哈密顿算符及其一般运算表达式的分析 General Expression Analysis of the Hamiltonian Operator and Its Formula

doi:10.12677/AAM.2020.98150

Advances in Applied Mathematics
Vol. 09 No. 08 ( 2020 ), Article ID: 37173 , 6 pages
10.12677/AAM.2020.98150

General Expression Analysis of the Hamiltonian Operator and Its Formula

Chaozhen Feng¹, Weilong Duan²

●How to Cite this Article

¹The Institute of Science and Engineering, Dehong Teacher’s College, Mangshi Yunnan

²City College, Kunming University of Science and Technology, Kunming Yunnan

Received: Jul. 27^th, 2020; accepted: Aug. 13^th, 2020; published: Aug. 20^th, 2020

ABSTRACT

The Hamiltonian operator $\nabla$ and the common expressions such as the Laplacian operator, gradient, divergence, and curl generated by it are not the same in different curve coordinate systems. This paper defines a three-dimensional orthogonal curve coordinate system $(u_{1}, u_{2}, u_{3})$ , introducing coordinate factor $h_{1}$ , $h_{2}$ , $h_{3}$ , deriving the general form of $\nabla$ , $\nabla ϕ$ , $\nabla \cdot A$ , $\nabla \times A$ , $\nabla^{2}$ , and the general expression of Poisson equation as well as Laplace equation.

Keywords:Hamiltonian Operators, Laplacian Operator, Gradient, Divergence, Curl

哈密顿算符及其一般运算表达式的分析

冯朝桢¹，段卫龙²

¹德宏师范高等专科学校理工学院，云南芒市

²昆明理工大学城市学院，云南昆明

收稿日期：2020年7月27日；录用日期：2020年8月13日；发布日期：2020年8月20日

摘要

哈密顿算符 $\nabla$ 及其产生的拉普拉斯算符、梯度、散度和旋度常见运算式在不同曲线坐标系中具体表达式不相同。本文通过定义一个三维正交曲线坐标系 $(u_{1}, u_{2}, u_{3})$ ，引入坐标因子 $h_{1}$ 、 $h_{2}$ 、 $h_{3}$ ，推导得到了关于 $\nabla$ 、 $\nabla ϕ$ 、 $\nabla \cdot A$ 、 $\nabla \times A$ 、 $\nabla^{2}$ 的一般形式及Poisson方程和Laplace方程的一般表达式。

关键词 :哈密顿算符，拉普拉斯算符，梯度，散度，旋度

This work is licensed under the Creative Commons Attribution International License (CC BY 4.0).

http://creativecommons.org/licenses/by/4.0/

1. 引言

哈密顿算符 $\nabla$ 是关于空间的一阶偏微分算子，在数学和物理学中发挥着重要作用。由 $\nabla$ 可以产生一系列的偏微分方程，如泊松方程、拉普拉斯方程等，这些方程在电磁学、热传导等方面经常用到 [1] [2] [3]。 $\nabla$ 及其产生的梯度、散度和旋度等常见运算式在笛卡尔坐标系、柱坐标系、球坐标系、和极坐标系下，其具体表达形式是不一样的，能不能给各种运算式找到一个通用的一般表达式来统一呢？本文针对这个问题做了一些分析，在三维正交曲线坐标系 $(u_{1}, u_{2}, u_{3})$ 下，得到了几个包含哈密顿算符运算式的一般表达式。

2. 正交曲线坐标系

考虑一个由三维正交曲线坐标系 $(u_{1}, u_{2}, u_{3})$ [4] 组成的空间，其中一个无限小的体积元可由 $u_{1}$ ， $u_{1} + d u_{1}$ ， $u_{2}$ ， $u_{2} + d u_{2}$ ， $u_{3}$ ， $u_{3} + d u_{3}$ 相围得到。一般通过 $u_{1}$ 、 $u_{2}$ 、 $u_{3}$ 乘以坐标因子 $h_{1}$ ， $h_{2}$ ， $h_{3}$ 来表示三个方向的距离， $h_{1}$ ， $h_{2}$ ， $h_{3}$ 是 $(u_{1}, u_{2}, u_{3})$ 的函数。

