# 量子力学到底是什么?

What exactly is quantum mechanics?译文简介

神秘的量子力学到底是什么呢？ 且看外国网友的通俗回答.

正文翻译

What exactly is quantum mechanics?

量子力学到底是什么?

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Devashish Singh, Ph.D. in Mathematical Physics

First things first -- let us begin with what the fundamental idea of a mechanics is. So, a mechanics is basically (i) a space of states for a given system with some (ii) dynamical laws or rules, for the evolution of the states, which are, in fact, the equations of motion. And the key idea is to be able to predict the state of the system at later times, by the evolution of the initial state of the system given at a particular instant of time.Classical mechanics applies to big particles -- a piece of stone, a steel ball, an wooden block, a dust particle, etc. The fundamental logic of classical mechanics is "set theory". The space of states is a set - the phase space, and the states are labeled by the phase space coordinates, so every state is a point in the phase space. The time-evolution of these states is governed by the Newton's equations (in vectorial formulation) or equivalently by the Euler-Lagrange / Hamilton's equations (in analytical formulation).

回答1:

首先，让我们从力学的基本概念开始。力学基本上是：一个给定系统的状态空间，带有一些动力学定律或规则，这些定律或规则是用来描述状态的演化的，实际上，就是运动方程了. 关键的思想是通过给定的某一时刻系统的初始状态的演化，能够预测系统演化之后的状态，

经典力学适用于大的粒子——一块石头，一个钢球，一个木块，一个尘埃粒子等等。经典力学的基本逻辑是“集合论”。状态空间是一个集合-相空间，状态由相空间坐标标记，所以每个状态都是相空间中的一个点。这些状态的时间演化由牛顿方程(矢量式)或欧拉-拉格朗日/汉密尔顿方程(解析式)决定。

First things first -- let us begin with what the fundamental idea of a mechanics is. So, a mechanics is basically (i) a space of states for a given system with some (ii) dynamical laws or rules, for the evolution of the states, which are, in fact, the equations of motion. And the key idea is to be able to predict the state of the system at later times, by the evolution of the initial state of the system given at a particular instant of time.Classical mechanics applies to big particles -- a piece of stone, a steel ball, an wooden block, a dust particle, etc. The fundamental logic of classical mechanics is "set theory". The space of states is a set - the phase space, and the states are labeled by the phase space coordinates, so every state is a point in the phase space. The time-evolution of these states is governed by the Newton's equations (in vectorial formulation) or equivalently by the Euler-Lagrange / Hamilton's equations (in analytical formulation).

回答1:

首先，让我们从力学的基本概念开始。力学基本上是：一个给定系统的状态空间，带有一些动力学定律或规则，这些定律或规则是用来描述状态的演化的，实际上，就是运动方程了. 关键的思想是通过给定的某一时刻系统的初始状态的演化，能够预测系统演化之后的状态，

经典力学适用于大的粒子——一块石头，一个钢球，一个木块，一个尘埃粒子等等。经典力学的基本逻辑是“集合论”。状态空间是一个集合-相空间，状态由相空间坐标标记，所以每个状态都是相空间中的一个点。这些状态的时间演化由牛顿方程(矢量式)或欧拉-拉格朗日/汉密尔顿方程(解析式)决定。

Quantum mechanics applies to small systems -- a molecule, an atom, an electron, etc. The fundamental logic of quantum mechanics is "linear algebra". The space of states is an abstract vector space (over a field of complex numbers) - the Hilbert space (a generalization of the Euclidean space), and the states are the abstract vectors living in the Hilbert space. The time-evolution of these state-vectors is governed by the Schrodinger's equation.The 'classical' mechanics differs from the 'quantum' mechanics because the space of states in the latter is not just an ordinary set like the former, but it is a vector space with two binary operations (satisfying certain axioms) defined on it: (i) vector addition and (ii) scalar multiplication, which allow the 'quantum' mechanics to have the possibility of quantum superposition (linear combination of states), which turns out to be both a surprising and a distinguishing key feature of it compared to the 'classical' mechanics.

