Relevant Research On Some Problems Of Quantum Mechanics Under Uncertainty Relation

Whether the fundamental laws of nature limit our ability to probe arbitrarily short distances? This problem that has plagued people for thousands of years still exists,and it is hoped that with the development of physics,people will eventually be able to understand the nature of time and space and answer this question fundamentally.Although so far,people still can not understand the essence of space-time,the exploration of this problem has never stopped.Especially with the successive proposed and establishment of the theory of relativity and quantum theory,people's cognition and understanding of physics have undergone a revolutionary change,and the two together form the basis of modern physics.In the course of more than one century of development,their theoretical correctness and reliability in their respective fields has been extensively tested and confirmed.Although they have achieved gratifying results in their respective applications,they are not perfect.At the same time,they all face some acute problems.Especially when dealing with problems of extremely large quality and extremely small scale,neither of them can give a reasonable explanation.This phenomenon all shows that neither the relativity theory nor quantum theory can be the ultimate theory of physical theory,but more like a specific description of a more basic theory under special conditions.Therefore,many researchers have sprouted the idea of fusing relativity theory and quantum theory,trying to construct a complete theoretical framework system-quantum gravity theory,which can contain both theories simultaneously.However,the quantized description of matter in quantum theory and the geometrical description of space-time in the theory of relativity are fundamentally incompatible.As a result,the theory of quantum gravity,which is designed to realize the unification of the relativity theory and quantum theory,is facing numerous difficulties,and how to achieve the organic combination of the two,is still one of the main challenges of theoretical physics.Since the 1970 s and 1980 s,people have continuously tried to find ways to realize the quantization of gravity from different aspects,and have successively proposed a series of related theoretical models.Among them,many theoretical models have given a common concept,that is,there is a minimum observable length on the Planck scale that plays a fundamental role in natural laws.Although no experiment can directly confirm this hypothesis so far,many theories provide indirect support for the existence of minimum observable length.Therefore,in the following decades,the research of minimum length correlation has been widely discussed as one of the hot topics.In order to enhance the knowledge and understanding of the uncertainty principle and generalized uncertainty principle in quantum systems,further explore the influence of this principle on the research of quantum system related issues.This thesis expands from the following aspects:(1)First,take the singular potential as an example to construct the generalized Dirac oscillator,and then discuss the generalized Dirac oscillator based on the Heisenberg uncertainty principle,and finally give the corresponding energy spectrum and wave function analytical expressions.Choosing the singular potential as an example to construct a generalized Dirac oscillator is mainly based on the following two considerations:First,the Dirac oscillator has been extensively studied in the fields of high energy physics and quantum optics;Second,the singular potential is a theoretical model with extraordinary significance in the research of atomic,molecular physics and chemical physics.In addition,there is a certain interaction between the two research fields,so the generalized Dirac oscillator constructed by singular potential has certain research value.(2)Based on the central premise of the existence of a minimum observable length,one of the existing generalized uncertainty principles is selected to study the related problems of the exponential potential function,so as to provide an easier,alternative and valid method to deal with exponential potential in a unified manner.Through the discussion of the Dirac equation under the generalized exponential potential under the background of minimum length,the analytical solution expression of the equation is further given.It is not difficult to find that the parameters in the generalized exponential potential can degenerate into various special exponential potential by appropriately selecting and adjusting the parameters,and the analytical solutions of various special exponential potential can also be degenerated from the analytical solutions of generalized exponential potential.Achieving the purpose of discussing multiple exponential potentials by processing the equations at one time,it is helpful to improve the efficiency and universality of work.(3)The research in this part is a further in-depth exploration on the basis of the previous two phases of research.Especially in the second stage of the study,it was noticed that almost all the generalized uncertainty principle proposed so far only modified one of between position and momentum operators,and did not consider the situation of modifying both operators at the same time.However,the curved space-time should be curved regardless of whether the momentum representation or the coordinate representation is used,and there is no coordinate transformation that can make the curvature of the momentum space disappear.Therefore,in order to make the generalized uncertainty principle more inclusive,a new type of generalized uncertainty principle is proposed by modifying the coordinate and momentum operator simultaneously,which could give a self-consistent phenomenological explanation for the existence of the minimum observable length.After that,using the basic principles of quantum mechanics to test the new generalized uncertainty principle,it can be found that the new generalized uncertainty principle better contains the conclusions of the generalized uncertainty principle,which seems to be helpful in discovering a richer physical connotation.(4)Based on the new generalized uncertainty principle,the exact solution of the Klein-Gordon equation with linear scalar potential and vector potential is discussed.In this research,the Klein-Gordon equation with linear scalar and vector potential under the new generalized uncertainty principle is discussed respectively through two different ways: direct solution and pure algebraic solution based on shape-invariant symmetry.The research results show that not only are the conclusions of the two different methods completely consistent,but also consistent with the conclusions under other generalized uncertainty principle in form,which supports the reliability and validity of the new generalized uncertainty principles to a certain extent.In addition,we also plan to use the Dirac equation,DKP equation and other related equations to further explore the new generalized uncertainty principle.If conditions permit,we will try to explore the intrinsic nature of minimum length by combining astronomy,black hole physics and other similar fields.(5)Finally,based on the new generalized uncertainty principle mentioned above and the generalized uncertainty principle proposed by Kempf et al,the Kemmer equation is studied.By comparing the analytical expressions of energy spectrum and wave function under two different correction models,the validity and reliability of new generalized uncertainty principle are further explored.In addition,in the discussion of the related problems of Kemmer oscillator under the minimum length scale,we also find that the new generalized uncertainty principle shows great advantages in mathematical processing.So far,this new model well contains the effective results of the existing model.To sum up,this thesis focuses on the uncertainty principle in quantum mechanics to carry out a series of related research.At present,the concept of a minimum observable length of the Planck scale has been widely recognized,but this concept is incompatible with the Heisenberg uncertainty principle in quantum mechanics.Therefore,people propose to extend the Heisenberg uncertainty principle to a generalized uncertainty principle by modifying the commutation relation between the position and momentum operators.However,it is worth noting that the existing generalized uncertainty principle is just modifying either momentum or position operator.Therefore,in order to give a more inclusive generalized uncertainty principle,this thesis achieves this goal by modifying the position and momentum operators simultaneously.And in the exploratory study of the new generalized uncertainty principle,we try to introduce the thinking of space-time effects,and hope that this attempt can bring some new enlightenment to the research of space-time related issues.