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%.     

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.     

变形对称双阱势

利用变形双曲函数,将对称双阱势推广为变形对称双阱势模型.证明了该势模型是五参量指数型势模型的特例.运用超对称WKB近似和五参量指数型势模型的能谱公式,获得了变形对称双阱势模型的能谱公式.     

量子力学中无限深势阱问题的教学研究

在一维问题的基础上探讨了二维无限深势阱的问题,发掘二维无限深势阱在不同边界约束情况下不同于一维问题的特征和应用.     

一维无限深势阱内粒子概率密度演示仪

设计了一维无限深势阱中粒子概率密度演示仪,利用粒子的态函数的驻波图像来演示粒子的概率密度,使学生能直观地观察一维无限深势阱内粒子的概率密度分布规律.     

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.     

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。     

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     

一维梯形势垒透射系数的计算

构造了在超晶格物理中具有潜在应用价值的一维梯形势垒模型并解析地得到了粒子隧穿势垒的透射系数.给出了该透射系数在低能近似和微斜近似下的近似表达式,并指出它可以视为方势阱透射系数在低能下的修正.此外,区别于方势垒模型,梯形势垒透射系数的峰值并不一定对应于共振透射,但是峰值处对应的粒子入射能量近似地满足势垒高度和相应一维无限深梯形势阱中粒子能级之和的规律.     

势阱中粒子能级与波函数微扰计算的代数递推公式

利用超位力定理（HVT）和Hellmann-Feynman定理（HFT）,导出了由有精确解的势阱的能级值用微扰法直接计算一维势阱的各级近似能级的普遍代数公式,并导出由能级近似值计算定态波函数近似表达式的代数公式,给出了代数公式具体应用的几个典型一维势阱实例,此法可推广到二维势阱与三维势阱的情形。     

Supersymmetric quantum mechanics for two-dimensional disk

The infinite square well potential in one dimension has a smooth supersymmetric partner potential which is shape invariant. In this paper, we study the generalization of this to two dimensions by constructing the supersymmetric partner of the disk billiard. We find that the property of shape invariance is lost in this case. Nevertheless, the WKB results are significantly improved when SWKB calculations are performed with the square of the superpotential. We also study the effect of inserting a singular flux line through the center of the disk.     

Supersymmetry and SWKB Approach to Dirac Equation with Hyperbolic Scarf Potential

By using the supersymmetric quantum mechanics and shape invariance concept, we study the Dirac equation with the hyperbolic Scarf potential and the exact energy spectrum is obtained. Also, we calculate the bound state energy eigenvalues by using the supersymmetric WKB approximation approach so that we get the same results.     

Exact solutions of equation of transverse vibrations for a bar with a specific cross section variation law

Flexural wave propagation along a bar whose thickness smoothly decreases down to zero within its end piece is considered. The propagation velocity tends to zero as the tapered end of the bar is approached, and the time of wave propagation to the tapered end is infinite. As a consequence, waves propagating along the bar are not reflected from the end. Previous quantitative study of the effect in the WKB approximation shows that, in the case of parabolic tapering, the WKB approximation yields a uniform asymptotics, which is valid (or invalid) for any of the bar’s cross sections. In the case of a bar with parabolic tapering, the equation of flexural vibrations of the bar has exact analytic solutions in the form of power functions. Based on these solutions, a modified WKB approximation is proposed to solve equations for bars with nonparabolic thickness variation laws. The input impedance of a bar with a parabolic tapering is calculated and analyzed.     

Nonlocal potentials and their exact and approximate local and velocity-dependent equivalents

We show that good approximations to the exact equivalent local potential (ELP) and damping factor of a nonlocal Perey-Buck potential can be calculated in the partial wave WKB approximation of Horiuchi. The exact ELP and damping factor are obtained by means of a method previously given by one of us. We also confirm that an approximate ELP proposed by Bauhoff et al. is of comparable accuracy as the Horiuchi approximation. Thesel-dependent ELP's exhibit reduced attraction in the interior and provide a test for higher order WKB approximations. We subsequently obtain an equivalent velocity dependent potential (EVDP) which is even exactly wave function equivalent to the original nonlocal potential. This almost local potential, unlike the trivial equivalent local potential, is smooth and well-behaved and is therefore particularly useful in nuclear reactions where the off-shell behaviour of the potential is important.     

Finite renormalizations of 0Sp(N, 4) supersymmetry

The calculation of the one-loop effective potential of the Wess-Zumino model is carried out using Green functions which propagate fields inn-dimensional anti-de Sitter space. The divergent parts of the amplitudes are independent of the choice of boundary conditions. The finite counterterms can be adjusted in such a way that the renormalized action be supersymmetric invariant. Addressing the question of the survival of the supersymmetry invariance of the vacuum state, we derive the result of the persistence of supersymmetry in the semiclassical approximation.     

量子系统的能量本征值在超对称性、形状不变性框架下的计算

在超对称性、形状不变性框架下，计算了一维修正Ｐｓｃｈｌ-Ｔｅｌｅｒ势、Ｎ维氢原子和Ｎ维各向同性谐振子的能量本征值．得到的能谱公式与用通常的因子化方法得到的严格解完全一致 关键词：     

Tight binding electrons on a disordered square lattice in a magnetic field

A semiclassical WKB treatment of the density of states spectrum of tight-binding electrons moving in a disordered two dimensional lattice in the presence of a transverse magnetic field is presented. The disorder is accounted for in the coherent potential approximation and analytical results are derived. For both ordered and disordered systems the line position of magnetic subbands as well as the cluster lineshape of the density of states agree quite well with exact numerical results.     

具有含时频率和边界条件的谐振子量子态的Berry相位

对具有一运动边界的一维无限深势阱内频率随时间变化的谐振子的含时Schrdinger方程连续进行两次规范变换，可以得到精确解和Lewis不变量算符.基于该精确解利用几何距离和曲线的几何长度概念计算了体系量子态的Berry相位. 关键词：