计算物理shooting for boundary value porblem 打靶法解决边界条件微分方程问题
解常微分方程组-边界值问题(Boundary
第十章解常微分方程組-邊界值問題(Boundary Value Problems for ODE)本章探討的邊界值問題模型如下:x''(t) = f(t, x(t), x'(t))x(a) = α , x(b) = β .考慮當函數f(t, x(t), x'(t))是線性(或非線性)時, 求解之方法:1. Discretization Method( finite difference approximations)2. Shooting Method在本章中包含Matlab 的m-file1. finitedf.m2.shoot.m3. shootnl.m將須要的m-file之檔案夾加入搜尋路徑中path('d:\numerical', path)註: 如果你有安裝Matlab Notebook 要執行下列input cells (綠色指令敘述)之前必須先執行上面的cell –[path (…) ]藍色的內容是Matlab [output cells]1. finitedf.m –利用finite difference approximationsx'(t) ~ (x(t+h) - x(t-h) ) / (2 h) ;x''(t) ~ (x(t+h) - 2 (t) + x(t-h) )/ h2 ;來估算邊界值問題x''(t) = f(t, x(t), x'(t))= u(t) t + v(t) x(t) + w(t) x'(t)x(a) = α , x(b) = β. 的數值解 .下列的m-file(finitedf.m)已經把例題中的u(t), v(t), w(t)定義了, 使用者套用此函數時, 記得更正為自己需要的.type finitedf.mfunction rs= finitedf(x0,xn, ta,tb,n)%to solve x''(t) = f(t, x(t), x'(t))=u(t) t + v(t) x(t) + w(t) x'(t)%by finite difference approximations, x0 and xn are the boundary values %t is in[ta,tb].ut = inline('exp(t) - 3*sin(t)', 't') ;vt = -1 ;wt = 1 ;h=(tb - ta)/n ;a=zeros(1,n); b=a; c=a; d=a;for i=1:n-1t = ta + i*h ;a(i) = -( 1+ h * wt/2) ;d(i) = 2 + h^2 * vt;c(i) = -( 1- h * wt/2) ;b(i) = -h^2* ut(t) ;end ;b(1) = b(1) - a(1)* x0 ;b(n-1) = b(n-1) - c(n-1)* xn ;for i=1:n-1a(i) = a(i+1) ;end ;y = trigauss(a, d, c,b,n-1) ;rs = y ;例題 1: 解邊界值問題x''(t) = e t - 3 sin(t) + x'(t) - x(t) ,x(1) = 1.09737491 , x(2) = 8.63749661ta = 1 ; tb = 2; n = 99;h = (tb- ta)/n ; error = zeros(1,n) ;x0 = 1.09737491 ; xn = 8.63749661 ;y = finitedf(x0, xn, ta, tb, n);for i = 1 : n-1t = ta + i * h ;error(i) = exp(t)- 3*cos(t)- y(i) ;end ;format longfprintf('\t t \t x(t) \t error\n') disp([1 x0 0])for i = 9 :9:n-1t = ta + i * h ;disp([t y(i) error(i)])end ;disp([2 xn 0])t x(t) error1.00000000000000 1.09737491000000 01.09090909090909 1.59194209918682 -0.000000365918851.181818181818182.12256804212776 -0.000000694314561.272727272727272.68955333563720 -0.000000963621881.36363636363636 3.29334396781902 -0.000001156328151.45454545454545 3.93457004989182 -0.000001259030391.54545454545455 4.61408706401417 -0.000001263391911.63636363636364 5.33301967061334 -0.000001167139561.72727272727273 6.09280813414278 -0.000000975096821.81818181818182 6.89525744460769 -0.000000700247811.90909090909091 7.74258923378422 -0.000000364826842.00000000000000 8.63749661000000 02. shoot.m –利用shooting method 來估算邊界值問題此方法是將邊界值問題轉為初值問題:x''(t) = f(t, x(t), x'(t)) = u(t) t + v(t) x(t) + w(t) x'(t)x(a) = α , x'(a) = z.其解x(t)在b的數值x(b)可視為z的函數φ(z) .希望好的z值能使得φ(z) =β.當φ(z)是線性時, 考慮函數g(t)=λx1(t) + (1-λ) x2(t) ,其中x1(t)與x2(t)分別是x'(a) = z1與.x'(a) = z2初值問題的解.解此初值問題可利用前面的函數rk4sys.mtype shoot.mfunction rs= shoot(x0,xn, ta,tb,n)%to solve x''(t) = f(t,x(t),x'(t))=u(t) t+ v(t) x(t)+ w(t) x'(t) %by finite difference approximations, x0 and xn are the%boundary values, t is in[ta,tb].ut = inline('exp(t) - 3*sin(t)', 't') ;vt = -1 ;wt = 1 ;h=(tb - ta)/n ; m=5 ;x=[1 x0 0 x0 1]';xt=zeros(m,n);xtnew=zeros(1,n) ;xt=rk4sysnew(x,ta,tb,n) ;xb1=xt(2,n) ;xb2=xt(4,n) ;p = (xn-xb2) / (xb1-xb2) ;q = 1-p ;xtnew=p*xt(2,1:n) +q*xt(4,1:n) ;%return the approximation solutionrs = xtnew ;type fxsys.mfunction fx=fxsys(t,X)%compute values of function F(t,X)%F(t,X) is defined by user%In this example, variable X includes t which is%the first component X(1)fx=zeros(length(X),1);fx(1) = 1 ;fx(2) = X(3);fx(3) = exp(X(1)) - 3*sin(X(1)) + X(3) -X(2);fx(4) = X(5) ;fx(5) = exp(X(1)) - 3*sin(X(1)) + X(5) -X(4); ;例題 2:利用shooting method解邊界值問題x''(t) = e t - 3 sin(t) + x'(t) - x(t) ,x(1) = 1.09737491 , x(2) = 8.63749661ta = 1 ; tb = 2; n = 99;h = (tb- ta)/n ; error = zeros(1,n) ;x0 = 1.09737491 ; xn = 8.63749661 ;y = shoot(x0, xn, ta, tb, n);for i = 1 : n-1t = ta + i * h ;error(i) = exp(t)- 3*cos(t)- y(i) ;end ;format longfprintf('\t t \t x(t) \t error\n') disp([1 x0 0])for i = 9 :9:n-1t = ta + i * h ;disp([t y(i) error(i)])end ; disp([2 xn 0])t x(t) error1.00000000000000 1.09737491000000 01.09090909090909 1.59194173254025 0.000000000727731.181818181818182.12256734722770 0.000000000585511.272727272727272.68955237158771 0.000000000427611.36363636363636 3.29334281123716 0.000000000253711.45454545454545 3.93456879079791 0.000000000063531.54545454545455 4.61408580076543 -0.000000000143171.63636363636364 5.33301850384033 -0.000000000366551.72727272727273 6.09280715965267 -0.000000000606711.81818181818182 6.89525674522358 -0.000000000863701.90909090909091 7.74258887009486 -0.000000001137472.00000000000000 8.63749661000000 03. shootnl.m –利用shooting method 來估算非線性的邊界值問題x''(t) = f(t, x(t), x'(t)) , x(a)= α , x(b)=β同樣地, 將邊界值問題轉為初值問題:x(a) = α , x'(a) = z.其解x(t)在b的數值x(b)可視為z的函數φ(z) .希望好的z值能使得φ(z) =β , 採用secant method .type shootnl.mfunction rs= shootnl(x0,xn, ta,tb,n)%to solve nonlinear x''(t) = f(t,x(t),x'(t)) boundary-Value problem%by Shooting method, x0 and xn are the boundary values,%t is in[ta, tb].%%x''(t)= -x'(t)^2 -x(t) + ln(t), t is in [1,2], x(1)=0, x(2) = ln2 .%exact solution x(t) = ln(t) .% x1(t)=t, x1'(t)=1, x2(t)= x(t), x2'(t)= x3(t)% x3'(t)= x2''(t)= -x3(t)^2 - x2(t) +ln(t)h=(tb - ta)/n ; m=3 ;%for some zz=[0.5 1] ; pz=[0 0] ; %initialepi=10^(-10) ;for j=1: 2x=[1 x0 z(j)]' ; %initialsxt=zeros(m,n) ;xt=rk4sysnew(x,ta,tb,n) ;pz(j)=xn -xt(2,n) ;if pz(j)< epirs = xt(2, 1:n) ;return %end ;end ;%compute the next z by secant methoditer= 15; tol = 10^-5 ; err=0.0 ;i =3 ;while(i <= iter & err < tol)slp= (z(i-1)-z(i-2))/(pz(i-1)-pz(i-2)) ;nextz = z(i-1)- slp*pz(i-1);err = abs(nextz - pz(i-1)) ;z=[z nextz] ;x=[1 x0 nextz]' ; %initialsxt=zeros(m,n) ;xt=rk4sysnew(x,ta,tb,n);pz(i)=xn -xt(2,n) ;if (err < tol | pz(i)<epi )rs = xt(2, 1:n ) ;returnend ;i= i+1 ;endif i>iterprintf(' Secant method fails in Shooting method \n' ) endrs = xt(2, 1:n); %returnstype fxsys.mfunction fx=fxsys(t,X)%compute values of function F(t,X)%F(t,X) is defined by user%In this example, variable X includes t which is%the first component X(1)fx=zeros(length(X),1);fx(1) = 1 ;fx(2) = X(3);fx(3) = -X(3).^2 - X(2) +log(X(1)) ;例題 3:利用shooting method 來估算非線性的邊界值問題x''(t) = -x'(t)2 - x(t) + ln(t) 1≤ t ≤ 2x(1) = 0 , x(2) = ln2 .ta = 1 ; tb = 2; n = 10;h = (tb- ta)/n ; error = zeros(1,n) ;x0 = 0.0 ; xn = log(2) ;y = shootnl(x0, xn, ta, tb, n);for i = 1 : n-1t = ta + i * h ;error(i) = log(t)- y(i) ;end ;format longfprintf('\t t \t x(t) \t error\n') fprintf('\t 1 \t 0 \t 0\n') for i = 1 :n-1t = ta + i * h ;disp([t y(i) error(i)])end ; disp([2 xn 0])t x(t) error1 0 01.10000000000000 0.09531000323697 0.000000176567361.20000000000000 0.18232131442583 0.000000242368121.30000000000000 0.26236400845782 0.000000256009671.40000000000000 0.33647199127437 0.000000245346841.50000000000000 0.40546488415551 0.000000223952651.60000000000000 0.47000343073393 0.000000198511811.70000000000000 0.53062807876778 0.000000172294391.80000000000000 0.58778651805356 0.000000146848561.90000000000000 0.64185376332331 0.000000122849092.00000000000000 0.69314718055995 0。
