,也可以不取定表象,正如几何学和经典力学中也可用矢量形式A来表示一个矢量,而不用具体坐标系中的分量(Ax,Ay,Az)表示一样。量子力学可以不涉及具体表象来讨论粒子的状态和运动规律。这种抽象的描述方法是由Dirac首先引用的,本质是一个线性泛函空间,所以该方法所使用的符号称为Dirac符号。2.态矢量(1).右矢空间力学量本征态构成完备系,所以本征函数所对应的右
2、矢空间中的右矢也组成该空间的完备右矢(或基组),即右矢空间中的完备的基本矢量(简称基矢)。右矢空间的任一矢量
3、ψ>可按该空间的某一完备基矢展开。例如:(2).左矢空间右矢空间中的每一个右矢量在左矢空间都有一个相对应的左矢量,记为<
4、。右矢空间和左矢空间称为伴空间或对偶空间,<ψ
5、和
6、ψ>称为伴矢量。
7、,8、,9、组成左矢空间的完备基组,任一左矢量可按其展开,即左矢空间的任一矢量可按左矢空间的完备基矢展开。(3).伴矢量<ψ10、和11、ψ>的关系12、ψ>按Q的左基矢13、Qn>展开:14、ψ>=a115、Q1>+a216、Q2>+...+a317、Q318、>+...展开系数即相当于Q表象中的表示:<ψ19、按Q的左基矢20、展开:<ψ21、=a*122、+a*223、+...+a*n24、+...展开系数即相当于Q表象中的表示:ψ+=(a*1,a*2,...,a*n,...)同理某一左矢量<φ25、亦可按Q的左基矢展开:<φ26、=b*127、+b*228、+...+b*n29、+...定义30、ψ>和<φ31、的标积为:。显然<φ32、ψ>*=<ψ33、φ>。对于满足归一化条件的内积有:。这样,本征态的归一化条件可以写为:muststrictlyenforcethehotexaminationandap34、provalsystem,holidays,hotworkapprovaltobeupgraded.(2)thefireoperation,guardianandextinguishingequipmentinplace.(3)keypartspersonregulation:weldingsite:trackguardian,dangerousgoods,warehouse:warehouseworkers,dormitory:1,theimplementationofpointsofdormitorycultureconstruct35、ion:construction,environmentalhealthintotheconstructionorganizationdesign,developedandcivilizedconstructionrequirements,noroughconstruction,projectsinaccordancewiththeconstructionprogrammeoftheDepartmentorganizetheimplementation,followedbytheprojectleader,thecompany...Ed36、ucationorpunishment.1,correctiveandpreventativemeasurescorrectivemeasures:(1)bytheprojectsecurityofficerinidentifyingcauses,inthecontextoffindingsmakerecommendationsforcorrectiveandpreventiveactions.(2)underthepresentproposal,formulatedbythesecuritystaffcorrectivemeasure37、s,forexaminationandapproval.(3)safetyofficersmonitortheimplementationofcorrectivemeasures,recordthecorrectivemeasuresprocess.2,preventivemeasures:(1)normaloperationsafetyensuringsystemintegrityandisfundamentaltoprevention.(2)implementingacomprehensive,standardizedmanagem38、entprocesses,staff,educationworkersandenhanceawarenessofselfprotection,由此可以看出:<ψ39、和40、ψ>满足:a)在同一确定表象中,各分量互
8、,9、组成左矢空间的完备基组,任一左矢量可按其展开,即左矢空间的任一矢量可按左矢空间的完备基矢展开。(3).伴矢量<ψ10、和11、ψ>的关系12、ψ>按Q的左基矢13、Qn>展开:14、ψ>=a115、Q1>+a216、Q2>+...+a317、Q318、>+...展开系数即相当于Q表象中的表示:<ψ19、按Q的左基矢20、展开:<ψ21、=a*122、+a*223、+...+a*n24、+...展开系数即相当于Q表象中的表示:ψ+=(a*1,a*2,...,a*n,...)同理某一左矢量<φ25、亦可按Q的左基矢展开:<φ26、=b*127、+b*228、+...+b*n29、+...定义30、ψ>和<φ31、的标积为:。显然<φ32、ψ>*=<ψ33、φ>。对于满足归一化条件的内积有:。这样,本征态的归一化条件可以写为:muststrictlyenforcethehotexaminationandap34、provalsystem,holidays,hotworkapprovaltobeupgraded.(2)thefireoperation,guardianandextinguishingequipmentinplace.(3)keypartspersonregulation:weldingsite:trackguardian,dangerousgoods,warehouse:warehouseworkers,dormitory:1,theimplementationofpointsofdormitorycultureconstruct35、ion:construction,environmentalhealthintotheconstructionorganizationdesign,developedandcivilizedconstructionrequirements,noroughconstruction,projectsinaccordancewiththeconstructionprogrammeoftheDepartmentorganizetheimplementation,followedbytheprojectleader,thecompany...Ed36、ucationorpunishment.1,correctiveandpreventativemeasurescorrectivemeasures:(1)bytheprojectsecurityofficerinidentifyingcauses,inthecontextoffindingsmakerecommendationsforcorrectiveandpreventiveactions.