量子力学适用于小系统——分子、原子、电子等。量子力学的基本逻辑是“线性代数”。状态空间是一个抽象的向量空间(在一个复数的域上)-希尔伯特空间(欧几里得空间的推广)，状态是生活在希尔伯特空间中的抽象向量。这些状态向量的时间演化由薛定谔方程决定。

“经典”力学不同于“量子”力学，因为后者的状态空间不只是一个普通的集合，不像前者，后者的状态空间是一个矢量空间，有两个二元操作(满足某些公理)定义:(i)向量加法和(ii)标量乘法，这允许“量子”力学有量子叠加(状态的线性组合)的可能性，这被证明是一个令人惊讶的和区别于“经典”力学的关键特征。

量子力学适用于小系统——分子、原子、电子等。量子力学的基本逻辑是“线性代数”。状态空间是一个抽象的向量空间(在一个复数的域上)-希尔伯特空间(欧几里得空间的推广)，状态是生活在希尔伯特空间中的抽象向量。这些状态向量的时间演化由薛定谔方程决定。

“经典”力学不同于“量子”力学，因为后者的状态空间不只是一个普通的集合，不像前者，后者的状态空间是一个矢量空间，有两个二元操作(满足某些公理)定义:(i)向量加法和(ii)标量乘法，这允许“量子”力学有量子叠加(状态的线性组合)的可能性，这被证明是一个令人惊讶的和区别于“经典”力学的关键特征。

The measurement itself affects the results of the experiment (determining whether the particle is particle-like or wave-like and determining what its precise energy, time, momentum, and/or position are).The trajectory of a particle is described by the Schrodinger equation or the Heisenberg equation (which are equivalent to one another), which describe the flow/evolution of the probability function.One uses the probability distribution to compute the most likely value that one will arrive at if one tries to measure a particular property and what the probability that the most likely value will be measured and what the probability is that some other value will be measured.Note that although quantum mechanics computes momentum, position, etc. based on a probability function, quantum mechanics is extremely precise.Those are the fundamental principles of quantum mechanics that I can think of at the moment, plus a little more added to make it a little more readable. It could be that I am forgetting something (if you ask the question in a different way or time I may give a different answer, since my wave function will have collapsed in a different way :) ).You may want to read Quantum mechanics - WikipediaI have not read it, but likely they have a lot of good information there, which may supplement anything important that I left out.

测量本身会影响实验结果(确定粒子是粒子还是类，并确定其精确的能量、时间、动量和/或位置)。

粒子的轨迹由薛定谔方程或海森堡方程描述(它们彼此等效)，薛定谔方程/海森堡方程式描述概率函数的流动/演化的.

探测者去测量一个粒子的动量、位置或其他的物理量时，我们使用概率分布来计算他可能得到的最有可能的值，概率分布公式会告诉探测者，最有可能的值被测量的概率以及其他值被测量的概率是多少.

请注意，尽管量子力学是基于概率函数计算动量、位置等的，但量子力学是极其精确的。

这些是我现在能想到的量子力学的基本原理，我用自己的语言把它说出来，让大家更容易读懂一些. 我可能也会忘记了一些量子力学知识了.(如果你用不同的方式问这个问题，或者我可能会给出不同的答案，因为我的波函数会以不同的方式坍缩:))。你们可以读一下维基百科上的量子力学，我还没有读过，但他们可能有很多有用的信息，可以补充我遗漏的重要信息。

测量本身会影响实验结果(确定粒子是粒子还是类，并确定其精确的能量、时间、动量和/或位置)。

粒子的轨迹由薛定谔方程或海森堡方程描述(它们彼此等效)，薛定谔方程/海森堡方程式描述概率函数的流动/演化的.

探测者去测量一个粒子的动量、位置或其他的物理量时，我们使用概率分布来计算他可能得到的最有可能的值，概率分布公式会告诉探测者，最有可能的值被测量的概率以及其他值被测量的概率是多少.