罗伯特方法提取边缘计算题
《聊聊罗伯特方法提取边缘计算题》嘿，朋友！今天咱来唠唠一个超有意思的玩意儿——罗伯特方法提取边缘计算题。
这名字听着是不是有点高大上？但别被它唬住啦，其实没那么难理解。
咱先说说啥是罗伯特方法。
这就像是一个魔法小技巧，能把图像的边缘给找出来。
就好像你在画画的时候，想把物体的轮廓画得更清楚，罗伯特方法就是那个能帮你把轮廓变得更明显的神奇招数。
那这个罗伯特方法咋用在计算题上呢？嘿嘿，这可就有点门道了。
比如说，有一个图像，咱想知道它的边缘在哪里。
通过一些数学运算，就能用罗伯特方法把边缘给算出来。
这就跟玩解谜游戏似的，一步一步地找到答案。
想象一下，你就像一个小侦探，拿着罗伯特方法这个秘密武器，去解开图像边缘的谜题。
每一步计算都像是在寻找线索，一点一点地接近真相。
可能一开始你会觉得有点晕乎，这都是正常的。
别着急，咱慢慢琢磨。
可以先从简单的例子开始，熟悉一下这个方法的套路。
等你掌握了窍门，就会发现它其实挺好玩的。
比如说，有一个小小的图形，咱用罗伯特方法去算它的边缘。
看着那些数字和公式，别害怕，把它们当成你的小伙伴。
跟它们聊聊天，问问它们该怎么组合才能找到边缘。
也许在计算的过程中会遇到一些小挫折，算错了或者卡住了。
没关系呀！这都是学习的过程。
就像你走路摔了一跤，爬起来拍拍土，继续往前走。
而且，当你终于算出了图像的边缘，那种成就感可别提了！就像你解开了一个超级难的谜语，心里那叫一个美。
所以说，别被罗伯特方法提取边缘计算题这个名字吓住。
大胆地去尝试，去探索。
说不定你会发现自己原来是个计算小天才呢！加油哦，小伙伴！让我们一起在罗伯特方法的世界里玩耍，解开一个又一个的边缘谜题。
泊松方程边值问题的弱形式
泊松方程边值问题的弱形式泊松方程是一种常见的偏微分方程，用于描述在无外力作用下，保持平衡的物理系统。
该方程在众多科学和工程领域中具有重要的应用，如电场分布、热传导和流体流动等。
泊松方程的基本形式如下：∇^2 Φ = f其中，Φ是待求函数，表示系统中某一物理量的势能或众多物理量之一，f是已知函数，表示泊松方程右侧的源项。
而边值问题是在给出了方程的某些边界条件条件下，求解泊松方程的解。
泊松方程的弱形式是一种通过假设解Φ与一个测试函数v之间的内积为零来定义的方程形式。
这可以通过将原方程乘以一个测试函数v，并对整个域Ω进行积分得到。
这样的话，泊松方程的弱形式可以表达为：∫∇^2 Φ v dV = ∫f v dV根据格林第一公式，上述弱形式可以写作：-∫∇Φ ∇v dV + ∫f v dV = 0通过将第一项应用于分部积分并丢弃边界积分项，我们可以将弱形式进一步简化为：∫∇Φ ∇v dV = ∫f v dV这样，泊松方程的弱形式就变成了一个求解一个变分问题的形式，其中Φ和v都是函数空间中的元素。
此时，我们可以使用适当的数学方法来求解这个边值问题。
对于给定的边界条件，我们可以选择合适的测试函数v，并将其施加在泊松方程的弱形式中。
然后，我们使用数学工具和算法对其进行离散化，转化为线性或非线性代数方程组，并求解出Φ的近似解。
泊松方程边值问题的弱形式在数学和工程领域中具有广泛的应用。
例如，在电场的模拟和建模中，可选择Φ为电势，f为电荷密度，以求解电场分布。
在热传导中，可将Φ视为温度场，f为热源密度，以求解温度分布。
此外，该方法还可应用于流体流动中的速度场、压力场等问题的求解。
总之，泊松方程边值问题的弱形式为求解复杂的偏微分方程提供了一种更便捷和高效的途径。
通过合适的方法和算法，我们可以将边值问题转化为数值方程，并求解出所需物理量的近似解，从而更好地理解和应用泊松方程。
物理实验技术中的数据处理算法和计算工具推荐
物理实验技术中的数据处理算法和计算工具推荐在物理实验中，数据处理是非常重要且不可或缺的步骤。
通过对实验数据的精确处理和分析，我们能够从中得出科学结论并验证理论模型。
为了提高数据处理的准确性和效率，使用适当的算法和计算工具非常重要。
本文将推荐几种常用的数据处理算法和计算工具，旨在帮助物理实验者提高数据处理和分析的水平。
一、数据处理算法1. 平均值算法：在实验中，我们通常需要重复测量同一物理量多次。
这时，计算平均值可以减小因外界干扰引起的误差，提高测量结果的精度。
平均值算法是将所有测量结果相加，然后除以测量次数。
2. 标准差算法：标准差是测量数据分散程度的一种指标。
它可以告诉我们测量结果的可靠性和精确性。
标准差算法是将每个测量结果与平均值的差的平方相加，然后除以测量次数的平方根。
3. 最小二乘法：最小二乘法用于拟合实验数据和理论模型之间的关系。
它通过最小化实际观测值与理论预测值之间的残差平方和，找到最佳拟合线或曲线。
4. 傅里叶变换：傅里叶变换是将一个函数或信号从时域转换到频域的一种数学方法。
在物理实验中，傅里叶变换广泛应用于信号分析、频谱分析和图像处理等领域。
二、计算工具推荐1. MATLAB：MATLAB是一种非常强大的科学计算软件，它提供了丰富的功能和工具箱，可以用于各种数据处理和分析任务。
MATLAB具有友好的界面和易于使用的语法，可以对实验数据进行快速处理、绘图和分析。
2. Python：Python是一种流行的编程语言，也被广泛应用于科学计算和数据处理。
Python拥有丰富的科学计算库，如NumPy和SciPy，可以支持各种数据处理算法和方法。
3. Origin：Origin是一种专业的数据分析和图形绘制软件，它可以用于各种科学数据的处理和分析。
Origin具有强大的绘图功能和可定制性，可以生成高质量的图形和图表。
4. Excel：Excel是一种常见的电子表格软件，它也可以用于简单的数据处理和分析。
球的外域poisson方程的第一边值问题
球的外域poisson方程的第一边值问题球的外域Poisson方程是一个描述球外电势分布的方程，它在物理学中有着广泛的应用。
第一边值问题是指需要给定边界条件的求解问题。
球的外域Poisson方程可以表示为：∇²V=-4πρ其中∇²是拉普拉斯算子，V是电势，ρ是球外的电荷分布。
在球的外域，我们需要给定边界条件，通常包括球体表面上的电势和球外无穷远处的电势。
具体来说，我们需要给定以下边界条件：1.对于球体表面上的电势，我们可以给定球上电势分布的函数形式。
例如，我们可以假设球体带有均匀分布的电荷，在球体上给定一个电势值。
2.对于球外无穷远处的电势，由于距离趋于无穷远时电势趋于0，我们可以给定球外无穷远处的电势为0。
求解球的外域Poisson方程的第一边值问题的一种常用方法是使用分离变量法。
我们可以假设球体电势的解具有球坐标形式，可表示为：V(r,θ,φ)=R(r)Y(θ,φ)通过将电势的球坐标形式代入Poisson方程，并分离变量，我们可以得到关于R(r)和Y(θ,φ)的两个独立的方程。
其中一个方程是径向方程，另一个方程是角向方程。
径向方程可以写为：1/r² d/dr (r² dR/dr) + k²R = 0其中k是积分常数。
通过求解径向方程，我们可以得到R(r)的解。
角向方程可以写为：1/Y sinθ d/dθ (sinθ dY/dθ) + [k² - m²/(sin²θ)] Y = 0其中m是另一个积分常数。
通过求解角向方程，我们可以得到Y(θ,φ)的解。
最终，将R(r)和Y(θ,φ)的解合并，就可以得到球体电势的解V(r,θ,φ)。
通过给定球体表面上的电势和球外无穷远处的电势，我们可以确定常数k和m的值。
具体而言，在球体表面上，根据边界条件，我们可以确定k的值。
在球外无穷远处，电势为0，因此可以确定m的值。
总结起来，球的外域Poisson方程的第一边值问题可以通过分离变量法求解。
unity中lambert计算公式
unity中lambert计算公式
lambert光照模型属于经验模型，主要用来简单模拟粗糙物体表面的光照现象
此模型假设物体表面为理想漫反射体（也就是只产生漫反射现象，也成为Lambert反射体），同时，场景中存在两种光，一种为环境光，一种为方向光，然后我们分别计算这两种光照射到粗糙物体表面所产生的光照现象，最后再将两个结果相加，得出反射后的光强值。
首先是计算环境光的公式：
I_inDirectionDiffuse = K_d * I_a;
其中，K_d为粗糙物体表面材质对光的反射系数，这个系数由程序编写者在宿主程序中给出，I_a为环境光的光强，也就是环境光的颜色数值，一般是一个float3型的变量然后是计算方向光的公式：
I_directionDiffuse = K_d * I_l * max(0，dot(N, L));
其中I_l为方向光的光强，也就是其颜色值；N为顶点的单位法向量；L为入射光的单位法向量（注意，光照向量是从顶点指向光源的向量；也就是，它与线的传播方向正好相反）。
这个公式与计算环境光的不同，对于环境光，我们不关心它的方向，因为环境光也没有方向，它给予物体的光照在各个顶点处均是一样的。
而方向光则需要关注其方向，例如一个聚光灯，灯从不同的角度来照射物体所产生的效果也是不一样的，光线方向越靠近法线，漫反射出来的光就越强，反之则越弱。
方向光的漫反射强度遵循Lambert余弦定理。
计算边界界面陷阱的方法
计算边界界面陷阱的方法边界界面陷阱（Boundary Interface Traps）是指半导体材料中的杂质或缺陷，可能会在材料的边界或界面上形成能量陷阱。
这些陷阱对于半导体器件的性能和可靠性都有很大影响，因此研究和计算边界界面陷阱的方法至关重要。
以下是几种常用的计算边界界面陷阱的方法：1.第一性原理方法：第一性原理方法是一种基于量子力学原理的计算方法，可以对材料的电子结构进行精确计算。
通过此方法，可以计算出杂质原子或缺陷在材料中的位置和能级分布。
常用的第一性原理计算方法包括密度泛函理论（DFT）和GW近似等。
这些方法可以预测杂质或缺陷对材料能带结构的影响，从而确定边界界面陷阱的性质。
2.电致发光谱（DLTS）：电致发光谱是一种通过测量材料在电场激励下的光发射来研究陷阱特性的实验方法。
通过测量电流与电压的响应，可以获得材料中的陷阱能级和浓度信息。
这种方法特别适用于测量有限结构（如PN结）中的陷阱，但对于界面陷阱的测量会更加复杂。
3.电感谱（DLTS）：电感谱是一种通过测量材料在变化的电场下的电容变化来研究陷阱特性的方法。
通过测量样品的电容随频率的变化，可以获得材料中的陷阱时间常数和浓度。
这种方法对于界面陷阱的测量也是有效的。
4.电波谱学：电波谱学是一种通过测量边界界面陷阱对电磁波的吸收和散射来研究其能级和浓度的方法。
常见的电波谱学方法包括微波谱、光反射和透射谱。
这些方法可以提供丰富的界面陷阱特性信息，如能级分布、捕获截面和陷阱分布等。
5.器件模拟：通过建立半导体器件的数值模型，结合材料电学参数和陷阱特性等信息，可以通过模拟器件的工作过程来研究边界界面陷阱的影响。
该方法通常基于Poisson方程和Drift-Diffusion方程，并采用隐式或显式数值方法求解。
器件模拟可以预测边界界面陷阱对器件的性能和可靠性的影响，帮助优化器件设计。
综上所述，计算边界界面陷阱的方法多种多样，可以从第一性原理计算、实验测量到数值模拟等多个层面来研究。
物理学专业英语
华中师范大学物理学院物理学专业英语仅供内部学习参考！2014一、课程的任务和教学目的通过学习《物理学专业英语》，学生将掌握物理学领域使用频率较高的专业词汇和表达方法，进而具备基本的阅读理解物理学专业文献的能力。
通过分析《物理学专业英语》课程教材中的范文，学生还将从英语角度理解物理学中个学科的研究内容和主要思想，提高学生的专业英语能力和了解物理学研究前沿的能力。
培养专业英语阅读能力，了解科技英语的特点，提高专业外语的阅读质量和阅读速度；掌握一定量的本专业英文词汇，基本达到能够独立完成一般性本专业外文资料的阅读；达到一定的笔译水平。
要求译文通顺、准确和专业化。
要求译文通顺、准确和专业化。
二、课程内容课程内容包括以下章节：物理学、经典力学、热力学、电磁学、光学、原子物理、统计力学、量子力学和狭义相对论三、基本要求1.充分利用课内时间保证充足的阅读量（约1200~1500词/学时），要求正确理解原文。
2.泛读适量课外相关英文读物，要求基本理解原文主要内容。
3.掌握基本专业词汇（不少于200词）。
4.应具有流利阅读、翻译及赏析专业英语文献，并能简单地进行写作的能力。