(2)underthepresentproposal,formulatedbythesecuritystaffcorrectivemeasure37、s,forexaminationandapproval.(3)safetyofficersmonitortheimplementationofcorrectivemeasures,recordthecorrectivemeasuresprocess.2,preventivemeasures:(1)normaloperationsafetyensuringsystemintegrityandisfundamentaltoprevention.(2)implementingacomprehensive,standardizedmanagem38、entprocesses,staff,educationworkersandenhanceawarenessofselfprotection,由此可以看出:<ψ39、和40、ψ>满足:a)在同一确定表象中,各分量互
9、组成左矢空间的完备基组,任一左矢量可按其展开,即左矢空间的任一矢量可按左矢空间的完备基矢展开。(3).伴矢量<ψ
10、和
11、ψ>的关系
12、ψ>按Q的左基矢
13、Qn>展开:
14、ψ>=a1
15、Q1>+a2
16、Q2>+...+a3
17、Q3
18、>+...展开系数即相当于Q表象中的表示:<ψ
19、按Q的左基矢20、展开:<ψ21、=a*122、+a*223、+...+a*n24、+...展开系数即相当于Q表象中的表示:ψ+=(a*1,a*2,...,a*n,...)同理某一左矢量<φ25、亦可按Q的左基矢展开:<φ26、=b*127、+b*228、+...+b*n29、+...定义30、ψ>和<φ31、的标积为:。显然<φ32、ψ>*=<ψ33、φ>。对于满足归一化条件的内积有:。这样,本征态的归一化条件可以写为:muststrictlyenforcethehotexaminationandap34、provalsystem,holidays,hotworkapprovaltobeupgraded.(2)thefireoperation,guardianandextinguishingequipmentinplace.(3)keypartspersonregulation:weldingsite:trackguardian,dangerousgoods,warehouse:warehouseworkers,dormitory:1,theimplementationofpointsofdormitorycultureconstruct35、ion:construction,environmentalhealthintotheconstructionorganizationdesign,developedandcivilizedconstructionrequirements,noroughconstruction,projectsinaccordancewiththeconstructionprogrammeoftheDepartmentorganizetheimplementation,followedbytheprojectleader,thecompany...Ed36、ucationorpunishment.1,correctiveandpreventativemeasurescorrectivemeasures:(1)bytheprojectsecurityofficerinidentifyingcauses,inthecontextoffindingsmakerecommendationsforcorrectiveandpreventiveactions.(2)underthepresentproposal,formulatedbythesecuritystaffcorrectivemeasure37、s,forexaminationandapproval.(3)safetyofficersmonitortheimplementationofcorrectivemeasures,recordthecorrectivemeasuresprocess.2,preventivemeasures:(1)normaloperationsafetyensuringsystemintegrityandisfundamentaltoprevention.(2)implementingacomprehensive,standardizedmanagem38、entprocesses,staff,educationworkersandenhanceawarenessofselfprotection,由此可以看出:<ψ39、和40、ψ>满足:a)在同一确定表象中,各分量互
20、展开:<ψ
21、=a*122、+a*223、+...+a*n24、+...展开系数即相当于Q表象中的表示:ψ+=(a*1,a*2,...,a*n,...)同理某一左矢量<φ25、亦可按Q的左基矢展开:<φ26、=b*127、+b*228、+...+b*n29、+...定义30、ψ>和<φ31、的标积为:。显然<φ32、ψ>*=<ψ33、φ>。对于满足归一化条件的内积有:。这样,本征态的归一化条件可以写为:muststrictlyenforcethehotexaminationandap34、provalsystem,holidays,hotworkapprovaltobeupgraded.(2)thefireoperation,guardianandextinguishingequipmentinplace.(3)keypartspersonregulation:weldingsite:trackguardian,dangerousgoods,warehouse:warehouseworkers,dormitory:1,theimplementationofpointsofdormitorycultureconstruct35、ion:construction,environmentalhealthintotheconstructionorganizationdesign,developedandcivilizedconstructionrequirements,noroughconstruction,projectsinaccordancewiththeconstructionprogrammeoftheDepartmentorganizetheimplementation,followedbytheprojectleader,thecompany...Ed36、ucationorpunishment.1,correctiveandpreventativemeasurescorrectivemeasures:(1)bytheprojectsecurityofficerinidentifyingcauses,inthecontextoffindingsmakerecommendationsforcorrectiveandpreventiveactions.