请注意，尽管量子力学是基于概率函数计算动量、位置等的，但量子力学是极其精确的。

这些是我现在能想到的量子力学的基本原理，我用自己的语言把它说出来，让大家更容易读懂一些. 我可能也会忘记了一些量子力学知识了.(如果你用不同的方式问这个问题，或者我可能会给出不同的答案，因为我的波函数会以不同的方式坍缩:))。你们可以读一下维基百科上的量子力学，我还没有读过，但他们可能有很多有用的信息，可以补充我遗漏的重要信息。

Akashneel Ray Chaudhuri, Physics Enthusiast

Well, it's certainly not a easy concept to catch on, so I'll use an example of how quantum mechanics developed.In earlier times, people used to believe that rest is a natural state of an obxt as any obxt, like a ball, practically comes to rest after a finite amount of time. This didn't cause any problem at that time because their scientific uses were mostly for practical situations. But then Galileo and Newton suggested that a obxt will continue in its state of rest or uniform motion unless a external force acts on it. This was important because it enables us to visualize ideal scenarios without friction and predict the effects of friction.Just like that, 17th century scientists also believed in things such as light being only a wave, the bohr model of the atom, matter being formed of only a particle, and most notably the ‘predictability’ of the universe.It was a common believe that with enough information, outcome of any event could be predicted with full certainty. Even in a so called random event like throwing a dice, the outcome can be predicted with enough information about the forces acting on the dice, the surface, etc., and any error would be due to insufficient information

回答3：

当然，这不是一个容易理解的概念，所以我用一个例子来说明量子力学是如何发展的。

在早期，人们常常认为静止是物体的一种自然状态，因为任何物体，比如一个球，在运动一定的时间之后都会静止下来，这在当时没有引起任何问题，因为当时的科学用途大多是“实用”的. 但后来伽利略和牛顿提出，除非有外力作用，否则物体将继续保持静止或匀速运动状态. 这一点很重要，因为它使我们能够想象没有摩擦的理想场景，并预测摩擦造成的影响.

像这样子，17世纪的科学家也相信一些东西，比如光只是一种波，玻尔原子模型，物质只是由一个个粒子组成等，最值得注意的是，当时他们都相信宇宙是“可预测性”的.

当时，人们普遍认为，只要有足够的信息，任何事件的结果都可以完全确定地预测出来。即使是掷骰子这样的随机事件，只要你提前掌握一切影响骰子运动的因素，那么，你就完全可以预测出扔出去的骰子会是几点. 当时，人们觉得，之所以不知道骰子扔出去之后是几点，原因是：信息不足，没完全掌握影响骰子的信息. 也就是说，当时人们相信：任何错误都是由于信息不足造成的。

Well, it's certainly not a easy concept to catch on, so I'll use an example of how quantum mechanics developed.In earlier times, people used to believe that rest is a natural state of an obxt as any obxt, like a ball, practically comes to rest after a finite amount of time. This didn't cause any problem at that time because their scientific uses were mostly for practical situations. But then Galileo and Newton suggested that a obxt will continue in its state of rest or uniform motion unless a external force acts on it. This was important because it enables us to visualize ideal scenarios without friction and predict the effects of friction.Just like that, 17th century scientists also believed in things such as light being only a wave, the bohr model of the atom, matter being formed of only a particle, and most notably the ‘predictability’ of the universe.It was a common believe that with enough information, outcome of any event could be predicted with full certainty. Even in a so called random event like throwing a dice, the outcome can be predicted with enough information about the forces acting on the dice, the surface, etc., and any error would be due to insufficient information

回答3：

当然，这不是一个容易理解的概念，所以我用一个例子来说明量子力学是如何发展的。

在早期，人们常常认为静止是物体的一种自然状态，因为任何物体，比如一个球，在运动一定的时间之后都会静止下来，这在当时没有引起任何问题，因为当时的科学用途大多是“实用”的. 但后来伽利略和牛顿提出，除非有外力作用，否则物体将继续保持静止或匀速运动状态. 这一点很重要，因为它使我们能够想象没有摩擦的理想场景，并预测摩擦造成的影响.

