叠加原理

来自集智百科 - 伊辛模型
跳到导航 跳到搜索

此词条暂由水流心不竞初译,翻译字数共,未经审校,带来阅读不便,请见谅。

{{简述}说明线性系统的物理解是线性的基本物理原理}}

模板:关于

文件:Anas platyrhynchos with ducklings reflecting water.jpg
Superposition of almost plane waves (diagonal lines) from a distant source and waves from the wake of the ducks. Linearity holds only approximately in water and only for waves with small amplitudes relative to their wavelengths.

[[文件:Anas platyrhynchos小鸭在倒影水.jpg|拇指|右|来自远源的几乎平面波s(对角线)和来自s的尾迹的波的叠加。线性仅在水中近似成立,并且仅适用于相对于其波长振幅较小的波。]]

Superposition of almost plane waves (diagonal lines) from a distant source and waves from the wake of the ducks. Linearity holds only approximately in water and only for waves with small amplitudes relative to their wavelengths.

来自远处震源的几乎是平面波(对角线)和鸭子尾迹的波的叠加。线性仅在水中近似成立,且仅适用于相对于波长振幅较小的波。


文件:Rolling animation.gif
Rolling motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: translation without rotation, and rotation without translation.

[[文件:Rolling animation.gif|右|拇指|滚动运动是两个运动的叠加。车轮的滚动运动可以描述为两个独立运动的组合:平移旋转和旋转无平移。]]

Rolling motion as superposition of two motions. The rolling motion of the wheel can be described as a combination of two separate motions: translation without rotation, and rotation without translation.

作为两个运动的叠加的滚动运动。车轮的滚动运动可以被描述为两个独立运动的组合: 无旋转的平动和无平移的旋转


The superposition principle,[1] also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y).

“‘叠加原理’”,[2]也称为“‘叠加性质’”,指出,对于所有的线性系统,两个或多个刺激引起的净反应是每个刺激单独引起的反应的总和。因此,如果输入“A”产生响应“X”,输入“B”产生响应“Y”,则输入(“A”+“B”)产生响应(“X”+“Y”)。

The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So that if input A produces response X and input B produces response Y then input (A + B) produces response (X + Y).

叠加原理,也被称为叠加性质,指出,对于所有线性系统,由两个或更多刺激引起的净反应是由每个刺激单独引起的反应的总和。因此,如果输入 a 产生响应 x,输入 b 产生响应 y,那么输入(a + b)产生响应(x + y)。


A function [math]\displaystyle{ F(x) }[/math] that satisfies the superposition principle is called a linear function. Superposition can be defined by two simpler properties; additivity and homogeneity

满足叠加原理的函数[math]\displaystyle{ F(x) }[/math]称为线性函数。叠加可以用两个简单的性质来定义;可加性齐性

A function [math]\displaystyle{ F(x) }[/math] that satisfies the superposition principle is called a linear function. Superposition can be defined by two simpler properties; additivity and homogeneity

一个函数 < math > f (x) </math > 满足叠加原理的称为线性函数。叠加可以用两个更简单的性质来定义: 可加性和均匀性

[math]\displaystyle{ F(x_1+x_2)=F(x_1)+F(x_2) \, }[/math] Additivity

[math]\displaystyle{ F(x_1+x_2)=F(x_1)+F(x_2) \, }[/math]Additivity

< math > f (x _ 1 + x _ 2) = f (x _ 1) + f (x _ 2) ,</math > 可加性


[math]\displaystyle{ F(a x)=a F(x) \, }[/math] Homogeneity

[math]\displaystyle{ F(a x)=a F(x) \, }[/math]Homogeneity

[math]\displaystyle{ F(a x)=a F(x) \, }[/math]Homogeneity


for scalar a.

for scalar .

对于纯量a


This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency domain linear transform methods such as Fourier, Laplace transforms, and linear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behaviour.

这一原理在物理工程中有许多应用,因为许多物理系统可以建模为线性系统。例如,可以建模为线性系统,其中输入激励是梁上的荷载,输出响应是梁的挠度。线性系统的重要性在于它们更易于数学分析;有大量的数学技术,频域线性变换方法,如傅立叶拉普拉斯变换线性算子理论,都是适用的。因为物理系统通常只是近似线性的,叠加原理只是真实物理行为的近似。

This principle has many applications in physics and engineering because many physical systems can be modeled as linear systems. For example, a beam can be modeled as a linear system where the input stimulus is the load on the beam and the output response is the deflection of the beam. The importance of linear systems is that they are easier to analyze mathematically; there is a large body of mathematical techniques, frequency domain linear transform methods such as Fourier, Laplace transforms, and linear operator theory, that are applicable. Because physical systems are generally only approximately linear, the superposition principle is only an approximation of the true physical behaviour.