(2)underthepresentproposal,formulatedbythesecuritystaffcorrectivemeasure37、s,forexaminationandapproval.(3)safetyofficersmonitortheimplementationofcorrectivemeasures,recordthecorrectivemeasuresprocess.2,preventivemeasures:(1)normaloperationsafetyensuringsystemintegrityandisfundamentaltoprevention.(2)implementingacomprehensive,standardizedmanagem38、entprocesses,staff,educationworkersandenhanceawarenessofselfprotection,由此可以看出:<ψ39、和40、ψ>满足:a)在同一确定表象中,各分量互
22、+a*223、+...+a*n24、+...展开系数即相当于Q表象中的表示:ψ+=(a*1,a*2,...,a*n,...)同理某一左矢量<φ25、亦可按Q的左基矢展开:<φ26、=b*127、+b*228、+...+b*n29、+...定义30、ψ>和<φ31、的标积为:。显然<φ32、ψ>*=<ψ33、φ>。对于满足归一化条件的内积有:。这样,本征态的归一化条件可以写为:muststrictlyenforcethehotexaminationandap34、provalsystem,holidays,hotworkapprovaltobeupgraded.(2)thefireoperation,guardianandextinguishingequipmentinplace.(3)keypartspersonregulation:weldingsite:trackguardian,dangerousgoods,warehouse:warehouseworkers,dormitory:1,theimplementationofpointsofdormitorycultureconstruct35、ion:construction,environmentalhealthintotheconstructionorganizationdesign,developedandcivilizedconstructionrequirements,noroughconstruction,projectsinaccordancewiththeconstructionprogrammeoftheDepartmentorganizetheimplementation,followedbytheprojectleader,thecompany...Ed36、ucationorpunishment.1,correctiveandpreventativemeasurescorrectivemeasures:(1)bytheprojectsecurityofficerinidentifyingcauses,inthecontextoffindingsmakerecommendationsforcorrectiveandpreventiveactions.(2)underthepresentproposal,formulatedbythesecuritystaffcorrectivemeasure37、s,forexaminationandapproval.(3)safetyofficersmonitortheimplementationofcorrectivemeasures,recordthecorrectivemeasuresprocess.2,preventivemeasures:(1)normaloperationsafetyensuringsystemintegrityandisfundamentaltoprevention.(2)implementingacomprehensive,standardizedmanagem38、entprocesses,staff,educationworkersandenhanceawarenessofselfprotection,由此可以看出:<ψ39、和40、ψ>满足:a)在同一确定表象中,各分量互
23、+...+a*n24、+...展开系数即相当于Q表象中的表示:ψ+=(a*1,a*2,...,a*n,...)同理某一左矢量<φ25、亦可按Q的左基矢展开:<φ26、=b*127、+b*228、+...+b*n29、+...定义30、ψ>和<φ31、的标积为:。显然<φ32、ψ>*=<ψ33、φ>。对于满足归一化条件的内积有:。这样,本征态的归一化条件可以写为:muststrictlyenforcethehotexaminationandap34、provalsystem,holidays,hotworkapprovaltobeupgraded.(2)thefireoperation,guardianandextinguishingequipmentinplace.(3)keypartspersonregulation:weldingsite:trackguardian,dangerousgoods,warehouse:warehouseworkers,dormitory:1,theimplementationofpointsofdormitorycultureconstruct35、ion:construction,environmentalhealthintotheconstructionorganizationdesign,developedandcivilizedconstructionrequirements,noroughconstruction,projectsinaccordancewiththeconstructionprogrammeoftheDepartmentorganizetheimplementation,followedbytheprojectleader,thecompany...Ed36、ucationorpunishment.1,correctiveandpreventativemeasurescorrectivemeasures:(1)bytheprojectsecurityofficerinidentifyingcauses,inthecontextoffindingsmakerecommendationsforcorrectiveandpreventiveactions.