幂次中心势阱中粒子能谱规律的分析

分析了幂次中心势阱中粒子的能谱，讨论了粒子能谱形状随幂次变化的规律     

Taylor公式在级数判敛中的应用

利用Taylor公式把一些级数的通项un近似表示成幂函数1/n^α和（-1）^n/n^β的线性组合，误差为高阶无穷小。根据级数∞∑n=1 1/n^α和∞∑n=1 （-1）^n/n^β的收敛情况比较容易地判别级数∞∑n=1 un的敛散性。     

Studies on structural safety in stochastically excited Duffing oscillator with double potential wells

The effects of the Gaussian white noise excitation on structural safety due to erosion of safe basin in Duffing oscillator with double potential wells are studied in the present paper. By employing the well-developed stochastic Melnikov condition and Monte–Carlo method, various eroded basins are simulated in deterministic and stochastic cases of the system, and the ratio of safe initial points (RSIP) is presented in some given limited domain defined by the system’s Hamiltonian for various parameters or first-passage times. It is shown that structural safety control becomes more difficult when the noise excitation is imposed on the system, and the fractal basin boundary may also appear when the system is excited by Gaussian white noise only. From the RSIP results in given limited domain, sudden discontinuous descents in RSIP curves may occur when the system is excited by harmonic or stochastic forces, which are different from the customary continuous ones in view of the first-passage problems. In addition, it is interesting to find that RSIP values can even increase with increasing driving amplitude of the external harmonic excitation when the Gaussian white noise is also present in the system. The project supported by the National Natural Science Foundation of China (10302025 and 10672140). The English text was polished by Yunming Chen.     

光子带隙结构理论的新方法

本文应用光流线量子论的基本思想，把介质中传输的光看作是能量流线而非电磁披，认为当光在介质中传输时，光同介质并非是彼此独立的，而是要发生相互作用，这种相互作用使光能量在介质中呈量子化的几率分布。将构成光子晶体的周期性介质模拟为周期性折射率势阱，通过求解光流线量子论的基本方程，讨论光子晶体中带隙产生的物理机制。     

一维无限深势阱中粒子的位置-动量不确定关系: 基于计算的讨论

不确定性原理是一个长期被误解和滥用了的概念.文章计算了一维无限深势阱中粒子的位置和动量的不确定度△x和△p,指出它们遵守不确定性关系△x·△p/2的一般性结论,但△x和△p总是同向变化的.所谓“此一量不确定度越大,彼一量不确定度越小”式的表述是没有根据的.类似地, △t和△E（注意, △意义已变） 的同向变化也早在实验上被观察到.     

一维有限深方势阱能量本征方程的泰勒级数解和二次近似解

一维有限深方势阱能量本征方程的求解受限于超越方程,无法严格求解.本文将奇、偶宇称情况的超越方程归结为一个方程,自洽地给出两种近似解析解：一阶泰勒级数解和二次近似解.分析二者适用范围并做误差分析,发现泰勒级数解可以很好地理解能谱随量子数n平方变化的数值解(即,所谓n平方律),但在特定参数R下失效,该参数正比于势阱宽度乘以势阱高度开方.二次近似解对所有参数R都适用,能谱在大R极限下,可退化为精确求解的无限深势阱情况.对于任意参数R,二次近似波函数的保真度始终大于99.7%.