谐振子

In this paper, the analytical solution of Smoluchowski equation, of which diffusive term is constant and drift term harmonic oscillator potential (including the well and the barrier potentials), is derived from a simple Gaussian distribution in a easy and wizard method.

当扩散项为常数、漂移项为一维谐振子势(包括谐振子势阱和谐振子位垒)的情况下，得到Smoluchowski方程的解析表达式。

This thesis analyzes the energy eigenvalue and of the Harmonic oscillator with a one-dimensional δ-function, makes a comparative study of those of the one-dimensional harmonic oscillator, and analyzes the reason which causes their different quality and the different influences to the eigenvalue of the even parity energy and to the wave function when the number of the "Z" is different.

本文研究了一维δ势场中的谐振子的能量本征值及其本征波函数，并与一维谐振子的能量本征值和波函数进行了比较，分析了它们的性质不同的原因以及取不同势参数Z值时对偶宇称能量本征值和波函数的不同影响。

In this paper, the current states of research about the basic theory of harmonic oscillator and hydrogen atom in quantum mechanics are firstly reviewed. On base of it, and the relationships between harmonic oscillator and hydrogen atom in quantum mechanics are studied. By use of the relationships between the hermite polynomial and the laguerre polynomial, the eigenequations of one-dimensional harmonic oscillator and hydrogen atom are conversed into the same equations in form. Therefore the relationships between energy levels and wave functions of one-dimensional harmonic oscillator and hydrogen atom are found. Through the coordinates transform, the relationships between energy levels and wave functions of two-dimensional harmonic oscillator and hydrogen atom are found.

首先综述了谐振子与氢原子的基本理论的研究现状，并在此基础上对谐振子与氢原子的关系展开了研究，通过厄密特方程与拉盖尔方程的相互转化，将一维谐振子与一维氢原子的本征值方程转化为相同形式的方程，从而比较得出它们能量及波函数间的关系，并通过坐标变换将直角坐标系下二维氢原子的本征值方程转化成与曲线坐标系下二维谐振子的本征值方程相同的形式，从而得出二维氢原子与二维谐振子的能量及波函数的关系。

Matrix element of coordinate operator ~l of harmonic oscillator is discussed by using coherent state and normal product.

利用相干态和正规乘积对谐振子任意次幂坐标算符^Xl矩阵元进行了讨论，导出了计算^Xl矩阵元的一般公式，为处理谐振子的微扰问题提供了一种新的方法。

Two basic theorems are given by researching Hamilton's two steps linear differential equations of bender equations which provide widespread application methods of normal differential operator correspond to ladder operators,then the solutions of general equations are obtained by taking advantage of the two above theorems: The energy eigenvalue equation of one-dimensional oscillator; the common state of ; the radial equation of a bondage state of hydrogen atom; the radial equation isotropic three-dimensional harmonic oscillator; the infinity deep potential well of ball

本文通过研究束缚态本征方程里的哈密顿算符的二阶线性常微分形式，给出两个基本定理并证明，提供构造二阶线性常微分算符相应的阶梯算符的普遍实用的方法，然后利用这两个定理构造量子力学中常见力学量的阶梯算符，并用以计算其能量本征值和对应本征函数：一维谐振子的能量本征方程；共同本征态；氢原子束缚态径向方程；三维各向同性谐振子的径向方程；无限深球势阱。

Using Extended Virial theorem , three recurrence relation of the average value of coordinate operator for linear harmonic oscillator, spherical harmonic oscillator and like-hydrogen ion were obtained respectively.

针对一维谐振子、类氢离子和三维各向同性谐振子三个量子体系，应用超维里定理，推导了幂坐标算符平均值的递推关系，方法较通常简洁明了。

Week 3 Simple vibrator model- spring vibrator, kinetic equation of simple vibrator and its solution, illustration method of simple vibration with rotating vector, energy of simple vibrator.

第3周 简谐振子模型——弹簧振子；简谐振子的动力学方程及其解；简谐振动的旋转矢量图示法。简谐振子的能量。

On one hand, profound understanding of harmonic oscillator model through various ways is crucial to understand the essence of quantum mechanics and research into micro physical models by means of quantum mechanics; on the other hand, the application of algebraic approach to solve the eigenvalues and wave function of mechanics is the main means to solve the actual problems in quantum mechanics.

简谐振子模型是量子力学中极其简单又重要的模型，其物理思想在其他相关的学科中都有着广泛的应用，通过多种途径去深入理解简谐振子模型，对理解量子力学的实质和运用量子力学作为工具去研究微观物理模型都有重要的意义；另一方面在实际工作中应用代数方法去求解力学量的本征值和波函数是研究量子力学的主要手段。

This research is built on a model that classifies the bio-molecules into two systems,harmonic oscillators and bipolar inharmonic oscillators. It discusses the local interaction of laser on certain links and bonds of bio-molecule by using quantum mechanics based on laser theory and bio-molecule structure.

在对激光和生物分子结构及其运动特性进行研究的基础上，将生物分子的按&谐振子&和&偶极非谐振子&模型处理，用量子力学讨论了激光与生物分子某些链、键的局域相互作用。

By using the laser-inharmonic oscillator system model, it was concluded that the bipolar inharmonic oscillator will be drastically stimulated when the laser reaches near-resonance frequency, the relative movements and total system momentum of the molecules will also rapidly increase, cause easy breaking of the bonds and links.

用得到的&激光-偶极非谐振子&相互作用的一些关系式，可导出当激光频率趋于&近共振频率&时，非谐振子将受到急剧激发，相对位移及动量也剧烈增大，振子震荡加剧，极易引起链、键的断裂。

On the other hand, the more important thing is because the urban housing is a kind of heterogeneity products.

另一方面，更重要的是由于城市住房是一种异质性产品。

Climate histogram is the fall that collects place measure calm value, cent serves as cross axle for a few equal interval, the area that the frequency that the value appears according to place is accumulated and becomes will be determined inside each interval, discharge the graph that rise with post, also be called histogram.

气候直方图是将所收集的降水量测定值，分为几个相等的区间作为横轴，并将各区间内所测定值依所出现的次数累积而成的面积，用柱子排起来的图形，也叫做柱状图。