邊界條件之理論再探討及其應用

Abstract

Abstract This theoretical investigation intends to study the boundary condition and its applications to mechanics of sediment transport, which is one of the most important components in the civil engineering. With a combined effort of quantum mechanics and similarity parameter, the partial differential equation of transient position-probability density is attained and can be applied to predict the electron’s position inside the boundary. Also, an appropriate set of the initial and the boundary conditions is set up in accordance to the actual electron behavior for solving this PDE of probability density function. Thereafter, a simple, closed-form solution for the probability density is obtained and expressed in terms of the error function for a new similarity variable. Note that this analytic similarity solution is easy to perform the calculation and suitable for any further mathematical operation, such as the optimization applications. In addition, it is shown that these predications are reasonable and in good agreement to the physical meanings, which are evaluated from both microscopic and macroscopic viewpoints. In conclusions, this is an innovative approach by using the Schrödinger equation directly to solve the boundary condition problem. Moreover, with the aids of this analytic position-probability-density solution, it is illustrated that the free single electron in the boundary can only appear at some specified regions, which are defined by a dimensionless parameter within a range of . This result can be served as a valuable consideration reference for setting the practical engineering. Keywords：boundary condition; velocity field; temperature field; concentration field; frontier orbital theory; Modern Physics