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所在平台: Udemy |
课程主页: https://www.udemy.com/course/mathematical-intuition-for-heisenberg-uncertainty-principle/
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Coursera 课程“偏微分方程：综合课程”旨在深入解析如何利用傅里叶变换这一强大工具来求解偏微分方程。 课程共分为三个部分： * **第一部分：** 从傅里叶级数入手，推导傅里叶变换及其逆变换，并将其应用于求解偏微分方程。此部分需要微积分和多变量微积分基础，特别是导数、积分、梯度、拉普拉斯算子和球坐标系。 * **第二部分：** 介绍笛卡尔坐标系和极坐标系下的热方程和拉普拉斯方程。将通过分离变量法解决带有不同边界条件的练习题。此部分独立于第一部分，但建议具备常微分方程 (ODE) 基础。 * **第三部分：** 专注于扩散/热方程，将从物理原理推导该方程并进行严格求解。此外，还包含关于海森堡不确定性原理数学推导的奖励性内容。 **课程收益：** * 全面掌握傅里叶变换及其在求解偏微分方程中的应用。 * 学习如何运用分离变量法解决不同边界条件的练习题。 * 深入理解扩散/热方程的原理及其求解方法。 * 通过海森堡不确定性原理的奖励性章节，加深对量子力学背后数学原理的理解。 **先修要求：** * 具备微积分和多变量微积分基础，特别是导数、积分、梯度、拉普拉斯算子和球坐标系。 * 建议具备常微分方程 (ODE) 基础。 * 具备一些复变函数与留数理论知识会有帮助。 **目标学员：** * 有数学或物理背景，希望深入了解如何使用傅里叶变换求解偏微分方程的学生和专业人士。 * 对量子力学和海森堡不确定性原理的数学原理感兴趣的学员。
Solving Partial Differential Equations using the Fourier Transform: A Step-by-Step GuideCourse Description:This course is designed to provide a comprehensive understanding of how the Fourier Transform can be used as a powerful tool to solve Partial Differential Equations (PDE). The course is divided into three parts, each building on the previous one, and includes bonus sections on the mathematical derivation of the Heisenberg Uncertainty Principle.Part 1: In this part, we will start with the basics of the Fourier series and derive the Fourier Transform and its inverse. We will then apply these concepts to solve PDE's using the Fourier Transform. Prerequisites for this section are Calculus and Multivariable Calculus, with a focus on topics related to derivatives, integrals, gradient, Laplacian, and spherical coordinates.Part 2: This section introduces the heat equation and the Laplace equation in Cartesian and polar coordinates. We will solve exercises with different boundary conditions using the Separation of Variables method. This section is self-contained and independent of the first one, but prior knowledge of ODEs is recommended.Part 3: This section is dedicated to the Diffusion/Heat equation, where we will derive the equation from physics principles and solve it rigorously. Bonus sections are included on the mathematical derivation of the Heisenberg Uncertainty Principle.Course Benefits:Gain a thorough understanding of the Fourier Transform and its application to solving PDE's.Learn how to apply Separation of Variables method to solve exercises with different boundary conditions.Gain insight into the Diffusion/Heat equation and how it can be solved.Bonus sections on the Heisenberg Uncertainty Principle provide a deeper understanding of the mathematical principles behind quantum mechanics.Prerequisites:Calculus and Multivariable Calculus with a focus on derivatives, integrals, gradient, Laplacian, and spherical coordinates.Prior knowledge of ODEs is recommended.Some knowledge of Complex Calculus and residues may be useful.Who is this course for?Students and professionals with a background in Mathematics or Physics looking to gain a deeper understanding of solving PDE's using the Fourier Transform.Those interested in the mathematical principles behind quantum mechanics and the Heisenberg Uncertainty Principle.