这个原理在物理学和工程学中有许多应用,因为许多物理系统可以被模拟为线性系统。例如,一个梁可以被建模为一个线性系统,其中输入的刺激是梁上的载荷,输出的响应是梁的挠度。线性系统的重要性在于它们更容易在数学上进行分析; 有大量的数学技术、频域线性变换方法,如傅立叶变换、拉普拉斯变换和线性算子理论,这些都是适用的。因为物理系统通常只是近似线性的,叠加原理只是真实物理行为的近似值。


The superposition principle applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum.

叠加原理适用于“任何”线性系统,包括这些形式的代数方程s、线性微分方程方程组。刺激和反应可以是数字、函数、向量、vector fields、时变信号或满足某些公理的任何其他对象。注意,当涉及向量或向量场时,叠加被解释为向量和

The superposition principle applies to any linear system, including algebraic equations, linear differential equations, and systems of equations of those forms. The stimuli and responses could be numbers, functions, vectors, vector fields, time-varying signals, or any other object that satisfies certain axioms. Note that when vectors or vector fields are involved, a superposition is interpreted as a vector sum.

叠加原理适用于任何线性系统,包括代数方程、线性微分方程和这些形式的方程组。刺激和反应可以是数字、函数、矢量、矢量场、时变信号或者任何其他满足某些公理的对象。注意,当涉及到向量或向量场时,叠加被解释为向量和。


Relation to Fourier analysis and similar methods与傅里叶分析及类似方法的关系

By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific and simple form, often the response becomes easier to compute.

通过将一个非常普遍的刺激(在线性系统中)写成一个特定和简单形式的刺激的叠加,反应往往变得更容易计算。

By writing a very general stimulus (in a linear system) as the superposition of stimuli of a specific and simple form, often the response becomes easier to compute.

通过写一个非常一般的刺激(在线性系统中)作为一个特定的和简单的形式的刺激叠加,通常反应变得更容易计算。


For example, in Fourier analysis, the stimulus is written as the superposition of infinitely many sinusoids. Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different amplitude and phase.) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses.

例如,在Fourier analysis中,刺激被写为无穷多个正弦波 sinusoid的叠加。由于叠加原理,每个正弦波可以单独分析,并且可以计算其单独的响应。(反应本身是一个正弦波,与刺激频率相同,但通常是不同的振幅相位)根据叠加原理,对原始刺激的反应是所有单个正弦波反应的总和(或积分)。

For example, in Fourier analysis, the stimulus is written as the superposition of infinitely many sinusoids. Due to the superposition principle, each of these sinusoids can be analyzed separately, and its individual response can be computed. (The response is itself a sinusoid, with the same frequency as the stimulus, but generally a different amplitude and phase.) According to the superposition principle, the response to the original stimulus is the sum (or integral) of all the individual sinusoidal responses.

例如,在《傅立叶变换家族中的关系,刺激信号就是无限多个正弦波的叠加。由于有了叠加原理,这些正弦波可以单独分析,也可以单独计算它们的响应。(反应本身是一个正弦波,与刺激频率相同,但一般是不同的振幅和相位。)根据叠加原理,对原始刺激的反应是所有单个正弦反应的总和(或积分)。


As another common example, in Green's function analysis, the stimulus is written as the superposition of infinitely many impulse functions, and the response is then a superposition of impulse responses.

另一个常见的例子是,在格林函数分析 Green's function analysis中,刺激被写成无穷多个脉冲函数impulse functions的叠加,然后响应就是脉冲响应s的叠加。

As another common example, in Green's function analysis, the stimulus is written as the superposition of infinitely many impulse functions, and the response is then a superposition of impulse responses.

作为另一个常见的例子,在格林函数分析中,刺激被写成无穷多个脉冲函数的叠加,而响应则是脉冲响应的叠加。


Fourier analysis is particularly common for waves. For example, in electromagnetic theory, ordinary light is described as a superposition of plane waves (waves of fixed frequency, polarization, and direction). As long as the superposition principle holds (which is often but not always; see nonlinear optics), the behavior of any light wave can be understood as a superposition of the behavior of these simpler plane waves.