如图1所示，定义三个线元：

Figure 1. The map of generalized coordinates volume element

图1. 广义坐标系体积元图

$d s_{1} = h_{1} d u_{1}$ ， $d s_{2} = h_{2} d u_{2}$ ， $d s_{3} = h_{3} d u_{3}$ (1.1)

体积元 $d V = d s_{1} d s_{2} d s_{3} = h_{1} h_{2} h_{3} d u_{1} d u_{2} d u_{3}$ ，则哈密顿算符可以表示为：

$\nabla = \frac{\partial}{\partial s_{1}} e_{1} + \frac{\partial}{\partial s_{2}} e_{2} + \frac{\partial}{\partial s_{3}} e_{3} = \frac{\partial}{h_{1} \partial u_{1}} e_{1} + \frac{\partial}{h_{2} \partial u_{2}} e_{2} + \frac{\partial}{h_{3} \partial u_{3}} e_{3}$ (1.2)

$e_{1}$ 、 $e_{2}$ 、 $e_{3}$ 分别是空间变量 $(u_{1}, u_{2}, u_{3})$ 的单位矢量。

1) 一标量 $ϕ$ 在正交曲线坐标系 $(u_{1}, u_{2}, u_{3})$ 连续，一阶微分存在，则根据梯度的定义式， $ϕ$ 的梯度分量：

$\frac{\partial ϕ}{\partial s_{1}} = \frac{\partial ϕ}{h_{1} \partial u_{1}}$ ， $\frac{\partial ϕ}{\partial s_{2}} = \frac{\partial ϕ}{h_{2} \partial u_{2}}$ ， $\frac{\partial ϕ}{\partial s_{3}} = \frac{\partial ϕ}{h_{3} \partial u_{3}}$ (1.3)

$\nabla ϕ = (\frac{\partial ϕ}{h_{1} \partial u_{1}}) e_{1} + (\frac{\partial ϕ}{h_{2} \partial u_{2}}) e_{2} + (\frac{\partial ϕ}{h_{3} \partial u_{3}}) e_{3}$ (1.4)

2) 一矢量 $A$ 在正交曲线坐标系 $(u_{1}, u_{2}, u_{3})$ 连续，一阶微分存在，那么矢量 $A$ 穿过面OCGB和ADFE的向外通量为：

$\frac{\partial}{\partial s_{1}} (A_{1} d s_{2} d s_{3}) d s_{1} = \frac{\partial}{\partial u_{1}} (h_{2} h_{3} A_{1}) d u_{1} d u_{2} d u_{3} = \frac{1}{h_{1} h_{2} h_{3}} \frac{\partial}{\partial u_{1}} (h_{2} h_{3} A_{1}) d V$ (1.5)

用同样的方法可以得到穿过其他4个平面的通量，则根据散度的定义， $A$ 的散度：

$div A = \nabla \cdot A = \frac{1}{h_{1} h_{2} h_{3}} [\frac{\partial}{\partial u_{1}} (h_{2} h_{3} A_{1}) + \frac{\partial}{\partial u_{2}} (h_{3} h_{1} A_{2}) + \frac{\partial}{\partial u_{3}} (h_{1} h_{2} A_{3})]$ (1.6)

3) 矢量 $A$ 是 $(u_{1}, u_{2}, u_{3})$ 的函数，在正交曲线坐标下连续可微， $A_{1}$ 、 $A_{2}$ 、 $A_{3}$ 是沿着 $u_{1}$ 、 $u_{2}$ 、 $u_{3}$ 三个变量方向上的分量，由Stokes定理，在面OADC上 $A_{1}$ 沿着边OA和DC的线积分等于：

$[h_{1} (u_{1}, u_{2}) A_{1} (u_{1}, u_{2}) - h_{1} (u_{1}, u_{2} + d u_{2}) A_{1} (u_{1}, u_{2} + d u_{2})] d u_{1} = - \frac{\partial (h_{1} A_{1})}{\partial u_{2}} d u_{1} d u_{2}$ (1.7)