四、参考书目录1 Physics 物理学 (1)Introduction to physics (1)Classical and modern physics (2)Research fields (4)V ocabulary (7)2 Classical mechanics 经典力学 (10)Introduction (10)Description of classical mechanics (10)Momentum and collisions (14)Angular momentum (15)V ocabulary (16)3 Thermodynamics 热力学 (18)Introduction (18)Laws of thermodynamics (21)System models (22)Thermodynamic processes (27)Scope of thermodynamics (29)V ocabulary (30)4 Electromagnetism 电磁学 (33)Introduction (33)Electrostatics (33)Magnetostatics (35)Electromagnetic induction (40)V ocabulary (43)5 Optics 光学 (45)Introduction (45)Geometrical optics (45)Physical optics (47)Polarization (50)V ocabulary (51)6 Atomic physics 原子物理 (52)Introduction (52)Electronic configuration (52)Excitation and ionization (56)V ocabulary (59)7 Statistical mechanics 统计力学 (60)Overview (60)Fundamentals (60)Statistical ensembles (63)V ocabulary (65)8 Quantum mechanics 量子力学 (67)Introduction (67)Mathematical formulations (68)Quantization (71)Wave-particle duality (72)Quantum entanglement (75)V ocabulary (77)9 Special relativity 狭义相对论 (79)Introduction (79)Relativity of simultaneity (80)Lorentz transformations (80)Time dilation and length contraction (81)Mass-energy equivalence (82)Relativistic energy-momentum relation (86)V ocabulary (89)正文标记说明：蓝色Arial字体（例如energy）：已知的专业词汇蓝色Arial字体加下划线（例如electromagnetism）：新学的专业词汇黑色Times New Roman字体加下划线（例如postulate）：新学的普通词汇1 Physics 物理学1 Physics 物理学Introduction to physicsPhysics is a part of natural philosophy and a natural science that involves the study of matter and its motion through space and time, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic disciplines, perhaps the oldest through its inclusion of astronomy. Over the last two millennia, physics was a part of natural philosophy along with chemistry, certain branches of mathematics, and biology, but during the Scientific Revolution in the 17th century, the natural sciences emerged as unique research programs in their own right. Physics intersects with many interdisciplinary areas of research, such as biophysics and quantum chemistry,and the boundaries of physics are not rigidly defined. New ideas in physics often explain the fundamental mechanisms of other sciences, while opening new avenues of research in areas such as mathematics and philosophy.Physics also makes significant contributions through advances in new technologies that arise from theoretical breakthroughs. For example, advances in the understanding of electromagnetism or nuclear physics led directly to the development of new products which have dramatically transformed modern-day society, such as television, computers, domestic appliances, and nuclear weapons; advances in thermodynamics led to the development of industrialization; and advances in mechanics inspired the development of calculus.Core theoriesThough physics deals with a wide variety of systems, certain theories are used by all physicists. Each of these theories were experimentally tested numerous times and found correct as an approximation of nature (within a certain domain of validity).For instance, the theory of classical mechanics accurately describes the motion of objects, provided they are much larger than atoms and moving at much less than the speed of light. These theories continue to be areas of active research, and a remarkable aspect of classical mechanics known as chaos was discovered in the 20th century, three centuries after the original formulation of classical mechanics by Isaac Newton (1642–1727) 【艾萨克·牛顿】.University PhysicsThese central theories are important tools for research into more specialized topics, and any physicist, regardless of his or her specialization, is expected to be literate in them. These include classical mechanics, quantum mechanics, thermodynamics and statistical mechanics, electromagnetism, and special relativity.Classical and modern physicsClassical mechanicsClassical physics includes the traditional branches and topics that were recognized and well-developed before the beginning of the 20th century—classical mechanics, acoustics, optics, thermodynamics, and electromagnetism.Classical mechanics is concerned with bodies acted on by forces and bodies in motion and may be divided into statics (study of the forces on a body or bodies at rest), kinematics (study of motion without regard to its causes), and dynamics (study of motion and the forces that affect it); mechanics may also be divided into solid mechanics and fluid mechanics (known together as continuum mechanics), the latter including such branches as hydrostatics, hydrodynamics, aerodynamics, and pneumatics.Acoustics is the study of how sound is produced, controlled, transmitted and received. Important modern branches of acoustics include ultrasonics, the study of sound waves of very high frequency beyond the range of human hearing; bioacoustics the physics of animal calls and hearing, and electroacoustics, the manipulation of audible sound waves using electronics.Optics, the study of light, is concerned not only with visible light but also with infrared and ultraviolet radiation, which exhibit all of the phenomena of visible light except visibility, e.g., reflection, refraction, interference, diffraction, dispersion, and polarization of light.Heat is a form of energy, the internal energy possessed by the particles of which a substance is composed; thermodynamics deals with the relationships between heat and other forms of energy.Electricity and magnetism have been studied as a single branch of physics since the intimate connection between them was discovered in the early 19th century; an electric current gives rise to a magnetic field and a changing magnetic field induces an electric current. Electrostatics deals with electric charges at rest, electrodynamics with moving charges, and magnetostatics with magnetic poles at rest.Modern PhysicsClassical physics is generally concerned with matter and energy on the normal scale of1 Physics 物理学observation, while much of modern physics is concerned with the behavior of matter and energy under extreme conditions or on the very large or very small scale.For example, atomic and nuclear physics studies matter on the smallest scale at which chemical elements can be identified.The physics of elementary particles is on an even smaller scale, as it is concerned with the most basic units of matter; this branch of physics is also known as high-energy physics because of the extremely high energies necessary to produce many types of particles in large particle accelerators. On this scale, ordinary, commonsense notions of space, time, matter, and energy are no longer valid.The two chief theories of modern physics present a different picture of the concepts of space, time, and matter from that presented by classical physics.Quantum theory is concerned with the discrete, rather than continuous, nature of many phenomena at the atomic and subatomic level, and with the complementary aspects of particles and waves in the description of such phenomena.The theory of relativity is concerned with the description of phenomena that take place in a frame of reference that is in motion with respect to an observer; the special theory of relativity is concerned with relative uniform motion in a straight line and the general theory of relativity with accelerated motion and its connection with