傅里叶分析对于s特别常见。例如,在电磁理论中,普通被描述为平面波s(固定频率偏振和方向的波)的叠加。只要叠加原理成立(这通常是但并不总是;见非线性光学),任何光波的行为都可以理解为这些更简单的平面波行为的叠加。

Fourier analysis is particularly common for waves. For example, in electromagnetic theory, ordinary light is described as a superposition of plane waves (waves of fixed frequency, polarization, and direction). As long as the superposition principle holds (which is often but not always; see nonlinear optics), the behavior of any light wave can be understood as a superposition of the behavior of these simpler plane waves.

傅立叶变换家族中的关系对于波浪来说尤其常见。例如,在电磁理论中,普通光被描述为平面波(具有固定频率、偏振和方向的波)的叠加。只要叠加原理能够持续(这种情况经常发生,但并不总是如此; 请参阅非线性光学) ,任何光波的行为都可以被理解为这些简单平面波行为的叠加。

Wave superposition波叠加

模板:Further

模板:进一步

文件:Standing wave 2.gif
Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a standing wave.

[[文件:驻波2.gif |拇指|右|在同一介质中以相反方向传播的两个波线性组合。在这个动画中,两个波的波长相同,振幅之和产生驻波.]] <!--见下文!-->

Two waves traveling in opposite directions across the same medium combine linearly. In this animation, both waves have the same wavelength and the sum of amplitudes results in a standing wave.

在同一介质中以相反方向运动的两个波线性结合。在这个动画中,两个波具有相同的波长,并且振幅之和导致了驻波. < ! -- 见下文!-->

文件:Standing waves1.gif
two waves permeate without influencing each other

拇指|两波互不影响

two waves permeate without influencing each other

两个波互相渗透而不相互影响


Waves are usually described by variations in some parameter through space and time—for example, height in a water wave, pressure in a sound wave, or the electromagnetic field in a light wave. The value of this parameter is called the amplitude of the wave, and the wave itself is a function specifying the amplitude at each point.

波通常由一些参数在空间和时间上的变化来描述,例如,水波中的高度、声波中的压力或光波中的电磁场。此参数的值称为波的振幅,波本身是一个指定每个点振幅的函数

Waves are usually described by variations in some parameter through space and time—for example, height in a water wave, pressure in a sound wave, or the electromagnetic field in a light wave. The value of this parameter is called the amplitude of the wave, and the wave itself is a function specifying the amplitude at each point.

波通常是通过某些参数在空间和时间上的变化来描述的,例如,水波中的高度、声波中的压力或光波中的电磁场。这个参数的值叫做波的振幅,波本身是指定每个点振幅的函数。


In any system with waves, the waveform at a given time is a function of the sources (i.e., external forces, if any, that create or affect the wave) and initial conditions of the system. In many cases (for example, in the classic wave equation), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at top.)

在任何有波浪的系统中,给定时间的波形是系统(即,产生或影响波浪的外力,如有)和初始条件的函数。在许多情况下(例如,在经典的波动方程),描述波动的方程是线性的。如果这是真的,可以应用叠加原理。这意味着,两个或多个波穿过同一空间所产生的净振幅是单独波产生的振幅之和。例如,两个相互传播的波将直接穿过对方,而另一侧没有任何变形。(见上图。)

In any system with waves, the waveform at a given time is a function of the sources (i.e., external forces, if any, that create or affect the wave) and initial conditions of the system. In many cases (for example, in the classic wave equation), the equation describing the wave is linear. When this is true, the superposition principle can be applied. That means that the net amplitude caused by two or more waves traversing the same space is the sum of the amplitudes that would have been produced by the individual waves separately. For example, two waves traveling towards each other will pass right through each other without any distortion on the other side. (See image at top.)