沿着边AD和CO的线积分等于

$[h_{2} (u_{1} + d u_{1}, u_{2}) A_{2} (u_{1} + d u_{1}, u_{2}) - h_{2} (u_{1}, u_{2}) A_{2} (u_{1}, u_{2})] d u_{2} = \frac{\partial (h_{2} A_{2})}{\partial u_{1}} d u_{1} d u_{2}$ (1.8)

根据Stokes定理， $A$ 沿着面OADC边界上的环路积分等于 $\nabla \times A$ 穿过平面OADC的通量，即：

${(\nabla \times A)}_{3} = \frac{1}{h_{1} h_{2}} [\frac{\partial (h_{2} A_{2})}{\partial u_{1}} - \frac{\partial (h_{1} A_{1})}{\partial u_{2}}]$ (1.9)

对于另外两个面OBGC和OAEB也可以得到

${(\nabla \times A)}_{1} = \frac{1}{h_{2} h_{3}} [\frac{\partial (h_{3} A_{3})}{\partial u_{2}} - \frac{\partial (h_{2} A_{2})}{\partial u_{3}}]$ (1.10)

${(\nabla \times A)}_{2} = \frac{1}{h_{3} h_{1}} [\frac{\partial (h_{1} A_{1})}{\partial u_{3}} - \frac{\partial (h_{3} A_{3})}{\partial u_{1}}]$ (1.11)

所以，矢量 $A$ 的旋度为

$\begin{matrix} curl A = \nabla \times A \\ = \frac{1}{h_{2} h_{3}} [\frac{\partial (h_{3} A_{3})}{\partial u_{2}} - \frac{\partial (h_{2} A_{2})}{\partial u_{3}}] e_{1} + \frac{1}{h_{3} h_{1}} [\frac{\partial (h_{1} A_{1})}{\partial u_{3}} - \frac{\partial (h_{3} A_{3})}{\partial u_{1}}] e_{2} \\ + \frac{1}{h_{1} h_{2}} [\frac{\partial (h_{2} A_{2})}{\partial u_{1}} - \frac{\partial (h_{1} A_{1})}{\partial u_{2}}] e_{3} \end{matrix}$ (1.12)

3. 几例含哈密顿算符的运算式

3.1. Poisson方程和Laplace方程

标量 $ϕ$ 在正交曲线坐标系 $(u_{1}, u_{2}, u_{3})$ 连续可微，存在二阶偏导数，且矢量 $A = ε \nabla ϕ$ (如电势和电场强度)，那么 $\nabla \cdot A$ 等于什么呢？将(1.4)式中3个分量乘以 $ε$ 代替(1.6)式中的 $A_{1}$ 、 $A_{2}$ 、 $A_{3}$ 有：

$\nabla \cdot A = \nabla \cdot ε \nabla ϕ = \frac{1}{h_{1} h_{2} h_{3}} [\frac{\partial}{\partial u_{1}} (\frac{h_{2} h_{3}}{h_{1}} ε \frac{\partial ϕ_{1}}{\partial u_{1}}) + \frac{\partial}{\partial u_{2}} (\frac{h_{3} h_{1}}{h_{2}} ε \frac{\partial ϕ_{2}}{\partial u_{2}}) + \frac{\partial}{\partial u_{3}} (\frac{h_{1} h_{2}}{h_{3}} ε \frac{\partial ϕ_{3}}{\partial u_{3}})]$ (2.1)

若 $\nabla \cdot A = f$ ，且 $f \neq 0$ 则(2.1)式称Poisson方程；若 $f \neq 0$ 则称为Laplace方程，令 $ε = 1$ ，可以得到拉普拉斯算符：

