Shut up and Calculate !

— N. David Mermin

# 前置知识

  • 高中函数和三角变换部分
  • 高等数学(微分表、积分表[1]
  • 线性代数
  • 数学物理方法(本科阶段通常是复变函数和微分方程)
  • 一点概率论(知道啥是概率密度[2]
  • 五小力学、分析力学、电动力学、热统(非物理科班:大学物理学)
  • 教材相关[3][4][5][6][7]

# 球坐标下拉普拉斯算符的表示

球坐标系

球坐标与直角坐标的关系:

x=rsinθcosφ,y=rsinθsinφ,z=rcosθ\begin{aligned} x=r\sin{\theta}\cos{\varphi},\quad y=r\sin{\theta}\sin{\varphi},\quad z=r\cos{\theta} \end{aligned}

r=x2+y2+z2,θ=arctanx2+y22,φ=arctanyx\begin{aligned} r=\sqrt{x^2+y^2+z^2},\quad \theta=\arctan{\frac{\sqrt{x^2+y^2}}{2}},\\ \varphi=\arctan{\frac{y}{x}} \end{aligned}

哈密顿算子(Nabla 算子):

=xi+yj+zk=e^rr+e^θ1rθ+e^φ1rsinθφ\begin{aligned} \nabla&=\frac{\partial}{\partial{x}}\vec{i}+\frac{\partial}{\partial{y}}\vec{j}+\frac{\partial}{\partial{z}}\vec{k}\\ &=\hat{e}_r\frac{\partial}{\partial{r}}+\hat{e}_\theta\frac{1}{r}\frac{\partial}{\partial\theta}+\hat{e}_\varphi\frac{1}{r\sin{\theta}}\frac{\partial}{\partial{\varphi}} \end{aligned}

拉普拉斯算符:

2=2x2+2y2+2z2=2r2+1r22r2+2rr+1r2cosθsinθθ+1r2sin2θ2φ2=1r2rr2r+1r2sinθθsinθθ+1r2sin2θ2φ2\begin{aligned} \nabla^2&=\frac{\partial^2}{\partial{x^2}}+\frac{\partial^2}{\partial{y^2}}+\frac{\partial^2}{\partial{z^2}}\\ &=\frac{\partial^2}{\partial{r^2}}+\frac{1}{r^2}\frac{\partial^2}{\partial{r^2}}+\frac{2}{r}\frac{\partial}{\partial{r}}+\frac{1}{r^2}\frac{\cos{\theta}}{\sin{\theta}}\frac{\partial}{\partial{\theta}}\\ &\;\;\quad+\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial{\varphi^2}}\\ &=\frac{1}{r^2}\frac{\partial}{\partial{r}}r^2\frac{\partial}{\partial{r}}+\frac{1}{r^2\sin{\theta}}\frac{\partial}{\partial{\theta}}\sin{\theta}\frac{\partial}{\partial{\theta}}\\ &\;\;\quad+\frac{1}{r^2\sin^2\theta}\frac{\partial^2}{\partial{\varphi^2}} \end{aligned}

# 柱坐标下拉普拉斯算符的表示

柱坐标系

柱坐标与直角坐标的关系:

x=ρcosφ,y=ρsinφ,z=zx=\rho\cos\varphi,\quad y=\rho\sin\varphi,\quad z=z

x2+y2=ρ2,yx=tanφ,z=zx^2+y^2=\rho^2,\quad \frac{y}{x}=\tan\varphi,\quad z=z

哈密顿算子(Nabla 算子):

=xi+yj+zk=e^ρρ+e^φ1ρφ+e^zz\begin{aligned} \nabla&=\frac{\partial}{\partial{x}}\vec{i}+\frac{\partial}{\partial{y}}\vec{j}+\frac{\partial}{\partial{z}}\vec{k}\\ &=\hat{e}_{\rho}\frac{\partial}{\partial{\rho}}+\hat{e}_{\varphi}\frac{1}{\rho}\frac{\partial}{\partial\varphi}+\hat{e}_z\frac{\partial}{\partial{z}} \end{aligned}

拉普拉斯算符:

2=2x2+2y2+2z2=2ρ2+1ρρ+1ρ22φ2+2z2=1ρρρρ+1ρ22φ2+2z2\begin{aligned} \nabla^2&=\frac{\partial^2}{\partial{x^2}}+\frac{\partial^2}{\partial{y^2}}+\frac{\partial^2}{\partial{z^2}}\\ &=\frac{\partial^2}{\partial\rho^2}+\frac{1}{\rho}\frac{\partial}{\partial{\rho}}+\frac{1}{\rho^2}\frac{\partial^2}{\partial\varphi^2}+\frac{\partial^2}{\partial{z^2}}\\ &=\frac{1}{\rho}\frac{\partial}{\partial\rho}\rho\frac{\partial}{\partial\rho}+\frac{1}{\rho^2}\frac{\partial^2}{\partial\varphi^2}+\frac{\partial^2}{\partial{z^2}} \end{aligned}

# 狄拉克 δ 函数

# 定义

Dirac 引入 δ\delta 函数定义,对于多维空间中:

δ(x)={,x=00,x0δ(x)dx=1\delta(x)=\begin{cases}\infty,&x=0\\[2ex]0,&x\neq 0\end{cases} \quad\text{且}\quad\int_{-\infty}^\infty{\delta(x)}\,\mathrm{d}x=1

δ\delta 函数在数学上可以通过分布理论[8]严格化。δ\delta 函数又叫脉冲函数,它是一个广义函数(泛函)。

狄拉克 δ 函数

# δ 函数的几种表达方式

实际应用中,δ\delta 函数常用某些函数的极限形式表达。

  1. limδ012πδex22δ=δ(x)\lim\limits_{\delta \to 0}\frac{1}{\sqrt{2\pi\delta}}e^{-\frac{x^2}{2\delta}}=\delta(x)
  2. limαsinαxπx=δ(x)\lim\limits_{\alpha\to\infty}\frac{\sin{\alpha x}}{\pi x}=\delta(x)
  3. 12πeikxdk=δ(x)\color{red}{\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{ikx}\,\rm{d}k=\delta(x)} 或者12πeikxdx=δ(k)\color{red}{\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-ikx}\,\mathrm{d}x=\delta(k)}
  4. limσ0+1πϵx2+ϵ2=δ(x)\lim\limits_{\sigma\to 0^+}\frac{1}{\pi}\frac{\epsilon}{x^2+\epsilon^2}=\delta(x)
  5. 12πei(kk)xdr=δ(kk)\color{red}{\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(k-k^{\prime})x}\,\mathrm{d}\vec{r}=\delta(k-k^{\prime})}
  6. 12πei(pp)xdx=δ(pp)\color{red}{\frac{1}{2\pi\hbar}\int_{-\infty}^{\infty}e^{\frac{i}{\hbar}(\vec{p}-\vec{p^{\prime}})x}\,\mathrm{d}x=\delta(\vec{p}-\vec{p^{\prime}})}(一维)
  7. 1(2π)3ei(pp)rdr=δ(pp)\color{red}{\frac{1}{(2\pi\hbar)^3}\int_{-\infty}^{\infty}e^{\frac{i}{\hbar}(\vec{p}-\vec{p^{\prime}})\vec{r}}\,\mathrm{d}\vec{r}=\delta(\vec{p}-\vec{p^{\prime}})}(三维)

# δ 函数的性质

  1. 对称性( δ\delta 函数是偶函数):δ(x)=δ(x)\delta(-x)=\delta(x)

  2. 挑选性(筛选性)

    对于任何连续函数 f(x)f(x)

    f(x)δ(xa)dx=f(a)\int_{\infty}^{-\infty}f(x)\delta(x-a)\,\mathrm{d}x=f(a)

    特别地,当 a=0a=0 时,f(x)δ(x)dx=f(0)\int_{\infty}^{-\infty}f(x)\delta(x)\,\mathrm{d}x=f(0)

    有一个 δ(x)\delta(x) 脱去一个积分号 \int 或求和号 \sum

  3. 放缩或相似性:

    δ(ax)=1aδ(x)\delta(ax)=\frac{1}{|a|}\delta(x)