gravitation.Both quantum theory and the theory of relativity find applications in all areas of modern physics.Difference between classical and modern physicsWhile physics aims to discover universal laws, its theories lie in explicit domains of applicability. Loosely speaking, the laws of classical physics accurately describe systems whose important length scales are greater than the atomic scale and whose motions are much slower than the speed of light. Outside of this domain, observations do not match their predictions.Albert Einstein【阿尔伯特·爱因斯坦】contributed the framework of special relativity, which replaced notions of absolute time and space with space-time and allowed an accurate description of systems whose components have speeds approaching the speed of light.Max Planck【普朗克】, Erwin Schrödinger【薛定谔】, and others introduced quantum mechanics, a probabilistic notion of particles and interactions that allowed an accurate description of atomic and subatomic scales.Later, quantum field theory unified quantum mechanics and special relativity.General relativity allowed for a dynamical, curved space-time, with which highly massiveUniversity Physicssystems and the large-scale structure of the universe can be well-described. General relativity has not yet been unified with the other fundamental descriptions; several candidate theories of quantum gravity are being developed.Research fieldsContemporary research in physics can be broadly divided into condensed matter physics; atomic, molecular, and optical physics; particle physics; astrophysics; geophysics and biophysics. Some physics departments also support research in Physics education.Since the 20th century, the individual fields of physics have become increasingly specialized, and today most physicists work in a single field for their entire careers. "Universalists" such as Albert Einstein (1879–1955) and Lev Landau (1908–1968)【列夫·朗道】, who worked in multiple fields of physics, are now very rare.Condensed matter physicsCondensed matter physics is the field of physics that deals with the macroscopic physical properties of matter. In particular, it is concerned with the "condensed" phases that appear whenever the number of particles in a system is extremely large and the interactions between them are strong.The most familiar examples of condensed phases are solids and liquids, which arise from the bonding by way of the electromagnetic force between atoms. More exotic condensed phases include the super-fluid and the Bose–Einstein condensate found in certain atomic systems at very low temperature, the superconducting phase exhibited by conduction electrons in certain materials,and the ferromagnetic and antiferromagnetic phases of spins on atomic lattices.Condensed matter physics is by far the largest field of contemporary physics.Historically, condensed matter physics grew out of solid-state physics, which is now considered one of its main subfields. The term condensed matter physics was apparently coined by Philip Anderson when he renamed his research group—previously solid-state theory—in 1967. In 1978, the Division of Solid State Physics of the American Physical Society was renamed as the Division of Condensed Matter Physics.Condensed matter physics has a large overlap with chemistry, materials science, nanotechnology and engineering.Atomic, molecular and optical physicsAtomic, molecular, and optical physics (AMO) is the study of matter–matter and light–matter interactions on the scale of single atoms and molecules.1 Physics 物理学The three areas are grouped together because of their interrelationships, the similarity of methods used, and the commonality of the energy scales that are relevant. All three areas include both classical, semi-classical and quantum treatments; they can treat their subject from a microscopic view (in contrast to a macroscopic view).Atomic physics studies the electron shells of atoms. Current research focuses on activities in quantum control, cooling and trapping of atoms and ions, low-temperature collision dynamics and the effects of electron correlation on structure and dynamics. Atomic physics is influenced by the nucleus (see, e.g., hyperfine splitting), but intra-nuclear phenomena such as fission and fusion are considered part of high-energy physics.Molecular physics focuses on multi-atomic structures and their internal and external interactions with matter and light.Optical physics is distinct from optics in that it tends to focus not on the control of classical light fields by macroscopic objects, but on the fundamental properties of optical fields and their interactions with matter in the microscopic realm.High-energy physics (particle physics) and nuclear physicsParticle physics is the study of the elementary constituents of matter and energy, and the interactions between them.In addition, particle physicists design and develop the high energy accelerators,detectors, and computer programs necessary for this research. The field is also called "high-energy physics" because many elementary particles do not occur naturally, but are created only during high-energy collisions of other particles.Currently, the interactions of elementary particles and fields are described by the Standard Model.●The model accounts for the 12 known particles of matter (quarks and leptons) thatinteract via the strong, weak, and electromagnetic fundamental forces.●Dynamics are described in terms of matter particles exchanging gauge bosons (gluons,W and Z bosons, and photons, respectively).