一维δ势阱中的相对论粒子

求解一维相对粒子的Dirac方程。对一维δ势阱，计算了束缚态能级与波函数；对一维双δ势阱，给出了束缚态能级所满足的超越方程，并确定了束缚态的数目，简单讨论了散射问题。     

方势阱中束缚态粒子能级的数值方法和波函数的图示

用数值方法求出了一维有限深不对称方势阱中束缚态粒子的能级和归一化波函数及其图示,所得结果在势阱深度趋于无穷大时与无限深势阱的结果一致.     

关于半壁无限深势阱束缚态存在条件的讨论

一维无限深势阱问题是量子力学中的经典问题,也是国内外量子力学初学者必学内容之一。一维半壁无限深势阱作为一维无限深势阱的变形,也是量子力学教学中常见的物理模型。然而关于半壁无限深势阱这一模型,许多教学和科研工作者甚至教材之中,都存在不严谨之处;尤其是对于束缚态存在条件的求解,许多教材与习题册没有对临界条件进行讨论验证。本文借助图解法对半壁无限深势阱束缚态存在条件进行了讨论并给出了严格解,纠正了长久以来人们对该问题的错误认识,对量子力学领域本科教学与实际科研工作有一定帮助。     

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%.     

线性变分法研究一维无限深势阱中粒子的能级和波函数

采用满足连续性条件的多项式独立基函数,应用线性变分法通过求解广义本征值方程研究了一维无限深势阱中粒子的能级和波函数.计算表明,随着试验基函数个数的增大,线性变分法的结果趋近于精确的解析结果.     

用基函数展开法求解一维有限深方势阱模型

用基函数展开法求解了一维有限深方势阱模型,所得结果与在坐标表象中直接求解薛定谔方程所得结果完全一致.重要的是这一求解过程为量子力学中基函数展开方法的教授与学习提供了一个范例,也为量子力学问题的计算机求解的教与学提供了参考.     

无限深球方势阱中粒子能级的一个简单近似解析式

本由ＷＫＢ方法和玻尔－索末菲量子化规则，求出了无限深球方势阱中粒子能级近似满足的超越方程；然后通过进一步近似，找到了粒子能级近似满足的解析式并对所得结果的精度进行了计算机分析。     

一维有限深方势阱的α0迭代求解法

介绍了用α０迭代方法求一维有限深方势阱的解析近似解，给出了解的通式和可求任意精度要求解的α０循环迭代计算程序流程。在新的基础上建立有限深方势阱的能级和波函数与无限深方势阱的能级和波函数的联系。     

无限深方势阱本征值和本征态的三种求解方法

一维无限深方势阱模型是量子力学理想模型,经典教材中势阱的边界一般取得比较特殊.或关于坐标原点具有对称性,或势阱左边界位于坐标原点.本文首先展示了如何利用3种方法求解一维任意边界无限深方势阱能量本征值和对应的本征态,不同方法得到的结果彼此之间等价,讨论分析了这3种方法的推导结果,然后得到关于一维任意边界无限深方势阱能量本征值和本征态的通式,从中比较容易看出这两个物理量均与阱宽有关,并且本征波函数与边界值有关,最后将一维结果拓展到二维和三维任意边界无限深方势阱情况.     

量子力学势垒穿透问题的光学类比

用量子态的线性叠加原理和波函数及其一阶导数的连续性,求解一维方势垒散射问题,得出各区域的波函数,最后求出了反射系数、透射系数以及势垒内部的概率流密度和概率分布,并得到由散射态过渡到束缚态的条件,另外,还将其自然过渡到一维对称方势阱的问题上去。     

氘-氘两体紧束缚态模型

提出了氘－氘紧束缚态模型．对两个氘原子核组成的体系,采用库仑势垒与核力方势阱组合的相互作用势,求解定态薛定谔方程,得到了两体紧束缚态的波函数和可能的能量本征值（核的库仑能数量级）,并说明了这种紧束缚态的存在条件． Deuterion deuterion tight bound states model about the mechanism of so called “nuclear cold fusion” is proposed. To solve the stationary Schrodinger equation with combination potential which contains Coulomb barrier and square well potential from nuclear force for the system containing two deuterions, the wave functions, the energy eigenvalues of the probable tight bound states and the existence condition of the tight bound states are obtained. In the case of ground...     

keV量级光子的线辐射源

在天体空间或实验室的等离子体中可能存在能量为若干keV量级的光子的线辐射,它们源于一种新的原子核物质结构形态(核核两体的松聚合)形成之时.对两个原子核组成的体系,采用库仑势垒与核力方势阱相组合的作用势,求解核核两体的定态薛定谔方程,得到了两体松束缚态的波函数和可能的能量本征量(核的长程库仑能的数量级)并说明了这种松束缚态的存在条件.对某些轻核,基态的结合能为若干keV.     