像这样子，17世纪的科学家也相信一些东西，比如光只是一种波，玻尔原子模型，物质只是由一个个粒子组成等，最值得注意的是，当时他们都相信宇宙是“可预测性”的.

当时，人们普遍认为，只要有足够的信息，任何事件的结果都可以完全确定地预测出来。即使是掷骰子这样的随机事件，只要你提前掌握一切影响骰子运动的因素，那么，你就完全可以预测出扔出去的骰子会是几点. 当时，人们觉得，之所以不知道骰子扔出去之后是几点，原因是：信息不足，没完全掌握影响骰子的信息. 也就是说，当时人们相信：任何错误都是由于信息不足造成的。

.But all this ended with the “Ultraviolet catastrophe”, when the prediction and visualization of the outcome was way different than reality. It basically was when scientists visualized a ideal black body and tried the predict its electromagnetic radiation. The results told them that a the intensity of the wave directly varied with the frequency of the wave emitted at a given temperature. So it showed that a body would release a high amount of ultraviolet heat. That basically meant that your toaster would toast you. But that doesn't happen right? In reality, a intensity vs frequency graph form a hill like curve, with the crest moving to a higher frequency with increase in temperature.So slowly and steadily new discoveries took place and theories were made that showed the wave-particle dual nature of both electromagnetic waves and matter. So basically that showed that all particles have a matter wave, which if composed of several different waves, each with a frequency and amplitude. These individual waves can be described by a complex number, which contains a real and a imaginary number.If the particle is unobserved, it is in all possible positions(determined by the amplitude of the wave) and has all the possible momentums(determined by frequency) of its matter wave. This is called its superposition. But upon observation, the matter wave collapses into one of its constituent waves; which one it collapses into is a game of probability, determined by the complex number associated with the wave.In the macro world, their are continously many observers, so a person doesn't randomly go into a superposition.Well, I explained to the best of my abilities, and I can't say I am a expert in the field, but if you don't mind please leave me a up vote?

但这一切（世间一切皆可预测的想法）在“紫外线灾难”爆发后，完全被摧毁了. 当时，科学家制作一个理想的黑体来实验时，得出的结果和预测值偏离得找不着北. 实验结果总结出来的公式显示，在给定的温度下，波的强度直接取决于波的频率。这表明，一个物体，它发出的紫外线，会释放大量的能量出去.（译注：世间任何物体都会发出紫外线的. 频率越高，光线越“紫”，若波的能量和频率挂钩的话，频率无穷大的“紫”外线，会射出无穷大的能量出来.）这基本上意味着你的烤面包机，会把站在旁边的你也烤熟. 但是，现实中，没有发生这种事吧，在现实中，强度与频率的关系图形成了一个类似小山的曲线，随着温度的升高，波峰会向更高的频率移动。

就这样，（由解决紫外灾难开始）缓慢而稳定地出现了新的发现，总结出了描述电磁波和物质的波粒双重性质的理论。基本上，量子力学表明，所有的粒子都有一个物质波，它由几个不同的波组成，每个波都有一个频率和振幅。这些单独的波可以用一个复数来描述，它包含一个实数和一个虚数。如果粒子没有被观测到，它就会处于所有可能的位置(由波的振幅决定)，并拥有物质波所有可能的动量(由频率决定)。

在宏观世界中，他们是连续的许多观察者，所以一个人不会随机进入叠加状态。

我已经尽我所能解释了，我不能说我是这方面的专家，但如果你不介意的话请给我点赞？

但这一切（世间一切皆可预测的想法）在“紫外线灾难”爆发后，完全被摧毁了. 当时，科学家制作一个理想的黑体来实验时，得出的结果和预测值偏离得找不着北. 实验结果总结出来的公式显示，在给定的温度下，波的强度直接取决于波的频率。这表明，一个物体，它发出的紫外线，会释放大量的能量出去.（译注：世间任何物体都会发出紫外线的. 频率越高，光线越“紫”，若波的能量和频率挂钩的话，频率无穷大的“紫”外线，会射出无穷大的能量出来.）这基本上意味着你的烤面包机，会把站在旁边的你也烤熟. 但是，现实中，没有发生这种事吧，在现实中，强度与频率的关系图形成了一个类似小山的曲线，随着温度的升高，波峰会向更高的频率移动。