●The Standard Model also predicts a particle known as the Higgs boson. In July 2012CERN, the European laboratory for particle physics, announced the detection of a particle consistent with the Higgs boson.Nuclear Physics is the field of physics that studies the constituents and interactions of atomic nuclei. The most commonly known applications of nuclear physics are nuclear power generation and nuclear weapons technology, but the research has provided application in many fields, including those in nuclear medicine and magnetic resonance imaging, ion implantation in materials engineering, and radiocarbon dating in geology and archaeology.University PhysicsAstrophysics and Physical CosmologyAstrophysics and astronomy are the application of the theories and methods of physics to the study of stellar structure, stellar evolution, the origin of the solar system, and related problems of cosmology. Because astrophysics is a broad subject, astrophysicists typically apply many disciplines of physics, including mechanics, electromagnetism, statistical mechanics, thermodynamics, quantum mechanics, relativity, nuclear and particle physics, and atomic and molecular physics.The discovery by Karl Jansky in 1931 that radio signals were emitted by celestial bodies initiated the science of radio astronomy. Most recently, the frontiers of astronomy have been expanded by space exploration. Perturbations and interference from the earth's atmosphere make space-based observations necessary for infrared, ultraviolet, gamma-ray, and X-ray astronomy.Physical cosmology is the study of the formation and evolution of the universe on its largest scales. Albert Einstein's theory of relativity plays a central role in all modern cosmological theories. In the early 20th century, Hubble's discovery that the universe was expanding, as shown by the Hubble diagram, prompted rival explanations known as the steady state universe and the Big Bang.The Big Bang was confirmed by the success of Big Bang nucleo-synthesis and the discovery of the cosmic microwave background in 1964. The Big Bang model rests on two theoretical pillars: Albert Einstein's general relativity and the cosmological principle (On a sufficiently large scale, the properties of the Universe are the same for all observers). Cosmologists have recently established the ΛCDM model (the standard model of Big Bang cosmology) of the evolution of the universe, which includes cosmic inflation, dark energy and dark matter.Current research frontiersIn condensed matter physics, an important unsolved theoretical problem is that of high-temperature superconductivity. Many condensed matter experiments are aiming to fabricate workable spintronics and quantum computers.In particle physics, the first pieces of experimental evidence for physics beyond the Standard Model have begun to appear. Foremost among these are indications that neutrinos have non-zero mass. These experimental results appear to have solved the long-standing solar neutrino problem, and the physics of massive neutrinos remains an area of active theoretical and experimental research. Particle accelerators have begun probing energy scales in the TeV range, in which experimentalists are hoping to find evidence for the super-symmetric particles, after discovery of the Higgs boson.Theoretical attempts to unify quantum mechanics and general relativity into a single theory1 Physics 物理学of quantum gravity, a program ongoing for over half a century, have not yet been decisively resolved. The current leading candidates are M-theory, superstring theory and loop quantum gravity.Many astronomical and cosmological phenomena have yet to be satisfactorily explained, including the existence of ultra-high energy cosmic rays, the baryon asymmetry, the acceleration of the universe and the anomalous rotation rates of galaxies.Although much progress has been made in high-energy, quantum, and astronomical physics, many everyday phenomena involving complexity, chaos, or turbulence are still poorly understood. Complex problems that seem like they could be solved by a clever application of dynamics and mechanics remain unsolved; examples include the formation of sand-piles, nodes in trickling water, the shape of water droplets, mechanisms of surface tension catastrophes, and self-sorting in shaken heterogeneous collections.These complex phenomena have received growing attention since the 1970s for several reasons, including the availability of modern mathematical methods and computers, which enabled complex systems to be modeled in new ways. Complex physics has become part of increasingly interdisciplinary research, as exemplified by the study of turbulence in aerodynamics and the observation of pattern formation in biological systems.Vocabulary★natural science 自然科学academic disciplines 学科astronomy 天文学in their own right 凭他们本身的实力intersects相交，交叉interdisciplinary交叉学科的，跨学科的★quantum 量子的theoretical breakthroughs 理论突破★electromagnetism 电磁学dramatically显著地★thermodynamics热力学★calculus微积分validity★classical mechanics 经典力学chaos 混沌literate 学者★quantum mechanics量子力学★thermodynamics and statistical mechanics热力学与统计物理★special relativity狭义相对论is concerned with 关注，讨论，考虑acoustics 声学★optics 光学statics静力学at rest 静息kinematics运动学★dynamics动力学ultrasonics超声学manipulation 操作，处理，使用University Physicsinfrared红外ultraviolet紫外radiation辐射reflection 反射refraction 折射★interference 干涉★diffraction 衍射dispersion散射★polarization 极化，偏振internal energy 内能Electricity电性Magnetism 磁性intimate 亲密的induces 诱导，感应scale尺度★elementary particles基本粒子★high-energy physics 高能物理particle accelerators 粒子加速器valid 有效的，正当的★discrete离散的continuous 连续的complementary 互补的★frame of reference 参照系★the special theory of relativity 狭义相对论★general theory of relativity 广义相对论gravitation 重力，万有引力explicit 详细的，清楚的★quantum field theory 量子场论★condensed matter physics凝聚态物理astrophysics天体物理geophysics地球物理Universalist博学多才者★Macroscopic宏观Exotic奇异的★Superconducting 超导Ferromagnetic铁磁质Antiferromagnetic 反铁磁质★Spin自旋Lattice 晶格，点阵，网格★Society社会，学会★microscopic微观的hyperfine splitting超精细分裂fission分裂，裂变fusion熔合，聚变constituents成分，组分accelerators加速器detectors 检测器★quarks夸克lepton 轻子gauge bosons规范玻色子gluons胶子★Higgs boson希格斯玻色子CERN欧洲核子研究中心★Magnetic Resonance Imaging磁共振成像，核磁共振ion implantation 离子注入radiocarbon dating放射性碳年代测定法geology地质学archaeology考古学stellar 恒星cosmology宇宙论celestial bodies 天体Hubble diagram 哈勃图Rival竞争的★Big Bang大爆炸nucleo-synthesis核聚合，核合成pillar支柱cosmological principle宇宙学原理ΛCDM modelΛ-冷暗物质模型cosmic inflation宇宙膨胀1 Physics 物理学fabricate制造，建造spintronics自旋电子元件，自旋电子学★neutrinos 中微子superstring 超弦baryon重子turbulence湍流，扰动，骚动catastrophes突变，灾变，灾难heterogeneous collections异质性集合pattern formation模式形成University Physics2 Classical mechanics 经典力学IntroductionIn physics, classical mechanics is one of the two major sub-fields of mechanics, which is concerned with the set of physical laws describing the motion of bodies under the action of a system of forces. The study of the motion of bodies is an ancient one, making classical mechanics one of the oldest and largest subjects in science, engineering and technology.Classical mechanics describes the motion of macroscopic objects, from projectiles to parts of machinery, as well as astronomical objects, such as spacecraft, planets, stars, and galaxies. Besides this, many specializations within the subject deal with gases, liquids, solids, and other specific sub-topics.Classical mechanics provides extremely accurate results as long as the domain of study is restricted to large objects and the