RESEARCH ON FUNCTIONAL ANALYTICAL FOUNDATION OF QUANTUM MECHANICS RELATING TO CANONICAL COMMUTATION RELATIONS

正则对易关系量子力学的泛函分析基础研究

基本信息

批准号：
12640146
负责人：
WATANABE Shuji
金额：
$ 0.83万
依托单位：
Aichi University of Technology
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2001
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640146/
关键词：
A GENERALIZED FOURIER TRANSFORM PARTIAL DIFFERENTIAL EQUATIONS WITH SINGULAR COEFFICIENTS EXPLICIT SOLUTIONS AN EMBEDDING THEOREM OF SOBOLEV TYPE INVERSE SQUARE POTENTIAL SELFADJOINTNESS OF DYNAMICAL VARIABLES QUANTUM MECHANICS ON MANIFOLDS

项目摘要

(1) I DEALT WITH AN OPERATOR COMPOSED OF A DIFFERENTIAL OPERATOR AND A SINGULAR TERM. I CONSTRUCTED A TRANSFORM, WHICH CONVERTS THE OPERATOR INTO A MULTIPLICATION OPERATOR. THIS TRANSFORM IS A GENERALIZED FOURIER TRANSFORM. I DERIVED A THEOREM OF PLANCHEREL TYPE RELATED TO THIS TRANSFORM. I APPLIED THIS TRANSFORM TO SOME CAUCHY PROBLEMS FOR PARTIAL DIFFERENTIAL EQUATIONS WITH SINGULAR COEFFICIENTS AND OBTAINED THE SOLUTIONS EXPLICITLY. USING THIS TRANSFORM I ALSO DEFINED A SPACE OF SOBOLEV TYPE AND SHOWED AN EMBEDDING THEOREM OF SOBOLEV TYPE CONCERNING THIS SPACE. IT IS WELL KNOWN THAT BY THE SOBOLEV EMBEDDING THEOREM WE OBTAIN INFORMATION CONCERNING SMOOTHNESS. ON THE OTHER HAND, THANKS TO OUR EMBEDDING THEOREM, WE CAN OBTAIN INFORMATION CONCERNING NOT ONLY SMOOTHNESS BUT ALSO SINGULARITY OF EACH ELEMENT IN THIS SPACE. THEN I APPLIED THIS TRANSFORM TO THE STUDY OF THE MOTION OF A QUANTUM MECHANICAL PARTICLE UNDER THE INVERSE SQUARE POTENTIAL. I FOUND OUT AN EASIER CONDITION UNDER WHICH THERE EXISTS THE MOTION OF SUCH A PARTICLE.(2) I DEALT WITH THE DYNAMICAL VARIABLES IN QUANTUM MECHANICS ON THE 1-SPHERE BASED ON DIRAC FORMALISM. I SHOWED SELFADJOINTNESS OF THE DYNAMICAL VARIABLES SUCH AS THE MOMONTUM OPERATORS AND THE HAMILTONIAN. AS A RESULT, I FOUND THAT THE DYNAMICAL SYSTEM IS ACTUALLY A QUANTUM MECHANICAL SYSYTEM. IT IS OF INTEREST THAT THE CONTINUOUS SPECTRUM OF EACH MOMENTUM OPERATOR COINCIDES WITH THE SET OF ALL REAL NUMBERS. I APPLIED THESE RESULTS TO THE CAUCHY PROBLEM FOR THE SCHRODINGER EQUATION FOR A FREE QUANTUM MECHANICAL PARTICLE MOVING ON THE 1-SPHERE.

（1）我与由差分运算符和单一术语组成的操作员打交道。我构建了一个转换，将操作员转换为乘法运算符。这种转换是广义的傅立叶变换。我得出了与此转换有关的Plancherel类型定理。我将这种转换应用于具有单数系数的部分微分方程的一些库奇问题，并明确获得了解决方案。使用此转换，我还定义了Sobolev类型的空间，并显示了有关此空间的Sobolev类型的嵌入定理。众所周知，通过嵌入定理的Sobolev我们获得了有关平滑度的信息。另一方面，由于我们的嵌入定理，我们不仅可以获取有关平滑度，而且可以获得该空间中每个元素的奇异性的信息。然后，我将此转换应用于反平方势下量子机械粒子的运动。我发现存在这种粒子的运动的一种更容易的条件。（2）我根据狄拉克形式上处理了1个速度上的量子力学中的动态变量。我展示了动态变量（例如Momontum操作员和Hamiltonian）的自相度。结果，我发现动态系统实际上是量子机械系统。有趣的是，每个动量操作员的连续光谱与所有实数的集合一致。我将这些结果应用于Schrodinger方程的Cauchy问题，用于在1球上移动的自由量子机械粒子。