在任何有波的系统中,在给定时间的波形是源(即,如果有,创造或影响波的外力)和系统的初始条件的函数。在许多情况下(例如,在经典波动方程中) ,描述波动的方程是线性的。如果这是真的,就可以应用叠加原理。这意味着两个或两个以上的波穿过同一空间所产生的净振幅是单个波所产生振幅的和。例如,两个互相走向的波会在另一边没有任何失真的情况下正好穿过对方。(见上图)


Wave diffraction vs. wave interference模板:Anchor模板:Anchor波衍射与波干涉模板:锚定模板:锚定

With regard to wave superposition, Richard Feynman wrote:[3]

关于波叠加,Richard Feynman写道:[4]

With regard to wave superposition, Richard Feynman wrote:

关于波叠加,理查德 · 费曼写道:

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

Other authors elaborate:[5]

其他作者阐述:[6]

Other authors elaborate:

其他作者详细说明:

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

模板:引述


Yet another source concurs:[7]

Yet another source concurs:

然而,另一位消息人士也表示同意:

/* Styling for Template:Quote */ .templatequote { overflow: hidden; margin: 1em 0; padding: 0 40px; } .templatequote .templatequotecite {

   line-height: 1.5em;
   /* @noflip */
   text-align: left;
   /* @noflip */
   padding-left: 1.6em;
   margin-top: 0;

}

Wave interference波的干涉

模板:主

The phenomenon of interference between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in noise-cancelling headphones, the summed variation has a smaller amplitude than the component variations; this is called destructive interference. In other cases, such as in a line array, the summed variation will have a bigger amplitude than any of the components individually; this is called constructive interference.

波与波之间的干涉现象就是基于这一思想。当两个或多个波穿过同一空间时,每个点的净振幅是单个波振幅的总和。在某些情况下,例如在噪声消除耳机中,求和的变化比分量变化具有更小的振幅;这被称为“相消干扰”。在其他情况下,例如在线阵列中,求和的变化将具有比单独的任何分量更大的幅度;这被称为“构造干扰”。

The phenomenon of interference between waves is based on this idea. When two or more waves traverse the same space, the net amplitude at each point is the sum of the amplitudes of the individual waves. In some cases, such as in noise-cancelling headphones, the summed variation has a smaller amplitude than the component variations; this is called destructive interference. In other cases, such as in a line array, the summed variation will have a bigger amplitude than any of the components individually; this is called constructive interference.

波与波之间的干涉现象正是基于这一思想。当两个或多个波穿过同一空间时,每个点的净振幅是单个波振幅的总和。在某些情况下,例如在噪声消除耳机中,求和的变化幅度小于分量变化幅度;这称为相消干扰。在其他情况下,例如在线阵中,求和的变化幅度将比单独的任何分量都大;这称为构造(相长?)干扰。


文件:Waventerference.gif
green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.

拇指|绿波向右移动,蓝波向左移动,每个点的红波净振幅是各个波振幅的总和。

green wave traverse to the right while blue wave traverse left, the net red wave amplitude at each point is the sum of the amplitudes of the individual waves.

绿波向右横移,蓝波向左横移,各点的净红波振幅为各波振幅之和。< ! -- 同上!-->

combined
waveform
combined
waveform
组合 < br > 波形 文件:Interference of two waves.svg File:Interference of two waves.svg 文件: 两个 waves.svg 的干涉
wave 1 wave 1

第一波

wave 2 wave 2

第二波



< br > Two waves in phase Two waves in phase

两个波同相位

Two waves 180° out
of phase
Two waves 180° out
of phase

两波相位差为180 °

Departures from linearity偏离线性

In most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles nonlinear optics and nonlinear acoustics.

在大多数实际的物理情况下,控制波浪的方程只是近似线性的。在这种情况下,叠加原理只能近似成立。通常,随着波的振幅变小,近似的精度趋于提高。有关叠加原理不完全成立时出现的现象的示例,请参见非线性光学非线性声学一文。

In most realistic physical situations, the equation governing the wave is only approximately linear. In these situations, the superposition principle only approximately holds. As a rule, the accuracy of the approximation tends to improve as the amplitude of the wave gets smaller. For examples of phenomena that arise when the superposition principle does not exactly hold, see the articles nonlinear optics and nonlinear acoustics.