  4. φ(x)\varphi(x) 是一个二次以上可导函数。设 {xi}\{x_i\} 为其单零点的集合,即在任意一点 xix_i 处,我们有 φ(xi)=0\varphi(x_i)=0φ(xi)0\varphi^{\prime}(x_i)\neq 0 ,那么有:

    δ(φ(x))=iNδ(xxi)φ(xi)\delta(\varphi(x))=\sum_{i}^{N}{\frac{\delta(x-x_i)}{|\varphi^{\prime}(x_i)|}}

  5. Rnf(x)[δ(xa)]dx=f(x)x=a\int_{R^n}f(x)[\nabla\delta(x-a)]\,\mathrm{d}x=\left.-\nabla f(x)\right|_{x=a}

  6. 位矢的微分:

    rr3=2(1r)=4πδ(r)\nabla\cdot\frac{\vec{r}}{r^3}=-\nabla^2\left(\frac{1}{r}\right)=4\pi\delta(\vec{r})

# 矢量分析

  1. (Av)=(A)v+(v)A+A×(×v)+v×(×A)\nabla(\vec{A}\cdot\vec{v})=(\vec{A}\cdot\nabla)\vec{v}+(\vec{v}\cdot\nabla)\vec{A}+\vec{A}\times(\nabla\times\vec{v})+\vec{v}\times(\nabla\times\vec{A})

  2. =xi+yj+zk\nabla=\frac{\partial}{\partial{x}}\vec{i}+\frac{\partial}{\partial{y}}\vec{j}+\frac{\partial}{\partial{z}}\vec{k}
    2==(xi+yj+zk)=2x2+2y2+2z2\nabla^2=\nabla\cdot\nabla=\nabla(\frac{\partial}{\partial{x}}\vec{i}+\frac{\partial}{\partial{y}}\vec{j}+\frac{\partial}{\partial{z}}\vec{k})=\frac{\partial^2}{\partial{x^2}}+\frac{\partial^2}{\partial{y^2}}+\frac{\partial^2}{\partial{z^2}}

# 无穷级数

函数直接展开为 xx 的幂级数:

  1. f(x)f(x) 的各阶导数 f(x),f(x),,f(n)(x),f^{\prime}(x),\;f^{\prime\prime}(x),\;\cdots\;,\;f^{(n)}(x),\;\cdots\;

  2. 求出函数及其各阶导数在 x=0x=0 处的值 f(0),f(0),f(0),,f(n)(0),f(0),\; f^{\prime}(0),\; f^{\prime\prime}(0),\;\cdots\;,\;f^{(n)}(0),\;\cdots\;

  3. 写出幂级数 f(0)+f(0)x+f(0)2!x2++f(n)(0)n!xn+f(0)+f^{\prime}(0)\,x+\frac{f^{\prime\prime}(0)}{2!}\,x^2+\cdots+\frac{f^{(n)}(0)}{n!}\,x^n+\cdots ,并求出收敛半径:R=limnanan+1R=\lim_{n\to\infty}\left|\frac{a_n}{a_{n+1}}\right|

  4. 验证余项 limnRn(x)=limn1(n+1)!f(n+1)(θx)xn+1\lim\limits_{n\to\infty}R_n(x)=\lim\limits_{n\to\infty}\frac{1}{(n+1)!}f^{(n+1)}(\theta x)x^{n+1} 是否为零,如果为零,则函数 f(x)f(x) 在区间 (R,R)(-R,R) 的幂级数展开式为:

    f(x)=f(0)+f(0)x+f(0)2!x2++fn(0)n!xn+,x(R,R)f(x)=f(0)+f^{\prime}(0)\,x+\frac{f^{\prime\prime}(0)}{2!}\,x^2+\cdots+\frac{f^{n}(0)}{n!}\,x^n+\cdots\,,\quad x\in(-R,R)

f(x)f(x)x0x_0 点的泰勒级数:

f(x)=n=01n!f(n)(x0)(xx0)n,xU(x0)f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}f^{(n)}(x_0)\,(x-x_0)^n,\quad x\in U(x_0)

x0=0x_0=0 时,级数

f(x)=n=01n!f(n)(0)xnf(x)=\sum_{n=0}^{\infty}\frac{1}{n!}f^{(n)}(0)\,x^n

称为麦克劳林级数。

如果 f(x)f(x)U(x0)U(x_0) 内具有各阶导数,则

f(x)=n=01n!f(n)(x0)(xx0)nlimnRn(x)=0f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}f^{(n)}(x_0)\,(x-x_0)^n\Longleftrightarrow\lim_{n\to\infty}R_n(x)=0