$\nabla^{2} = [\frac{\partial}{\partial u_{1}} (\frac{h_{2} h_{3}}{h_{1}} \frac{\partial}{\partial u_{1}}) + \frac{\partial}{\partial u_{2}} (\frac{h_{3} h_{1}}{h_{2}} \frac{\partial}{\partial u_{2}}) + \frac{\partial}{\partial u_{3}} (\frac{h_{1} h_{2}}{h_{3}} \frac{\partial}{\partial u_{3}})]$ (2.2)

3.2. $\nabla \cdot (μ A) = (\nabla μ) \cdot A + μ (\nabla \cdot A)$

$μ (u_{1}, u_{2}, u_{3})$ 连续可微，由(1.4)和(1.6)式可得到：

$\begin{matrix} \nabla \cdot (μ A) = (\nabla μ) \cdot A + μ (\nabla \cdot A) \\ = [(\frac{\partial μ}{h_{1} \partial u_{1}}) A_{1} + (\frac{\partial μ}{h_{2} \partial u_{2}}) A_{2} + (\frac{\partial μ}{h_{3} \partial u_{3}}) A_{3}] \\ + \frac{μ}{h_{1} h_{2} h_{3}} [\frac{\partial}{\partial u_{1}} (h_{2} h_{3} A_{1}) + \frac{\partial}{\partial u_{2}} (h_{3} h_{1} A_{2}) + \frac{\partial}{\partial u_{3}} (h_{1} h_{2} A_{3})] \end{matrix}$ (2.3)

3.3. $\nabla^{2} A = (\nabla^{2} A_{1}) e_{1} + (\nabla^{2} A_{2}) e_{2} + (\nabla^{2} A_{3}) e_{3}$

由(2.2)式可得：

$\begin{matrix} \nabla^{2} A = (\nabla^{2} A_{1}) e_{1} + (\nabla^{2} A_{2}) e_{2} + (\nabla^{2} A_{3}) e_{3} \\ = [\frac{\partial}{\partial u_{1}} (\frac{h_{2} h_{3}}{h_{1}} \frac{\partial A_{1}}{\partial u_{1}}) + \frac{\partial}{\partial u_{2}} (\frac{h_{3} h_{1}}{h_{2}} \frac{\partial A_{1}}{\partial u_{2}}) + \frac{\partial}{\partial u_{3}} (\frac{h_{1} h_{2}}{h_{3}} \frac{\partial A_{1}}{\partial u_{3}})] e_{1} \\ + [\frac{\partial}{\partial u_{1}} (\frac{h_{2} h_{3}}{h_{1}} \frac{\partial A_{2}}{\partial u_{1}}) + \frac{\partial}{\partial u_{2}} (\frac{h_{3} h_{1}}{h_{2}} \frac{\partial A_{2}}{\partial u_{2}}) + \frac{\partial}{\partial u_{3}} (\frac{h_{1} h_{2}}{h_{3}} \frac{\partial A_{2}}{\partial u_{3}})] e_{2} \\ + [\frac{\partial}{\partial u_{1}} (\frac{h_{2} h_{3}}{h_{1}} \frac{\partial A_{3}}{\partial u_{1}}) + \frac{\partial}{\partial u_{2}} (\frac{h_{3} h_{1}}{h_{2}} \frac{\partial A_{3}}{\partial u_{2}}) + \frac{\partial}{\partial u_{3}} (\frac{h_{1} h_{2}}{h_{3}} \frac{\partial A_{3}}{\partial u_{3}})] e_{3} \end{matrix}$ (2.4)