在大多数实际的物理情况下,控制波的方程只是近似线性的。在这种情况下,叠加原理只能近似成立。作为一个规律,随着波幅的变小,近似的精度趋于提高。如果你想了解一些由于叠加原理不完全成立而引起的现象的例子,请参阅非线性光学和非线性声学的文章。

Quantum superposition态叠加原理

模板:主

In quantum mechanics, a principal task is to compute how a certain type of wave propagates and behaves. The wave is described by a wave function, and the equation governing its behavior is called the Schrödinger equation. A primary approach to computing the behavior of a wave function is to write it as a superposition (called "quantum superposition") of (possibly infinitely many) other wave functions of a certain type—stationary states whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.[8]

量子力学中,主要任务是计算某种类型的波传播及其行为。波由波函数描述,控制其行为的方程称为薛定谔方程。计算波函数行为的一种主要方法是把它写成某种类型的其他波函数的叠加(称为“量子叠加”),这些波函数的行为非常简单。由于薛定谔方程是线性的,因此可以用叠加原理计算原始波函数的行为。[8]

In quantum mechanics, a principal task is to compute how a certain type of wave propagates and behaves. The wave is described by a wave function, and the equation governing its behavior is called the Schrödinger equation. A primary approach to computing the behavior of a wave function is to write it as a superposition (called "quantum superposition") of (possibly infinitely many) other wave functions of a certain type—stationary states whose behavior is particularly simple. Since the Schrödinger equation is linear, the behavior of the original wave function can be computed through the superposition principle this way.

在量子力学中,一项主要任务是计算某种波的传播和行为。波用波函数来描述,控制其行为的方程称为薛定谔方程。计算波函数行为的一个主要方法是把它写成一个叠加(称为“量子叠加”),它是某一类定态的其他波函数的叠加(可能是无穷多个),这些定态的行为特别简单。由于薛定谔方程是线性的,因此可以用叠加原理计算原始波函数的行为。


模板:AnchorThe projective nature of quantum-mechanical-state space makes an important difference: it does not permit superposition of the kind that is the topic of the present article. A quantum mechanical state is a ray in projective Hilbert space, not a vector. The sum of two rays is undefined. To obtain the relative phase, we must decompose or split the ray into components

The projective nature of quantum-mechanical-state space makes an important difference: it does not permit superposition of the kind that is the topic of the present article. A quantum mechanical state is a ray in projective Hilbert space, not a vector. The sum of two rays is undefined. To obtain the relative phase, we must decompose or split the ray into components

量子力学状态空间的投影性质有一个重要的区别:它不允许像本文所讨论的那样的叠加。量子力学状态是射影Hilbert空间中的射线,而不是向量。两条光线之和未定义。为了获得相对相位,我们必须将光线分解成不同的分量

[math]\displaystyle{ |\psi_i\rangle = \sum_{j}{C_j}|\phi_j\rangle, }[/math]

[math]\displaystyle{ |\psi_i\rangle = \sum_{j}{C_j}|\phi_j\rangle, }[/math]

where the [math]\displaystyle{ C_j\in \textbf{C} }[/math] and the [math]\displaystyle{ |\phi_j\rangle }[/math] belongs to an orthonormal basis set. The equivalence class of [math]\displaystyle{ |\psi_i\rangle }[/math] allows a well-defined meaning to be given to the relative phases of the [math]\displaystyle{ C_j }[/math].[9]

其中[math]\displaystyle{ C_j\in \textbf{C} }[/math][math]\displaystyle{ |\phi_j\rangle }[/math]属于正交基集。[math]\displaystyle{ |\psi_i\rangle }[/math]的等价类允许对[math]\displaystyle{ C_j }[/math]的相应相位给出明确的含义。[10]

where the [math]\displaystyle{ C_j\in \textbf{C} }[/math] and the [math]\displaystyle{ |\phi_j\rangle }[/math] belongs to an orthonormal basis set. The equivalence class of [math]\displaystyle{ |\psi_i\rangle }[/math] allows a well-defined meaning to be given to the relative phases of the [math]\displaystyle{ C_j }[/math].

其中文本中的 < math > c j 和 < math > | phi rangle </math > 属于标准正交基集合。这个等价类可以给出一个明确的定义,用来表示相关的相位。


There are some likenesses between the superposition presented in the main on this page, and quantum superposition. Nevertheless, on the topic of quantum superposition, Kramers writes: "The principle of [quantum] superposition ... has no analogy in classical physics." According to Dirac: "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory [italics in original]."[11]

在这一页的主要部分中所呈现的叠加和量子叠加之间有一些相似之处。然而,关于量子叠加的话题, Kramers写道:“[量子]叠加原理。。。在经典物理学中没有类比。“根据 Dirac:”“量子力学中发生的叠加与经典理论中发生的任何叠加具有本质上不同的性质”“[斜体原文]。”[12]

There are some likenesses between the superposition presented in the main on this page, and quantum superposition. Nevertheless, on the topic of quantum superposition, Kramers writes: "The principle of [quantum] superposition ... has no analogy in classical physics." According to Dirac: "the superposition that occurs in quantum mechanics is of an essentially different nature from any occurring in the classical theory [italics in original]."