(2)underthepresentproposal,formulatedbythesecuritystaffcorrectivemeasure37、s,forexaminationandapproval.(3)safetyofficersmonitortheimplementationofcorrectivemeasures,recordthecorrectivemeasuresprocess.2,preventivemeasures:(1)normaloperationsafetyensuringsystemintegrityandisfundamentaltoprevention.(2)implementingacomprehensive,standardizedmanagem38、entprocesses,staff,educationworkersandenhanceawarenessofselfprotection,由此可以看出:<ψ39、和40、ψ>满足:a)在同一确定表象中,各分量互
24、+...展开系数即相当于Q表象中的表示:ψ+=(a*1,a*2,...,a*n,...)同理某一左矢量<φ
25、亦可按Q的左基矢展开:<φ
26、=b*127、+b*228、+...+b*n29、+...定义30、ψ>和<φ31、的标积为:。显然<φ32、ψ>*=<ψ33、φ>。对于满足归一化条件的内积有:。这样,本征态的归一化条件可以写为:muststrictlyenforcethehotexaminationandap34、provalsystem,holidays,hotworkapprovaltobeupgraded.(2)thefireoperation,guardianandextinguishingequipmentinplace.(3)keypartspersonregulation:weldingsite:trackguardian,dangerousgoods,warehouse:warehouseworkers,dormitory:1,theimplementationofpointsofdormitorycultureconstruct35、ion:construction,environmentalhealthintotheconstructionorganizationdesign,developedandcivilizedconstructionrequirements,noroughconstruction,projectsinaccordancewiththeconstructionprogrammeoftheDepartmentorganizetheimplementation,followedbytheprojectleader,thecompany...Ed36、ucationorpunishment.1,correctiveandpreventativemeasurescorrectivemeasures:(1)bytheprojectsecurityofficerinidentifyingcauses,inthecontextoffindingsmakerecommendationsforcorrectiveandpreventiveactions.(2)underthepresentproposal,formulatedbythesecuritystaffcorrectivemeasure37、s,forexaminationandapproval.(3)safetyofficersmonitortheimplementationofcorrectivemeasures,recordthecorrectivemeasuresprocess.2,preventivemeasures:(1)normaloperationsafetyensuringsystemintegrityandisfundamentaltoprevention.(2)implementingacomprehensive,standardizedmanagem38、entprocesses,staff,educationworkersandenhanceawarenessofselfprotection,由此可以看出:<ψ39、和40、ψ>满足:a)在同一确定表象中,各分量互
27、+b*228、+...+b*n29、+...定义30、ψ>和<φ31、的标积为:。显然<φ32、ψ>*=<ψ33、φ>。对于满足归一化条件的内积有:。这样,本征态的归一化条件可以写为:muststrictlyenforcethehotexaminationandap34、provalsystem,holidays,hotworkapprovaltobeupgraded.(2)thefireoperation,guardianandextinguishingequipmentinplace.(3)keypartspersonregulation:weldingsite:trackguardian,dangerousgoods,warehouse:warehouseworkers,dormitory:1,theimplementationofpointsofdormitorycultureconstruct35、ion:construction,environmentalhealthintotheconstructionorganizationdesign,developedandcivilizedconstructionrequirements,noroughconstruction,projectsinaccordancewiththeconstructionprogrammeoftheDepartmentorganizetheimplementation,followedbytheprojectleader,thecompany...Ed36、ucationorpunishment.1,correctiveandpreventativemeasurescorrectivemeasures:(1)bytheprojectsecurityofficerinidentifyingcauses,inthecontextoffindingsmakerecommendationsforcorrectiveandpreventiveactions.(2)underthepresentproposal,formulatedbythesecuritystaffcorrectivemeasure37、s,forexaminationandapproval.(3)safetyofficersmonitortheimplementationofcorrectivemeasures,recordthecorrectivemeasuresprocess.2,preventivemeasures:(1)normaloperationsafetyensuringsystemintegrityandisfundamentaltoprevention.(2)implementingacomprehensive,standardizedmanagem38、entprocesses,staff,educationworkersandenhanceawarenessofselfprotection,由此可以看出:<ψ39、和40、ψ>满足:a)在同一确定表象中,各分量互
