标题： 时空量子化的薛定谔方程的自由粒子本征函数 显示英文标题

标题： Free Particle Eigenfunctions of Schrodinger Equation with Quantized Space-time

摘要： 众所周知，将坐标作为连续变量，由 0 到$L$之间所有点组成的集合，与测量的可观测性相矛盾。 换言之，在自然界可能存在一个基本长度，例如普朗克长度$\lambda_P$，因此不可能以小于这个基本长度的精度来测量位置坐标。 因此，有必要用离散变量作为坐标来研究量子力学的表述。 当然，从这种方法出发研究整个量子力学或物理学的任何分支都是一项艰巨的任务，因此值得考虑一个具体的简单问题，以便阐述基本思想。 在这篇短文中，我们比较了在一维自由粒子的薛定谔方程在通常的空间-时间连续情况下和当假设空间-时间被量化时所得到的解。 为此，我们将薛定谔方程中的导数替换为相应的离散导数。 我们在两种情况下计算概率密度（以及概率流）；结果发现这两种情况下的结果有很大不同。 我们还假设空间-时间被量化，并对比了算符的对易关系$[p,x]^q$和通常的对易关系$[p,x]$。 显示英文摘要

摘要： It is well-known that the coordinate as a continuous variable, consisting of a set of all points between 0 and $L$ contradicts the observability of measurement. In other words there might exist a fundamental length in nature, such as the Planck length $\lambda_P$, so that it is not possible to measure a position coordinate with accuracy smaller than this fundamental length. It is therefore necessary to investigate the formulation of quantum mechanics using only discrete variables as coordinates. To investigate all of quantum mechanics or any branch of physics from this approach is of course a daunting task and thus it is worthwhile to consider a specific simple problem in order to formulate the basic ideas. In this note we compare the solutions of Schrodinger equation for one-dimensional free particle under the usual space-time continuum with those that are obtained when space-time is assumed to be quantized using a simple model. For this purpose, we replace the derivatives occurring in Schrodinger equation with the corresponding discrete derivatives. We compute the probability density (and the probability current) under the two scenarios; they turn out to be quite different in the two cases. We also obtain the operator identity for the commutator $[p,x]^q$ under the assumption of quantized space-time and contrast it with the usual commutator $[p,x]$.