speeds involved do not approach the speed of light. When the objects being dealt with become sufficiently small, it becomes necessary to introduce the other major sub-field of mechanics, quantum mechanics, which reconciles the macroscopic laws of physics with the atomic nature of matter and handles the wave–particle duality of atoms and molecules. In the case of high velocity objects approaching the speed of light, classical mechanics is enhanced by special relativity. General relativity unifies special relativity with Newton's law of universal gravitation, allowing physicists to handle gravitation at a deeper level.The initial stage in the development of classical mechanics is often referred to as Newtonian mechanics, and is associated with the physical concepts employed by and the mathematical methods invented by Newton himself, in parallel with Leibniz【莱布尼兹】, and others.Later, more abstract and general methods were developed, leading to reformulations of classical mechanics known as Lagrangian mechanics and Hamiltonian mechanics. These advances were largely made in the 18th and 19th centuries, and they extend substantially beyond Newton's work, particularly through their use of analytical mechanics. Ultimately, the mathematics developed for these were central to the creation of quantum mechanics.Description of classical mechanicsThe following introduces the basic concepts of classical mechanics. For simplicity, it often2 Classical mechanics 经典力学models real-world objects as point particles, objects with negligible size. The motion of a point particle is characterized by a small number of parameters: its position, mass, and the forces applied to it.In reality, the kind of objects that classical mechanics can describe always have a non-zero size. (The physics of very small particles, such as the electron, is more accurately described by quantum mechanics). Objects with non-zero size have more complicated behavior than hypothetical point particles, because of the additional degrees of freedom—for example, a baseball can spin while it is moving. However, the results for point particles can be used to study such objects by treating them as composite objects, made up of a large number of interacting point particles. The center of mass of a composite object behaves like a point particle.Classical mechanics uses common-sense notions of how matter and forces exist and interact. It assumes that matter and energy have definite, knowable attributes such as where an object is in space and its speed. It also assumes that objects may be directly influenced only by their immediate surroundings, known as the principle of locality.In quantum mechanics objects may have unknowable position or velocity, or instantaneously interact with other objects at a distance.Position and its derivativesThe position of a point particle is defined with respect to an arbitrary fixed reference point, O, in space, usually accompanied by a coordinate system, with the reference point located at the origin of the coordinate system. It is defined as the vector r from O to the particle.In general, the point particle need not be stationary relative to O, so r is a function of t, the time elapsed since an arbitrary initial time.In pre-Einstein relativity (known as Galilean relativity), time is considered an absolute, i.e., the time interval between any given pair of events is the same for all observers. In addition to relying on absolute time, classical mechanics assumes Euclidean geometry for the structure of space.Velocity and speedThe velocity, or the rate of change of position with time, is defined as the derivative of the position with respect to time. In classical mechanics, velocities are directly additive and subtractive as vector quantities; they must be dealt with using vector analysis.When both objects are moving in the same direction, the difference can be given in terms of speed only by ignoring direction.University PhysicsAccelerationThe acceleration , or rate of change of velocity, is the derivative of the velocity with respect to time (the second derivative of the position with respect to time).Acceleration can arise from a change with time of the magnitude of the velocity or of the direction of the velocity or both . If only the magnitude v of the velocity decreases, this is sometimes referred to as deceleration , but generally any change in the velocity with time, including deceleration, is simply referred to as acceleration.Inertial frames of referenceWhile the position and velocity and acceleration of a particle can be referred to any observer in any state of motion, classical mechanics assumes the existence of a special family of reference frames in terms of which the mechanical laws of nature take a comparatively simple form. These special reference frames are called inertial frames .An inertial frame is such that when an object without any force interactions (an idealized situation) is viewed from it, it appears either to be at rest or in a state of uniform motion in a straight line. This is the fundamental definition of an inertial frame. They are characterized by the requirement that all forces entering the observer's physical laws originate in identifiable sources (charges, gravitational bodies, and so forth).A non-inertial reference frame is one accelerating with respect to an inertial one, and in such a non-inertial frame a particle is subject to acceleration by fictitious forces that enter the equations of motion solely as a result of its accelerated motion, and do not originate in identifiable sources. These fictitious forces are in addition to the real forces recognized in an inertial frame.A key concept of inertial frames is the method for identifying them. For practical purposes, reference frames that are un-accelerated with respect to the distant stars are regarded as good approximations to inertial frames.Forces; Newton's second lawNewton was the first to mathematically express the relationship between force and momentum . Some physicists interpret Newton's second law of motion as a definition of force and mass, while others consider it a fundamental postulate, a law of nature. Either interpretation has the same mathematical consequences, historically known as "Newton's Second Law":a m t v m t p F ===d )(d d dThe quantity m v is called the (canonical ) momentum . The net force on a particle is thus equal to rate of change of momentum of the particle with time.So long as the force acting on a particle is known, Newton's second law is sufficient to。