28、+...+b*n29、+...定义30、ψ>和<φ31、的标积为:。显然<φ32、ψ>*=<ψ33、φ>。对于满足归一化条件的内积有:。这样,本征态的归一化条件可以写为:muststrictlyenforcethehotexaminationandap34、provalsystem,holidays,hotworkapprovaltobeupgraded.(2)thefireoperation,guardianandextinguishingequipmentinplace.(3)keypartspersonregulation:weldingsite:trackguardian,dangerousgoods,warehouse:warehouseworkers,dormitory:1,theimplementationofpointsofdormitorycultureconstruct35、ion:construction,environmentalhealthintotheconstructionorganizationdesign,developedandcivilizedconstructionrequirements,noroughconstruction,projectsinaccordancewiththeconstructionprogrammeoftheDepartmentorganizetheimplementation,followedbytheprojectleader,thecompany...Ed36、ucationorpunishment.1,correctiveandpreventativemeasurescorrectivemeasures:(1)bytheprojectsecurityofficerinidentifyingcauses,inthecontextoffindingsmakerecommendationsforcorrectiveandpreventiveactions.(2)underthepresentproposal,formulatedbythesecuritystaffcorrectivemeasure37、s,forexaminationandapproval.(3)safetyofficersmonitortheimplementationofcorrectivemeasures,recordthecorrectivemeasuresprocess.2,preventivemeasures:(1)normaloperationsafetyensuringsystemintegrityandisfundamentaltoprevention.(2)implementingacomprehensive,standardizedmanagem38、entprocesses,staff,educationworkersandenhanceawarenessofselfprotection,由此可以看出:<ψ39、和40、ψ>满足:a)在同一确定表象中,各分量互
29、+...定义
30、ψ>和<φ
31、的标积为:。显然<φ
32、ψ>*=<ψ
33、φ>。对于满足归一化条件的内积有:。这样,本征态的归一化条件可以写为:muststrictlyenforcethehotexaminationandap
34、provalsystem,holidays,hotworkapprovaltobeupgraded.(2)thefireoperation,guardianandextinguishingequipmentinplace.(3)keypartspersonregulation:weldingsite:trackguardian,dangerousgoods,warehouse:warehouseworkers,dormitory:1,theimplementationofpointsofdormitorycultureconstruct
35、ion:construction,environmentalhealthintotheconstructionorganizationdesign,developedandcivilizedconstructionrequirements,noroughconstruction,projectsinaccordancewiththeconstructionprogrammeoftheDepartmentorganizetheimplementation,followedbytheprojectleader,thecompany...Ed
36、ucationorpunishment.1,correctiveandpreventativemeasurescorrectivemeasures:(1)bytheprojectsecurityofficerinidentifyingcauses,inthecontextoffindingsmakerecommendationsforcorrectiveandpreventiveactions.(2)underthepresentproposal,formulatedbythesecuritystaffcorrectivemeasure
37、s,forexaminationandapproval.(3)safetyofficersmonitortheimplementationofcorrectivemeasures,recordthecorrectivemeasuresprocess.2,preventivemeasures:(1)normaloperationsafetyensuringsystemintegrityandisfundamentaltoprevention.(2)implementingacomprehensive,standardizedmanagem
38、entprocesses,staff,educationworkersandenhanceawarenessofselfprotection,由此可以看出:<ψ
39、和
40、ψ>满足:a)在同一确定表象中,各分量互
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