就这样，（由解决紫外灾难开始）缓慢而稳定地出现了新的发现，总结出了描述电磁波和物质的波粒双重性质的理论。基本上，量子力学表明，所有的粒子都有一个物质波，它由几个不同的波组成，每个波都有一个频率和振幅。这些单独的波可以用一个复数来描述，它包含一个实数和一个虚数。如果粒子没有被观测到，它就会处于所有可能的位置(由波的振幅决定)，并拥有物质波所有可能的动量(由频率决定)。

在宏观世界中，他们是连续的许多观察者，所以一个人不会随机进入叠加状态。

我已经尽我所能解释了，我不能说我是这方面的专家，但如果你不介意的话请给我点赞？

Richard Muller, Prof Physics, UC Berkeley; author "Physics for Future Presidents"

Quantum mechanics, at its heart, is simply the recognition that there are no particles and no waves, only something that has properties of both. Sometimes this is called a wave function, but that term typically applies to the wave aspects - not to the particle ones. For this post, let me refer to them as wavicles (combination of wave and particle).When we see a classical wave, what we are seeing is a large number of wavicles acting together, in such a way that the "wave" aspect of the wavicles dominates our measurements. When we detect a wavicle with a position detector, the energy is absorbed abruptly, the wavicle might even disappear; we then get the impression that we are observing the "particle" nature. A large bunch of wavicles, all tied together by their mutual attraction, can be totally dominated by its particle aspect; that is, for example, what a baseball is.There is no paradox, unless you somehow think that particles and waves really do exist separately. Then you wonder about this "duality" - it is really a particle or a wave?But it is really neither.

回答4：

量子力学的核心就是认识到没有单纯的粒子和波，而是，所有的微观“东西”都是粒子和波的结合. 有时这被称为波函数，但这个术语通常适用于波的方面，而不是粒子的方面。在这篇文章中，让我把它称为波(波和粒子的结合)。 当我们看到一个经典波时，我们看到的是大量的波一起作用，在这种情况下，波的“波”方面支配着我们的测量。当我们用位置探测器探测一个波形时，能量突然被吸收，波形甚至可能消失;然后我们得到这样的印象:我们正在观察“粒子”的性质。大量的波，由于相互吸引而结合在一起，可以完全由它的粒子性质支配;

微观的“东西”，既是粒子，又是波，这并不矛盾，除非你以某种方式认为粒子和波确实是分开存在的。然后你会对这种“二象性”感到疑惑——它到底的是粒子还是波? 但实际上两者都不是。

Quantum mechanics, at its heart, is simply the recognition that there are no particles and no waves, only something that has properties of both. Sometimes this is called a wave function, but that term typically applies to the wave aspects - not to the particle ones. For this post, let me refer to them as wavicles (combination of wave and particle).When we see a classical wave, what we are seeing is a large number of wavicles acting together, in such a way that the "wave" aspect of the wavicles dominates our measurements. When we detect a wavicle with a position detector, the energy is absorbed abruptly, the wavicle might even disappear; we then get the impression that we are observing the "particle" nature. A large bunch of wavicles, all tied together by their mutual attraction, can be totally dominated by its particle aspect; that is, for example, what a baseball is.There is no paradox, unless you somehow think that particles and waves really do exist separately. Then you wonder about this "duality" - it is really a particle or a wave?But it is really neither.