常见函数的泰勒级数:

  1. cosα=n=0(1)kα2k(2k)!,n=0,2,4,\cos\alpha=\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^{k}\,\alpha^{2k}}{(2k)!},\quad n=0,2,4,\ldots
  2. sinα=n=1(1)kα2k+1(2k+1)!,n=1,3,5,\sin\alpha=\displaystyle\sum_{n=1}^{\infty}\frac{(-1)^{k}\,\alpha^{2k+1}}{(2k+1)!},\quad n=1,3,5,\ldots
  3. ex=n=0xnn!e^{x}=\displaystyle\sum_{n=0}^{\infty}\frac{x^n}{n!}\,
  4. ax=exlna=n=0(lna)nn!xna^{x}=e^{x\ln{a}}=\displaystyle\sum_{n=0}^{\infty}\frac{(\ln{a})^n}{n!}\,x^n
  5. ln(1+x)=n=1(1)n1xnn,x(1,1]\ln{(1+x)}=\displaystyle\sum_{n=1}^{\infty}(-1)^{n-1}\frac{x^n}{n},\quad x\in (-1,1]
  6. (1+x)m=n=1m(m1)(mn+1)n!xn,x(1,1)(1+x)^m=\displaystyle\sum_{n=1}^{\infty}\frac{m(m-1)\cdots(m-n+1)}{n!}\,x^n,\quad x\in (-1,1)
  7. 11x=n=0xn,x(1,1)\frac{1}{1-x}=\displaystyle\sum_{n=0}^{\infty} x^n,\quad x\in (-1,1)
  8. arctanx=n=0(1)n2n+1x2n+1,x[1,1]\arctan{x}=\displaystyle\sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1}\,x^{2n+1},\quad x\in [-1,1]

# 补充

# 三角变换

sin(α±β)=sinαcosβ±cosαsinβ\sin(\alpha\pm \beta)=\sin\alpha\cos\beta\pm \cos\alpha\sin\beta

cos(α±β)=cosαcosβsinαsinβ\cos(\alpha\pm \beta)=\cos\alpha\cos\beta\mp \sin\alpha\sin\beta

# 其它

  1. 0xneax=n!an+1\int_{0}^{\infty} x^{n}\,e^{-ax}=\frac{n!}{a^{n+1}}\,

  2. 高斯积分公式:

    eax2dx=na\int_{-\infty}^{\infty}e^{-ax^2}\,\mathrm{d}x=\sqrt{\frac{n}{a}}

    或者

    0x2neax2dx=135(2n1)2n+1anna\int_{0}^{\infty}x^{2n}\,e^{-ax^2}\,\mathrm{d}x=\frac{1\cdot 3\cdot 5\cdots(2n-1)}{2^{n+1}a^n}\sqrt{\frac{n}{a}}

  3. η=ξtanξ\eta=\xi\tan{\xi}ξ2+η2=Q2\xi^{2}+\eta^{2}=Q^{2}

η=ξtgξ、η=ξctgξ

# 参考资料


  1. 微积分:常用公式、微分方程、级数 ↩︎

  2. 概率函数 P(x)P(x) 、概率分布函数 F(x)F(x) 、概率密度函数 f(x)f(x) ↩︎

  3. 《The Historical Development of Quantum Theory》J. Mehra, H. Rechenberg ↩︎

  4. 《量子力学 卷一(四)》 曾谨言 ↩︎

  5. 《量子力学习题精选与剖析》 曾谨言,钱伯初 ↩︎

  6. 《Quantum Mechanics Non-relativistic Theory》 L.D. Landau, E.M. Lifschitz ↩︎

  7. 《Modern Quantum Mechanics》 J.J. Sakurai ↩︎

  8. 分布理论 ↩︎