在这一页的重叠部分和态叠加原理之间有一些相似之处。然而,关于态叠加原理的话题,kramer 写道: “[量子]叠加原理... 在经典物理学中没有类比。”按照狄拉克的说法: “量子力学的叠加态本质上不同于古典理论中的叠加态。”

Boundary value problems边值问题

模板:Further

模板:进一步

A common type of boundary value problem is (to put it abstractly) finding a function y that satisfies some equation

A common type of boundary value problem is (to put it abstractly) finding a function y that satisfies some equation

一种常见的边值问题是(抽象地说)找到一个满足某个方程的函数 y

[math]\displaystyle{ F(y)=0 }[/math]

[math]\displaystyle{ F(y)=0 }[/math]


with some boundary specification

with some boundary specification

一些边界条件为

[math]\displaystyle{ G(y)=z }[/math]

[math]\displaystyle{ G(y)=z }[/math]


For example, in Laplace's equation with Dirichlet boundary conditions, F would be the Laplacian operator in a region R, G would be an operator that restricts y to the boundary of R, and z would be the function that y is required to equal on the boundary of R.

例如,在具有 Dirichlet边界条件的拉普拉斯Laplace方程中,“F”将是区域“R”中的Laplacian算子,“G”将是将“y”限制在“R”边界上的算子,“z”是“y”在“R”边界上必须相等的函数。

For example, in Laplace's equation with Dirichlet boundary conditions, F would be the Laplacian operator in a region R, G would be an operator that restricts y to the boundary of R, and z would be the function that y is required to equal on the boundary of R.

例如,在拉普拉斯的 Dirichlet 边界条件方程中,F 是区域 R 中的 拉普拉斯算子,G 是限制 y 到 R 边界的算子,z 是要求 y 在 R 边界上相等的函数。


In the case that F and G are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation:

In the case that F and G are both linear operators, then the superposition principle says that a superposition of solutions to the first equation is another solution to the first equation:

在 F 和 G 都是线性算子的情况下,叠加原理称第一个方程的解的叠加是第一个方程的另一个解:

[math]\displaystyle{ F(y_1)=F(y_2)=\cdots=0\ \Rightarrow\ F(y_1+y_2+\cdots)=0 }[/math]

[math]\displaystyle{ F(y_1)=F(y_2)=\cdots=0\ \Rightarrow\ F(y_1+y_2+\cdots)=0 }[/math]

< math > f (y _ 1) = f (y _ 2) = cdots = 0 right tarrow f (y _ 1 + y _ 2 + cdots) = 0 </math >

while the boundary values superpose:

while the boundary values superpose:

当边界值叠加时:

[math]\displaystyle{ G(y_1)+G(y_2) = G(y_1+y_2) }[/math]

[math]\displaystyle{ G(y_1)+G(y_2) = G(y_1+y_2) }[/math]


Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary value problems.

Using these facts, if a list can be compiled of solutions to the first equation, then these solutions can be carefully put into a superposition such that it will satisfy the second equation. This is one common method of approaching boundary value problems.

利用这些事实,如果可以列出第一个方程的解,那么这些解就可以仔细地叠加起来,使它满足第二个方程。这是处理边值问题的一种常用方法。

Additive state decomposition加性状态分解

模板:主

Consider a simple linear system :

Consider a simple linear system :

考虑一个简单的线性系统: < br >

[math]\displaystyle{ \dot{x}=Ax+B(u_{1}+u_{2}), x(0)=x_{0}. }[/math]

[math]\displaystyle{ \dot{x}=Ax+B(u_{1}+u_{2}), x(0)=x_{0}. }[/math]

By superposition principle, the system can be decomposed into

By superposition principle, the system can be decomposed into

利用叠加原理,将系统分解为

[math]\displaystyle{ \dot{x}_{1}=Ax_{1}+Bu_{1}, x_{1}(0)=x_{0}. }[/math]