3.4. $\nabla \times (\nabla \times A) = \nabla (\nabla \cdot A) - \nabla^{2} A$

由(1.4)，(1.6)，(2.4)三式可以得到

$\begin{array}{l} \nabla \times (\nabla \times A) \\ = \nabla (\nabla \cdot A) - \nabla^{2} A \\ = \frac{\partial}{h_{1} \partial u_{1}} {\frac{1}{h_{1} h_{2} h_{3}} [\frac{\partial}{\partial u_{1}} (h_{1} h_{2} A_{1}) + \frac{\partial}{\partial u_{2}} (h_{3} h_{1} A_{2}) + \frac{\partial}{\partial u_{3}} (h_{1} h_{2} A_{3})]} e_{1} \\ + \frac{\partial}{h_{2} \partial u_{2}} {\frac{1}{h_{1} h_{2} h_{3}} [\frac{\partial}{\partial u_{1}} (h_{1} h_{2} A_{1}) + \frac{\partial}{\partial u_{2}} (h_{3} h_{1} A_{2}) + \frac{\partial}{\partial u_{3}} (h_{1} h_{2} A_{3})]} e_{2} \\ + \frac{\partial}{h_{3} \partial u_{3}} {\frac{1}{h_{1} h_{2} h_{3}} [\frac{\partial}{\partial u_{1}} (h_{1} h_{2} A_{1}) + \frac{\partial}{\partial u_{2}} (h_{3} h_{1} A_{2}) + \frac{\partial}{\partial u_{3}} (h_{1} h_{2} A_{3})]} e_{3} \end{array}$

$\begin{array}{l} - [\frac{\partial}{\partial u_{1}} (\frac{h_{2} h_{3}}{h_{1}} \frac{\partial A_{1}}{\partial u_{1}}) + \frac{\partial}{\partial u_{2}} (\frac{h_{3} h_{1}}{h_{2}} \frac{\partial A_{1}}{\partial u_{2}}) + \frac{\partial}{\partial u_{3}} (\frac{h_{1} h_{2}}{h_{3}} \frac{\partial A_{1}}{\partial u_{3}})] e_{1} \\ - [\frac{\partial}{\partial u_{1}} (\frac{h_{2} h_{3}}{h_{1}} \frac{\partial A_{2}}{\partial u_{1}}) + \frac{\partial}{\partial u_{2}} (\frac{h_{3} h_{1}}{h_{2}} \frac{\partial A_{2}}{\partial u_{2}}) + \frac{\partial}{\partial u_{3}} (\frac{h_{1} h_{2}}{h_{3}} \frac{\partial A_{2}}{\partial u_{3}})] e_{2} \\ - [\frac{\partial}{\partial u_{1}} (\frac{h_{2} h_{3}}{h_{1}} \frac{\partial A_{3}}{\partial u_{1}}) + \frac{\partial}{\partial u_{2}} (\frac{h_{3} h_{1}}{h_{2}} \frac{\partial A_{3}}{\partial u_{2}}) + \frac{\partial}{\partial u_{3}} (\frac{h_{1} h_{2}}{h_{3}} \frac{\partial A_{3}}{\partial u_{3}})] e_{3} \end{array}$ (2.5)

3.5. 动量算符 $- i ℏ \nabla$ 与Schrödinger方程

由(1.2)式可知动量算符 $- i ℏ \nabla = - i ℏ (e_{1} \frac{\partial}{h_{1} \partial u_{1}} + e_{2} \frac{\partial}{h_{2} \partial u_{2}} + e_{3} \frac{\partial}{h_{3} \partial u_{3}})$ ，自由粒子波函数 $ψ$ 的波动方程为

$- i ℏ \frac{\partial ψ}{\partial t} = - \frac{ℏ^{2}}{2 m} [\frac{\partial}{\partial u_{1}} (\frac{h_{2} h_{3}}{h_{1}} \frac{\partial ψ}{\partial u_{1}}) + \frac{\partial}{\partial u_{2}} (\frac{h_{3} h_{1}}{h_{2}} \frac{\partial ψ}{\partial u_{2}}) + \frac{\partial}{\partial u_{3}} (\frac{h_{1} h_{2}}{h_{3}} \frac{\partial ψ}{\partial u_{3}})] + V (u_{1}, u_{2}, u_{3}) ψ$ (2.6)

4. 在笛卡尔坐标系、柱坐标系和球坐标系的具体形式

4.1. 笛卡尔坐标系

在笛卡尔坐标系中，坐标因子 $h_{1} = 1$ ， $h_{2} = 1$ ， $h_{3} = 1$ ， $d s_{1} = x$ ， $d s_{2} = y$ ， $d s_{3} = z$ ，则：