Second approximation of the generalized wkb method. I

Solutions are obtained for the one-dimensional Schrödinger equation corresponding to the second approximation of the generalized Wentzel-Kramers-Brillouin method (Petrashen-Miller-Good method). In the same approximation, the energy quantization conditions are found for bound particle states in a potential well.     

Local density of states of vortex state in mesoscopic superconductors

We study local density of states (LDOS) of vortex state in mesoscopic square superconductors with Bogoliubov-de Gennes (BdG) equation. We develop one effective numerical method based on the finite element method to self-consistently solve the BdG equation. The LDOS for various vortex states is obtained. Our results about the single vortex show that the LDOS has the particle-hole asymmetry and the results for one- and two-vortex state agree very well with the experimental observation. Besides, we predict the LDOS of multi-vortex states, which is crucial for the further STM/STS experimental study of vortex state in mesoscopic superconducting system.     

Particle clustering and Mott transitions in nuclear matter at finite temperature: (I). Method and general aspects

The thermodynamic state of nuclear matter as regards dependence on density and temperature is considered. Expressions for the association degree are derived describing the ratio of nuclear matter which is clustered to bound states. The problem of two nucleons imbedded in the surrounding nuclear matter is considered with the help of the Bethe-Goldstone equation for thermodynamic Green functions. The two-particle energy shift due to the effective nuclear matter hamiltonian is considered in a Hartree-Fock approximation, and a Mott density is obtained so that for densities of nuclear matter higher than the Mott density bound states cannot exist. With a simplified effective two-nucleon interaction the association degree is calculated as a function of the nucleon density and the temperature.     

On infinite walls in deformation quantization

We examine the deformation quantization of a single particle moving in one dimension (i) in the presence of an infinite potential wall, (ii) confined by an infinite square well, and (iii) bound by a delta function potential energy. In deformation quantization, considered as an autonomous formulation of quantum mechanics, the Wigner function of stationary states must be found by solving the so-called ?-genvalue (“stargenvalue”) equation for the Hamiltonian. For the cases considered here, this pseudo-differential equation is difficult to solve directly, without an ad hoc modification of the potential. Here we treat the infinite wall as the limit of a solvable exponential potential. Before the limit is taken, the corresponding ?-genvalue equation involves the Wigner function at momenta translated by imaginary amounts. We show that it can be converted to a partial differential equation, however, with a well-defined limit. We demonstrate that the Wigner functions calculated from the standard Schrödinger wave functions satisfy the resulting new equation. Finally, we show how our results may be adapted to allow for the presence of another, non-singular part in the potential.     

精确的量子化条件和不变量

提出并证明了一维量子系统和三维球对称量子系统的一个精确的量子化条件.在此精确量子化条件中, 除了通常的Nπ项外, 还有一积分项, 称为修正项. 发现该修正项正是在超对称量子力学中所谓的有形状不变势的量子系统的一个不变量，它不依赖于波函数的节点数.对这些系统, 可用基态能级和波函数确定此不变量的值, 从而由精确的量子化条件容易算出全部束缚态的能级. 计算得到能级的正确性又反过来验证了在有形状不变势的量子系统中此修正项确实是不变量.计算的有形状不变势的量子系统, 包括一维的有限方势阱、Morse势及其变形、R 关键词： 量子化条件 超对称量子力学 形状不变势 不变量     

无限深势阱中粒子的态密度

对于自由粒子在有限容器中的能态密度,热力学统计教材一般根据半经典量子图像,由驻波条件和德布罗意关系,以动量分立值为基础出发得到;然而根据量子理论,无限深势阱中的粒子存在能量本征态,而非动量本征态.本文以能量本征态为统计对象推导了有限体积中的自由粒子的能态密度,结果与教材一致.但是我们的处理方式显得更为自然.