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.6 Boundary- value and eigenvalue problemsAnother class of problems in physics requires knowledge of solutions to differential equations with values of physical quantities or their derivatives given at the boundaries of a specified region. This applies to the solution of the Poisson equation with a given charge distribution and known boundary values of the electrostatic potential or of the stationary Schrodinger equation with a given potential and boundary conditionsA typical boundary- value problem in physics us usually given in a second- order differential equation(3.43)Where is a function of x, and are the first-order and second-order derivatives of with respect to x , and is a function of , , and x . Either u or is given at each boundary. Note that one can always choose a coordinate system so that the boundaries of the system are at x=0 and x=1 without losing any generality.For example，if the actual boundaries are at x= and for a given problem one call always bring them back to with a transformation(3.44)For problems in one dimension，one can have a total of four possible types ofboundary conditions：(1) and(2) and(3) and(4) andThe boundary-value problem is more difficult to solve than the similar initial—value problem with the same differential equation For example，if we want to solve, (3.45)with the initial conditions u(0)= and , we can first transform the differential equation into a set of first—order differential equations with a redefinition of the first order derivative into a new variable. The solution will follow if we adopt one of the algorithms discussed earlier in this chapter．However for the boundary-value problem，we know only or ，which is not sufficient to start any algorithms for the initial．value problems without ally further work. Typical eigenvalue problems are even more complicated because at least one more parameter ,that is．the eigenvalue is involved 1n the equation For example，with a set of given boundary conditions，defines an eigenvalue problem. Here the eigenvalue can have only some selected values in order to yield acceptable solutions of the equation under the given boundary conditions.Let us take the longitudinal vibrations along an elastic rod as an example here. The equation describing the stationary solution of elastic waves is given bywhere u(x)is the displacement from the equilibrium at z and the allowed values of the wavevector k are the eigenvalues of the problem The wavevector in the equation is related to the sound speed c along the rod and the allowed angular frequency by the dispersion relationIf both ends (x=0 , x=1) of the rod are fixed，the boundary conditions are then If one end (x=O)is fixed and the other end (x=1) is free，the boundary conditions are then u(0)=0 and For this problem，one can obtain an analytical solution For example，if both ends off the rod are fixed，one will have the eigenfunctions(3 49) as possible-solutions of the differential equation. Here the eigenvalues are given by(3 50)where n= The complete solution of the longitudinal waves along the elastic rod is given by a linear combination of all the eigen-functions with their associated initial solutions，(3 51)Where are coefficients to be determined by the initial conditions We will come back to this problem in Chapter 6whenwe discuss the solutions of a partial differential equation.7The shooting methodA simple method for solving the boundary-value problem of Eq.(3.43) and the eigenvalue Problem of Eq.(3.46) with a set of given boundary conditions is the so-called shooting method. We will first discuss how it works for the boundary-value Problem and then generalize it to the eigenvalue problem.We first convert the second-order differential equation into two first-order differential equations by defining and ; that is,(3.52)(3.53) with a given set of boundary conditions. To illustrate the method, let us assume that the boundary conditions are and . Any other types of Boundary conditions can be solved in a similar manner. The key here is to make the problem look like an initial-value problem by introducing an adjustable parameter；the solution is then obtained by varying the parameter Since u(0) is given already ，we can make a guess for the first-0rder derivative at x=0, for example , . Here is the parameter to be adjusted. For a specific , we can integrate the equation to x=1 with any of the algorithms discussed earlier for the initial-value problem . Since the initial choice of can hardly be the actual derivative at x=0 , the value of the function (1), resulting from the integration with to x=1 , would not be the same as . The idea of the shooting method is to use one of the root search algorithms to find the appropriate that ensures within a given toleranceLet us take an actual numerical example to illustrate the scheme . Assume that we want to solve the following differential equation;(3.54) With the given boundary conditions and u(1)=1, we can define the newvariables and ; then we have(3.55)r (3.56) Now assume that this equation set has the initial values and . Here is a parameter to be adjusted in order to have , with being the tolerance of the solution . We can combine the secant method for root search and the fourth-order Runge-Kutta method for the initial-value problem to solve the above equation .The following program is an implementation of such a combined approach to the boundary-value problem defined in Eq. (3.54) or Eqs. (3.55) and (3.56) with the given boundary conditions.CThe boundary—value problem solved from the above program can also be solvedexactly with all analytical solution(3 57) One can easily check that the above expression does satisfy the equation and theboundary conditions Here we plot both the numerical result obtained from the shooting method and the analytical solution in Fig 3．4 As one can see，the shooting method provides a very accurate solution of the boundary—value problem It is also a very general method for the boundary—value problem.Fig 3．4 Tire numerical solution of the boundary value problem of Eq.