回答4：

量子力学的核心就是认识到没有单纯的粒子和波，而是，所有的微观“东西”都是粒子和波的结合. 有时这被称为波函数，但这个术语通常适用于波的方面，而不是粒子的方面。在这篇文章中，让我把它称为波(波和粒子的结合)。 当我们看到一个经典波时，我们看到的是大量的波一起作用，在这种情况下，波的“波”方面支配着我们的测量。当我们用位置探测器探测一个波形时，能量突然被吸收，波形甚至可能消失;然后我们得到这样的印象:我们正在观察“粒子”的性质。大量的波，由于相互吸引而结合在一起，可以完全由它的粒子性质支配;

微观的“东西”，既是粒子，又是波，这并不矛盾，除非你以某种方式认为粒子和波确实是分开存在的。然后你会对这种“二象性”感到疑惑——它到底的是粒子还是波? 但实际上两者都不是。

Once you have this concept (and it was created by Einstein and deBroglie) then you can ask the question: how does this wavicle move? How is it affected by forces? Those were the questions answered by Schrodinger, Heisenberg, Dirac and others, with their quantum mechanical equations. “What about fields, such as the electric, magnetic, and strong force fields?” you would ask. Those were also recognized as being (what I, for lack of a better term in this answer) call wavicles. Yukawa explained the nuclear force as the exchange of a virtual pi meson. Schwinger and Feynman explained the electromagnetic force as the exchange of a virtual photon.Once you have accepted that the only things that really exist are wavicles, and that they change abruptly when measured - once you accept that the momentum is given by the wavelength and the energy by the frequency, then the Heisenberg uncertainty principle is a mathematical consequence.

一旦你有了这个概念(它是由爱因斯坦和德布罗意提出的)你就可以问这个问题:这个波柱是如何运动的?它是如何受力影响的?薛定谔、海森堡、狄拉克和其他人用他们的量子力学方程回答了这些问题。

“那么电场，比如电场、磁场和强磁场呢?”你会问。那些也被认为是(我，因为没有更好的术语来回答这个问题)所谓的波浪。汤川解释说，核力是一个虚拟介子的交换。施温格和费曼将电磁力解释为虚光子的交换。

一旦你接受了真正存在的东西只有波，而且它们在测量时突然发生变化——一旦你接受了动量是由波长给出的，能量是由频率给出的，那么海森堡测不准原理就是一个数学推导出来的结果.

一旦你有了这个概念(它是由爱因斯坦和德布罗意提出的)你就可以问这个问题:这个波柱是如何运动的?它是如何受力影响的?薛定谔、海森堡、狄拉克和其他人用他们的量子力学方程回答了这些问题。

“那么电场，比如电场、磁场和强磁场呢?”你会问。那些也被认为是(我，因为没有更好的术语来回答这个问题)所谓的波浪。汤川解释说，核力是一个虚拟介子的交换。施温格和费曼将电磁力解释为虚光子的交换。

一旦你接受了真正存在的东西只有波，而且它们在测量时突然发生变化——一旦你接受了动量是由波长给出的，能量是由频率给出的，那么海森堡测不准原理就是一个数学推导出来的结果.

Unmesh Guragol

Quantum mechanics (QM) is a set of scientific principles describing the known behavior of energy and matter that predominates at the atomic and subatomic scales. The name derives from the observation that some physical quantities — such as the energy of an electron bound into an atom or molecule — can be changed only by discrete amounts, or quanta, rather than being capable of varying by any amount. The wave–particle duality of energy and matter at the atomic scale provides a unified view of the behavior of particles such as photons and electrons. Photons are the quanta of light, and have energy values proportional to their frequency via the Planck constant. An electron bound in an atomic orbital has quantized values of angular momentum and energy. The unbound electron does not exhibit quantized energy levels, but is associated with a quantum mechanical wavelength, as are all massive particles. The full significance of the Planck constant is expressed in physics through the abstract mathematical notion of action.