[math]\displaystyle{ \dot{x}_{1}=Ax_{1}+Bu_{1}, x_{1}(0)=x_{0}. }[/math]


[math]\displaystyle{ \dot{x}_{2}=Ax_{2}+Bu_{2}, x_{2}(0)=0. }[/math]

[math]\displaystyle{ \dot{x}_{2}=Ax_{2}+Bu_{2}, x_{2}(0)=0. }[/math]


with

with

用 < br >

[math]\displaystyle{ x=x_{1}+x_{2}. }[/math]

[math]\displaystyle{ x=x_{1}+x_{2}. }[/math]


Superposition principle is only available for linear systems. However, the Additive state decomposition can be applied not only to linear systems but also nonlinear systems. Next, consider a nonlinear system

Superposition principle is only available for linear systems. However, the Additive state decomposition can be applied not only to linear systems but also nonlinear systems. Next, consider a nonlinear system

叠加原理只适用于线性系统。然而,可加状态分解不仅适用于线性系统,也适用于非线性系统。接下来,考虑一个非线性系统

[math]\displaystyle{ \dot{x}=Ax+B(u_{1}+u_{2})+ \phi (c^Tx), x(0)=x_{0}. }[/math]

[math]\displaystyle{ \dot{x}=Ax+B(u_{1}+u_{2})+ \phi (c^Tx), x(0)=x_{0}. }[/math]

[math]\displaystyle{ \dot{x}=Ax+B(u_{1}+u_{2})+ \phi (c^Tx), x(0)=x_{0}. }[/math]

where [math]\displaystyle{ \phi }[/math] is a nonlinear function. By the additive state decomposition, the system can be ‘additively’ decomposed into

where [math]\displaystyle{ \phi }[/math] is a nonlinear function. By the additive state decomposition, the system can be ‘additively’ decomposed into

其中 phi 是一个非线性函数。通过可加状态分解,系统可以“可叠加地”分解为“ < br > ”

[math]\displaystyle{ \dot{x}_{1}=Ax_{1}+Bu_{1}+ \phi (y_{d}), x_{1}(0)=x_{0}. }[/math]

[math]\displaystyle{ \dot{x}_{1}=Ax_{1}+Bu_{1}+ \phi (y_{d}), x_{1}(0)=x_{0}. }[/math]


[math]\displaystyle{ \dot{x}_{2}=Ax_{2}+Bu_{2}+ \phi (c^Tx_{1}+c^Tx_{2})- \phi (y_{d}), x_{2}(0)=0. }[/math]

[math]\displaystyle{ \dot{x}_{2}=Ax_{2}+Bu_{2}+ \phi (c^Tx_{1}+c^Tx_{2})- \phi (y_{d}), x_{2}(0)=0. }[/math]


with

with

用 < br >

[math]\displaystyle{ x=x_{1}+x_{2}. }[/math]

[math]\displaystyle{ x=x_{1}+x_{2}. }[/math]


This decomposition can help to simplify controller design.

This decomposition can help to simplify controller design.

这种分解有助于简化控制器的设计。

Other example applications其他示例应用程序

  • In electrical engineering, in a linear circuit, the input (an applied time-varying voltage signal) is related to the output (a current or voltage anywhere in the circuit) by a linear transformation. Thus, a superposition (i.e., sum) of input signals will yield the superposition of the responses. The use of Fourier analysis on this basis is particularly common. For another, related technique in circuit analysis, see Superposition theorem.
  • 电气工程中,在线性电路中,输入(施加的时变电压信号)通过线性变换与输出(电路中任何位置的电流或电压)相关。因此,输入信号的叠加(即总和)将产生响应的叠加。在此基础上使用Fourier analysis尤其常见。另一方面,电路分析中的相关技术,参见叠加定理
  • In physics, Maxwell's equations imply that the (possibly time-varying) distributions of charges and currents are related to the electric and magnetic fields by a linear transformation. Thus, the superposition principle can be used to simplify the computation of fields which arise from a given charge and current distribution. The principle also applies to other linear differential equations arising in physics, such as the heat equation.
  • 物理学中,麦克斯韦方程暗示电荷电流的分布(可能是时变的)通过线性变换与电场磁场有关。因此,叠加原理可用于简化由给定电荷和电流分布产生的场的计算。这一原理也适用于物理学中出现的其他线性微分方程,如热方程
  • In mechanical engineering, superposition is used to solve for beam and structure deflections of combined loads when the effects are linear (i.e., each load does not affect the results of the other loads, and the effect of each load does not significantly alter the geometry of the structural system).[13] Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.[14]
  • 机械工程中,叠加用于解决组合荷载作用下梁和结构的挠度,当效应为线性时(即每个荷载不影响其他荷载的结果,并且每个荷载的效应不会显著改变结构系统的几何结构)。[15] Mode superposition method uses the natural frequencies and mode shapes to characterize the dynamic response of a linear structure.[16]
  • The superposition principle can be applied when small deviations from a known solution to a nonlinear system are analyzed by linearization.
  • 当用线性化分析非线性系统已知解的微小偏差时,可以应用叠加原理。