(3.54) by the Shooting method(+) compared with the analytical solution(solid line)of the same ProblemBoundary value problems with other types of boundary conditions can be solved in a similar manner , For example，if and u(1)= are given， we can make a guess on and then integrate the equation set of and to x=1 the root to be sought is from Here is the numerical result of the equation with . If given，the equation ， is solved Instead.One can apply the shooting method to solve the eigenvalue problem too. The parameter to be adjusted in the eigenvalue problem is no longer a parameter introduced but the eigenvalue of the problem. For example，if and are given，we can integrate the equation with with as a small quantity. Then we search for the root of . When ，we obtain an approximate eigenvalue and the corresponding eigenvector from the normalized solution of . The introduced parameter d is not relevant here because it will be automatically modified to be the first-order derivative here solution is normalized In other words，one can choose the first-order derivative at the boundary arbitrarily and it will not affect the results as long as the solutions are made orthonormal (orthogonal).3．8 Linear equations and the Sturm-Liouville problemMany eigenvalue or boundary value problems are in the form of linear equations, (3.59) Where d(x), q (x), and s(x) are functions of x. Assume that the boundary conditions are and . If all d(x)，q(x)，and s(x)are smooth，one can solve the equation with the shooting method developed in the preceding section. In fact，one can show that an extensive search for the parameter from . 0 is not necessary in this case, because of the superposition principle of the linear equation. Any linear combination of the solutions is still a solution of the equation. one needs only two trial solutions and ，with and being two different parameters The correct solution of the equation is given bywith a and b determined from u(0)= and u(1)=.Note that . So we havea+b=1 (36.0)(3.61) which can easily be solved to give(3.62)(3.63) With a and b given from the above equation，we have the solution of the differential equation from Eq(3 59). An important class of linear equations in physics is referred to as the Sturm-Liouville problem，defined by(3 64)which has the first-order derivative term combined with the second-order derivative term. P(x) , q(x) , and s(x) are the coefficient functions of x. For most actual problems，s(x)=0 and q(x)= ，with being the eigenvalueof the equation r(x)and w(x)are the redefined coefficient functions. The Legendre equation，the Bessel equation，and the related equations in physics are examples of the Sturm-Liouville problem. Our goal here is to construct an accurate algorithm that can integrate Sturm-Liouville equation，that is，Eq(3 64)In Chapter 2，we obtained the three-point(3.65)And(3.36)Now if we multiply Eq. (3 65) by and Eq(3 66) by and add them together, we have(3 67)If we replace the first term on the right-hand side with and drop the second term, we obtain the simplest numerical algorithm for the Sturm-LiouvlIle equation(3.68) which is accurate up to O. One has to be very careful with this apparent local accuracy in the algorithm. In reality, because of the repeated use of the three-point formula, the local accuracy delivered is usually lower than O. Before we discuss how to improve the accuracy of this algorithm，let us illustrate it with all example The Legendre equation is given by(3.69) With and x The solutions of the Legendre equation are the Legendre polynomials . Let us assume that we do not know the value of l but know the first two points of =x：then we can treat the problem as an eigenvalue problem. The following program is an implementation of the simplest algorithm for the Sturm-Liouville equation in combination with the bisection method for the root search to solve for the eigenvalue =1 of the Legendre equationThe eigenvalue obtained from the above program is 1.000091, which contains an error of 9 in comparison with the exact result of . The error comes from both the rounding error and the inaccuracy of the algorithm. If we want to have higher accuracy in the algorithm for the Sturm-Liouvile Equation , we can differentiate the equation twice , Then we have, (3.70)Where on the right-hand side can be replaced withWhich is the result of differentiating the Sturm-Liouville equation once . If we combine Eqs.(3.67) , (3.70) , and (3.71) , we obtain a betteralgorithm(3.72)where and are given by(3.73)(3.74)(3.75)(3.76)which can be evaluated easily if p(x)，q(x)，and s(x) are explicitly given In the case where some the derivatives that are needed are not easily obtained analytically，one can evaluate them numerically In order to maintain the high accuracy of the algorithm，one needs to use comparable numerical formulas. For the special case with p(x)=1，the above coefficients reduce to much simpler forms Without sacrificing the high accuracy of the algorithm，we can apply the three—point formulas to the first order and second order derivatives of q(x) and s(x) . Then we have(3.77)(3.78)(3.79)(3.80)which are slightly different from the Numerov algorithm which is an extremely accurate scheme for linear differential equations without the first-order derivative term, that is , Eq.(3.58) with d(x)=0 or the Sturm-Liouville equation with p(x)=1.Many equations in physics have this form , for example , the Poisson equation with Spherical symmetry or the one-dimensional Schrodinger equation. The Numerov algorithm is derived from the three-point formula for the second-Order derivative from Eq, (3.66) with the fourth-orderderivative of u(x) from taking the second-order derivative of the differential equation.(3.81)If we apply the three-point formula to the above equation by keeping all the terms on the right-hand side together , then we obtain the Numerov algorithm (Koonin, 1986, pp. 50-1) with the recursion of Eq.(3.72) and the coefficients given by(3.82)(3.83)(3.84)(3.85)Note that even though the apparent local accuracy of the Numerov algorithm and the algorithm w4e derived earlier in this section for the Sturm-Liouville equation is O() , the actual global accuracy of the algorithm is only O() because of the repeated use of the three-point formulas . For more discussion on this issue , see Simos (1993). The algorithms discussed here can be applied to initial-value problems as well as to boundary-value or eigenvalue problems . The Numerov algorithm and the algorithm for the Sturm-Liouville equation usually have lover accuracy than the fourth-order Runge-Kutta algorithm when applied to the same problem , and this is different from what the apparent local accuracies imply . For more numerical examples of the Sturm-Liouville equation and the Numerov algorithm in the related problems , see Pryce(1993) and Onodera(1994)。