$\nabla = \frac{\partial}{\partial s_{1}} e_{1} + \frac{\partial}{\partial s_{2}} e_{2} + \frac{\partial}{\partial s_{3}} e_{3} = \frac{\partial}{\partial x} i + \frac{\partial}{\partial y} j + \frac{\partial}{\partial z} k$

$\nabla ϕ = (\frac{\partial ϕ}{\partial x}) i + (\frac{\partial ϕ}{\partial y}) j + (\frac{\partial ϕ}{\partial z}) k$

$div A = \nabla \cdot A = \frac{\partial A_{x}}{\partial x} + \frac{\partial A_{y}}{\partial y} + \frac{\partial A_{z}}{\partial z}$

$curl A = \nabla \times A = (\frac{\partial A_{z}}{\partial y} - \frac{\partial A_{y}}{\partial z}) i + (\frac{\partial A_{x}}{\partial z} - \frac{\partial A_{z}}{\partial x}) j + (\frac{\partial A_{y}}{\partial x} - \frac{\partial A_{x}}{\partial y}) k$

$\nabla^{2} = \frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial z^{2}}$

4.2. 柱坐标系

在柱坐标系中，坐标因子 $h_{1} = 1$ ， $h_{2} = ρ$ ， $h_{3} = 1$ ， $d s_{1} = d ρ$ ， $d s_{2} = ρ d φ$ ， $d s_{3} = d z$ 则：

$\nabla = \frac{\partial}{\partial s_{1}} e_{1} + \frac{\partial}{\partial s_{2}} e_{2} + \frac{\partial}{\partial s_{3}} e_{3} = \frac{\partial}{\partial ρ} e_{ρ} + \frac{1}{ρ} \frac{\partial}{\partial φ} e_{φ} + \frac{\partial}{\partial z} e_{z}$

$\nabla ϕ = (\frac{\partial ϕ}{\partial ρ}) e_{ρ} + \frac{1}{ρ} (\frac{\partial ϕ}{\partial φ}) e_{φ} + (\frac{\partial ϕ}{\partial z}) e_{z}$

$div A = \nabla \cdot A = \frac{1}{ρ} \frac{\partial (ρ A_{ρ})}{\partial ρ} + \frac{1}{ρ} \frac{\partial A_{φ}}{\partial φ} + \frac{\partial A_{z}}{\partial z}$

$curl A = \nabla \times A = (\frac{1}{ρ} \frac{\partial A_{z}}{\partial φ} - \frac{\partial A_{φ}}{\partial z}) e_{ρ} + (\frac{\partial A_{ρ}}{\partial z} - \frac{\partial A_{z}}{\partial ρ}) e_{φ} + \frac{1}{ρ} (\frac{\partial (ρ A_{φ})}{\partial ρ} - \frac{\partial A_{ρ}}{\partial φ}) e_{z}$

$\nabla^{2} = \frac{1}{ρ} \frac{\partial}{\partial ρ} (ρ \frac{\partial}{\partial ρ}) + \frac{1}{ρ^{2}} \frac{\partial^{2}}{\partial φ^{2}} + \frac{\partial^{2}}{\partial z^{2}}$

4.3. 球坐标系

球坐标系中，坐标因子 $h_{1} = 1$ ， $h_{2} = r$ ， $h_{3} = r \sin θ$ ， $d s_{1} = d r$ ， $d s_{2} = r d θ$ ， $d s_{3} = r \sin θ d φ$ ，则：

$\nabla = \frac{\partial}{\partial r} e_{r} + \frac{1}{r} \frac{\partial}{\partial θ} e_{θ} + \frac{1}{r \sin θ} \frac{\partial}{\partial φ} e_{φ}$

$\nabla ϕ = (\frac{\partial ϕ}{\partial r}) e_{r} + \frac{1}{r} (\frac{\partial ϕ}{\partial θ}) e_{θ} + \frac{1}{r \sin θ} (\frac{\partial ϕ}{\partial φ}) e_{φ}$