History历史

According to Léon Brillouin, the principle of superposition was first stated by Daniel Bernoulli in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected by Leonhard Euler and then by Joseph Lagrange. Later it became accepted, largely through the work of Joseph Fourier.[17]

根据Léon Brillouin的说法,叠加原理最早是由丹尼尔·伯努利Daniel Bernoulli在1753年提出的:“振动系统的一般运动是由其固有振动的叠加给出的。”这一原理先后被Leonhard Euler Joseph Lagrange拒绝。后来主要是通过约瑟夫傅立叶的工作,它被人们接受。

According to Léon Brillouin, the principle of superposition was first stated by Daniel Bernoulli in 1753: "The general motion of a vibrating system is given by a superposition of its proper vibrations." The principle was rejected by Leonhard Euler and then by Joseph Lagrange. Later it became accepted, largely through the work of Joseph Fourier.

按照 l é on Brillouin 的说法,叠加原理最早是由丹尼尔·伯努利在1753年提出的: “一个振动系统的一般运动是由它本身的振动的叠加而得到的。”这个原理被欧拉和拉格朗日先后否定。后来,主要通过约瑟夫 · 傅立叶的工作,这种理论被人们所接受。

See also请参阅

References参考文献

  1. The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London
  2. The Penguin Dictionary of Physics, ed. Valerie Illingworth, 1991, Penguin Books, London
  3. Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [1]
  4. Lectures in Physics, Vol, 1, 1963, pg. 30-1, Addison Wesley Publishing Company Reading, Mass [2]
  5. N. K. VERMA, Physics for Engineers, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [3]
  6. N. K. VERMA, Physics for Engineers, PHI Learning Pvt. Ltd., Oct 18, 2013, p. 361. [4]
  7. Tim Freegarde, Introduction to the Physics of Waves, Cambridge University Press, Nov 8, 2012. [5]
  8. 8.0 8.1 Quantum Mechanics, Kramers, H.A. publisher Dover, 1957, p. 62
  9. Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics. 23 (2): 185–195. Bibcode:1993FoPh...23..185S. doi:10.1007/BF01883623.
  10. Solem, J. C.; Biedenharn, L. C. (1993). "Understanding geometrical phases in quantum mechanics: An elementary example". Foundations of Physics. 23 (2): 185–195. Bibcode:1993FoPh...23..185S. doi:10.1007/BF01883623.
  11. Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK, p. 14.
  12. Dirac, P.A.M. (1958). The Principles of Quantum Mechanics, 4th edition, Oxford University Press, Oxford UK, p. 14.
  13. Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192
  14. Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785
  15. Mechanical Engineering Design, By Joseph Edward Shigley, Charles R. Mischke, Richard Gordon Budynas, Published 2004 McGraw-Hill Professional, p. 192
  16. Finite Element Procedures, Bathe, K. J., Prentice-Hall, Englewood Cliffs, 1996, p. 785
  17. Brillouin, L. (1946). Wave propagation in Periodic Structures: Electric Filters and Crystal Lattices, McGraw–Hill, New York, p. 2.

Further reading延伸阅读

  • Haberman, Richard (2004). Applied Partial Differential Equations. Prentice Hall. ISBN 978-0-13-065243-0. 

External links外部链接

模板:Authority control 模板:权限控制 类别:概念物理

Category:Concepts in physics

分类: 物理概念

类别:波

Category:Systems theory

范畴: 系统论


This page was moved from wikipedia:en:Superposition principle. Its edit history can be viewed at 叠加原理/edithistory

此页摘自维基百科:英文:叠加原理。其编辑历史记录可在叠加原理/编辑历史记录查阅