spherical harmonics

简明释义

球函数

英英释义

例句

1.In quantum mechanics, the wave functions of particles in a central potential are expressed using spherical harmonics 球面谐波.

在量子力学中，处于中心势阱中的粒子的波函数使用球面谐波 spherical harmonics表示。

2.The visualization of molecular orbitals can be enhanced by employing spherical harmonics 球面谐波 to represent their shapes.

通过使用球面谐波 spherical harmonics来表示分子轨道的形状，可以增强分子轨道的可视化效果。

3.When analyzing the radiation patterns of antennas, engineers utilize spherical harmonics 球面谐波 to accurately describe the emission characteristics.

在分析天线的辐射模式时，工程师利用球面谐波 spherical harmonics准确描述发射特性。

4.The spherical harmonics 球面谐波 are crucial in the field of geophysics for modeling the Earth's gravitational field.

在地球物理学领域，球面谐波 spherical harmonics对于建模地球的引力场至关重要。

5.In computational physics, we often use spherical harmonics 球面谐波 to solve problems involving angular momentum.

在计算物理中，我们经常使用球面谐波 spherical harmonics来解决涉及角动量的问题。

作文

In the field of mathematics and physics, the concept of spherical harmonics plays a crucial role in solving problems that involve spherical geometries. These mathematical functions are defined on the surface of a sphere and can be used to represent complex shapes and phenomena in three dimensions. Understanding spherical harmonics is essential for various applications, including quantum mechanics, electromagnetism, and geophysics. To begin with, spherical harmonics can be thought of as a generalization of the idea of Fourier series, which are used to represent functions on a flat surface. While Fourier series break down periodic functions into sine and cosine components, spherical harmonics decompose functions defined on the surface of a sphere into a set of orthogonal basis functions. This orthogonality property makes them particularly useful for solving partial differential equations in spherical coordinates.One of the most significant applications of spherical harmonics is in the field of quantum mechanics, specifically in the study of atomic orbitals. The shapes of these orbitals, which dictate how electrons are arranged around an atom, can be described using spherical harmonics. Each orbital corresponds to a specific set of quantum numbers, which determine the energy levels and spatial distribution of electrons. By utilizing spherical harmonics, physicists can accurately model the behavior of electrons in atoms, leading to a deeper understanding of chemical bonding and reactions.Moreover, spherical harmonics are also employed in geophysics to analyze the Earth's gravitational and magnetic fields. The Earth's shape is not a perfect sphere; it is an oblate spheroid, which means that its equatorial radius is larger than its polar radius. However, by using spherical harmonics, scientists can create models that approximate the Earth's gravitational field with high precision. This modeling is vital for satellite navigation systems and for understanding geological processes such as tectonic movements and volcanic activity.In computer graphics, spherical harmonics are utilized for rendering lighting effects on 3D models. By representing light sources and their interactions with surfaces using spherical harmonics, artists and developers can achieve realistic shading and illumination in virtual environments. This technique allows for efficient computation of lighting without sacrificing visual quality, making it an essential tool in modern computer graphics.Furthermore, the application of spherical harmonics extends to signal processing and data analysis. In these fields, they are used to analyze signals that have a spatial component, such as images or sound captured from different directions. By transforming these signals into the spherical harmonics domain, researchers can extract meaningful features and patterns, which can then be used for classification or reconstruction purposes.In conclusion, the study of spherical harmonics is a fascinating intersection of mathematics, physics, and engineering. Their ability to model complex phenomena on spherical surfaces makes them invaluable in various scientific and technological fields. As research continues to advance, the applications of spherical harmonics will likely expand, leading to new discoveries and innovations that can further enhance our understanding of the universe around us. Therefore, gaining a solid grasp of spherical harmonics is not only beneficial for students and professionals in these domains but also for anyone interested in the fundamental principles that govern our world.

在数学和物理学领域，spherical harmonics（球谐函数）的概念在解决涉及球形几何的问题中起着至关重要的作用。这些数学函数定义在球面上，可以用来表示三维空间中的复杂形状和现象。理解spherical harmonics对于包括量子力学、电磁学和地球物理学在内的各种应用至关重要。首先，spherical harmonics可以被视为傅里叶级数的推广，傅里叶级数用于表示平面上的函数。虽然傅里叶级数将周期函数分解为正弦和余弦成分，但spherical harmonics则将定义在球面上的函数分解为一组正交基函数。这种正交性使得它们在求解球坐标系中的偏微分方程时特别有用。spherical harmonics最重要的应用之一是在量子力学领域，特别是在研究原子轨道方面。这些轨道的形状决定了电子在原子周围的排列方式，可以使用spherical harmonics进行描述。每个轨道对应一组特定的量子数，这些量子数决定了能级和电子的空间分布。通过利用spherical harmonics，物理学家能够准确模拟原子中电子的行为，从而加深对化学键合和反应的理解。此外，spherical harmonics还被用于地球物理学中，以分析地球的重力场和磁场。地球的形状并不是一个完美的球体；它是一个扁球体，这意味着其赤道半径大于极半径。然而，通过使用spherical harmonics，科学家可以创建高精度的模型来近似地球的重力场。这种建模对于卫星导航系统以及理解地质过程（如构造运动和火山活动）至关重要。在计算机图形学中，spherical harmonics被用于渲染3D模型上的光照效果。通过使用spherical harmonics来表示光源及其与表面的相互作用，艺术家和开发者能够在虚拟环境中实现逼真的阴影和照明。这种技术允许高效计算光照而不牺牲视觉质量，使其成为现代计算机图形学中的重要工具。此外，spherical harmonics的应用还扩展到信号处理和数据分析。在这些领域，它们用于分析具有空间分量的信号，如从不同方向捕获的图像或声音。通过将这些信号转换到spherical harmonics域，研究人员可以提取有意义的特征和模式，然后可以用于分类或重建目的。总之，spherical harmonics的研究是数学、物理和工程的迷人交集。它们在球面上建模复杂现象的能力使得它们在各个科学和技术领域中不可或缺。随着研究的不断推进，spherical harmonics的应用可能会扩展，带来新的发现和创新，进一步增强我们对周围宇宙的理解。因此，掌握spherical harmonics不仅对这些领域的学生和专业人士有益，也对任何对支配我们世界的基本原则感兴趣的人都有帮助。