回答5：

量子力学(QM)是一套科学原理，描述已知:行为的能量和物质，在原子和亚原子的微观层次上占主导地位。它的名字来自于观察到的一些物理量——比如，电子进入原子或分子之后的能量（交换）——只能通过离散量或量子来改变，而不能随任何量变化。原子尺度上能量和物质的波粒二象性提供了光子和电子等粒子行为的统一观点。

光子是光的量子，光子的能量与频率成正比，是频率和普朗克常数的乘积. 被束缚在原子轨道上的电子具有量子化的角动量和能量，未束缚的电子就没有表现出量子化的能级，（它的能量是连续的）就像所有的大质量粒子一样。普朗克常数的意义在物理学中通过抽象数学的概念来表达。

Quantum mechanics (QM) is a set of scientific principles describing the known behavior of energy and matter that predominates at the atomic and subatomic scales. The name derives from the observation that some physical quantities — such as the energy of an electron bound into an atom or molecule — can be changed only by discrete amounts, or quanta, rather than being capable of varying by any amount. The wave–particle duality of energy and matter at the atomic scale provides a unified view of the behavior of particles such as photons and electrons. Photons are the quanta of light, and have energy values proportional to their frequency via the Planck constant. An electron bound in an atomic orbital has quantized values of angular momentum and energy. The unbound electron does not exhibit quantized energy levels, but is associated with a quantum mechanical wavelength, as are all massive particles. The full significance of the Planck constant is expressed in physics through the abstract mathematical notion of action.

回答5：

量子力学(QM)是一套科学原理，描述已知:行为的能量和物质，在原子和亚原子的微观层次上占主导地位。它的名字来自于观察到的一些物理量——比如，电子进入原子或分子之后的能量（交换）——只能通过离散量或量子来改变，而不能随任何量变化。原子尺度上能量和物质的波粒二象性提供了光子和电子等粒子行为的统一观点。

光子是光的量子，光子的能量与频率成正比，是频率和普朗克常数的乘积. 被束缚在原子轨道上的电子具有量子化的角动量和能量，未束缚的电子就没有表现出量子化的能级，（它的能量是连续的）就像所有的大质量粒子一样。普朗克常数的意义在物理学中通过抽象数学的概念来表达。

The mathematical formulation of quantum mechanics is abstract and its implications are often non-intuitive. The centerpiece of this mathematical system is the wave function. The wave function is a mathematical function of time and space that can provide information about the position and momentum of a particle, but only as probabilities, as dictated by the constraints imposed by the uncertainty principle. Mathematical manipulations of the wave function usually involve the bra-kt notation, which requires an understanding of complex numbers and linear functional s. Many of the results of QM can only be expressed mathematically and do not have models that are as easy to visualize as those of classical mechanics. For instance, the ground state in quantum mechanical model is a non-zero energy state that is the lowest permitted energy state of a system, rather than a more traditional system that is thought of as simple being at rest with zero kinetic energy.

量子力学的数学公式很抽象，其含义往往是非直观的。这个数学系统的核心是波函数。波函数是时间和空间的数学函数，它可以提供关于一个粒子的位置和动量的信息，但它提供的是粒子的各种物理量的分布概率信息. 各种物理量都受制于不确定原理. 波函数的数学运算通常涉及到狄拉克符号.

要理解量子力学的数学，需要掌握复数和线性泛函，量子力学的许多结果只能用数学表达，而没有像经典力学那样容易可视化的模型。例如，在量子力学模型中，基态是一个非零的能量状态，它是系统允许的最低能量状态，而不是传统的力学那样基态是静止的、动能为零的状态.

量子力学的数学公式很抽象，其含义往往是非直观的。这个数学系统的核心是波函数。波函数是时间和空间的数学函数，它可以提供关于一个粒子的位置和动量的信息，但它提供的是粒子的各种物理量的分布概率信息. 各种物理量都受制于不确定原理. 波函数的数学运算通常涉及到狄拉克符号.

要理解量子力学的数学，需要掌握复数和线性泛函，量子力学的许多结果只能用数学表达，而没有像经典力学那样容易可视化的模型。例如，在量子力学模型中，基态是一个非零的能量状态，它是系统允许的最低能量状态，而不是传统的力学那样基态是静止的、动能为零的状态.

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