$div A = \nabla \cdot A = \frac{1}{r^{2}} \frac{\partial (r^{2} A_{r})}{\partial r} + \frac{1}{r \sin θ} \frac{\partial (A_{θ} \sin θ)}{\partial θ} + \frac{1}{r \sin θ} \frac{\partial A_{φ}}{\partial φ}$

$curl A = \nabla \times A = \frac{1}{r \sin θ} (\frac{\partial (A_{φ} \sin θ)}{\partial θ} - \frac{\partial A_{θ}}{\partial φ}) e_{r} + \frac{1}{r} (\frac{1}{\sin θ} \frac{\partial A_{r}}{\partial φ} - \frac{\partial (r A_{φ})}{\partial r}) e_{θ} + \frac{1}{r} (\frac{\partial (r A_{θ})}{\partial r} - \frac{\partial A_{r}}{\partial θ}) e_{φ}$

$\nabla^{2} = \frac{1}{r^{2}} \frac{\partial}{\partial r} (r^{2} \frac{\partial}{\partial r}) + \frac{1}{r^{2} \sin θ} \frac{\partial}{\partial θ} (\sin θ \frac{\partial}{\partial θ}) + \frac{1}{r^{2} \sin^{2} θ} \frac{\partial^{2}}{\partial φ^{2}}$

5. 结论

通过定义一个三维正交曲线坐标系 $(u_{1}, u_{2}, u_{3})$ 组成的空间，以及用 $u_{1}$ 、 $u_{2}$ 、 $u_{3}$ 乘以坐标因子 $h_{1}$ 、 $h_{2}$ 、 $h_{3}$ 来表示三个方向的线长， $h_{1}$ 、 $h_{2}$ 、 $h_{3}$ 是 $(u_{1}, u_{2}, u_{3})$ 的函数。定义三个线元 $d s_{1} = h_{1} d u_{1}$ ， $d s_{2} = h_{2} d u_{2}$ ， $d s_{3} = h_{3} d u_{3}$ ，可以得到关于哈密顿算符 $\nabla$ 、 $\nabla ϕ$ 、 $\nabla \cdot A$ 、 $\nabla \times A$ 、 $\nabla^{2}$ 及的一般表达式Poisson方程和Laplace方程的一般表达式。在这些表达式中 $h_{1}$ 、 $h_{2}$ 、 $h_{3}$ 取不同的值，可以演化得到在相应坐标系下的对应运算式。

基金项目

云南省教育厅科学研究基金项目(批准号：2016ZZX047)。

文章引用

冯朝桢,段卫龙. 哈密顿算符及其一般运算表达式的分析
General Expression Analysis of the Hamiltonian Operator and Its Formula[J]. 应用数学进展, 2020, 09(08): 1286-1291. https://doi.org/10.12677/AAM.2020.98150

参考文献

1. Morales, M., Diaz, R.A. and Herrera, W.J. (2015) Solutions of Laplace’s Equation with Simple Boundary Conditions, and Their Applications for Capacitors with Multiple Symmetries. Journal of Electrostatics, 78, 31-45. https://doi.org/10.1016/j.elstat.2015.09.006

2. Polyakov, P.A., Rusakova, N.E. and Samukhina, Y.V. (2015) New Solutions for Charge Distribution on Conductor Surface. Journal of Electrostatics, 77, 147-152. https://doi.org/10.1016/j.elstat.2015.08.003

3. Frąckowiaka, A., Wolfersdorfb, J.V. and Ciałkowski, M. (2011) Solution of the Inverse Heat Conduction Problem Described by the Poisson Equation for a Cooled Gas-Turbine Blade. International Journal of Heat and Mass Transfer, 54, 1236-1243. https://doi.org/10.1016/j.ijheatmasstransfer.2010.11.001

4. Smythe, W.R. (1989) Static and Dynamic Electricity. 3rd Edition, Taylor & Francis, Abingdon, 50-53.

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