厄米算符本征函数完备性的一般证明

从一组完备的本征函数开始,以表象原理为前提,建立一个唯象的公式,利用反证法证明了力学量算符的完备性,即这一组完备的本征函数可以投影到力学量算符的本征函数空间中.从而给出了力学量算符本征函数完备性的一般证明.     

关于《对〈厄米算符本征函数完备性的一般证明〉的质疑》的回答

正尊敬的编辑:您好!首先,我们感到非常的高兴,有人对我们的工作产生兴趣,并且提出见解,说明我们的工作也是有一定价值与意义的.下面我们就读者的质疑作一下说明:首先,我们工作的目的是对基础量子力学的理论进行补充,是为了使量子力学能有更明确、更简单的框架.对于量子力学中关于厄米算符本征矢完备性的探讨,前人也做过相关的工作,如参考文献[6-8]中所作出的处理,将本征矢分为有限和无限分别     

Molecular dynamics simulations on the dynamics of two-dimensional rounded squares

The collective motion of rounded squares with different corner-roundness ζ is studied by molecular dynamics(MD)simulation in this work. Three types of translational collective motion pattern are observed, including gliding, hopping and a mixture of gliding and hopping. Quantitatively, the dynamics of each observed ordered phase is characterized by both mean square displacement and van Hove functions for both translation and rotation. The effect of corner-roundness on the dynamics is further studied by comparing the dynamics of the rhombic crystal phases formed by different corner-rounded particles at a same surface fraction. The results show that as ζ increases from 0.286 to 0.667, the translational collective motion of particles changes from a gliding-dominant pattern to a hopping-dominant pattern, whereas the rotational motion pattern is hopping-like and does not change in its type, but the rotational hopping becomes much more frequent as ζincreases(i.e., as particles become more rounded). A simple geometrical model is proposed to explain the trend of gliding motion observed in MD simulations.     

Phase behavior of rotationally asymmetric Brownian kites containing 90° internal angles

Previous Monte Carlo simulations have shown that ordered tetratic phases can emerge in a dense two-dimensional Brownian system of rotationally asymmetric hard kites having 90° internal angles. However, there have been no experimental investigations yet to compare with these simulation results. Here, we have fabricated two types of micron-sized kites having internal angles of 72°-90°-108°-90° and 72°-99°-90°-99°, respectively, and we have experimentally studied their phase behavior in two-dimensional systems. Interestingly and in contrast to the Monte Carlo simulations, the experimental results show a phase sequence of isotropic fluid-hexagonal rotator crystal-square crystal as the area fraction φ_A increases for both types of kites. The observed square crystal displays not only a quasi-long-range translational order but also (quasi-)long-range 4-fold bond- and molecular-orientational order; these characteristics confirm that tetratic order can emerge even in dense Brownian systems of rotationally asymmetric particles. A model based on local polymorphic configurations (LPCs) is proposed to understand the origin of the square lattice order in these dense kite systems. The results in this study provide a new route to realize custom-designed self-assembly of colloids by controlling LPCs.     

Solid-to-molecular-orientational-hexatic melting induced by local environment determined defect proliferations

Two-dimensional (2D) melting is a fundamental research topic in condensed matter physics, which can also provide guidance on fabricating new functional materials. Nevertheless, our understanding of 2D melting is still far from being complete due to existence of possible complicate transition mechanisms and absence of effective analysis methods. Here, using Monte Carlo simulations, we investigate 2D melting of 60° rhombs which melt from two different surface-fully-coverable crystals, a complex hexagonal crystal (cHX) whose primitive cell contains three rhombs, and a simple rhombic crystal (RB) whose primitive cell contains one rhomb. The melting of both crystals shows a sequence of solid, hexatic in molecular orientation (H^mo), and isotropic phases which obey the Berezinskii-Kosterlitz-Thouless-Halperin-Nelson-Young (BKTHNY) theory. However, local polymorphic configuration (LPC) based analysis reveals different melting mechanisms: the cHX-H^mo transition is driven by the proliferation of point-like defects during which defect-associated LPCs are generated sequentially, whereas the RB-H^mo transition is driven by line defects where defect-associated LPCs are generated simultaneously. These differences result in the observed different solid-H^mo transition points which are φ_A=0.812 for the cHX-H^mo and φ_A=0.828 for the RB-H^mo. Our work will shed light on the initial-crystal-dependence of 2D melting behavior.