薛定谔方程

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{{简述}线性偏微分方程,其解描述了量子力学系统。}}

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文件:Grave Schroedinger (detail).png
Schrödinger's equation inscribed on the gravestone of Annemarie and Erwin Schrödinger. (Newton's dot notation for the time derivative is used.)

[[档案:格雷夫·施罗德(细节).png | alt=| thumb | Schrödinger方程,刻在安妮玛丽和埃尔文·薛定谔的墓碑上。(牛顿点符号表示时间导数。)]]

Newton's dot notation for the time derivative is used.)

使用了牛顿的点符号来表示时间导数

The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system.[1]:1–2 It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933.[2][3]


In classical mechanics, Newton's second law (F = ma)[note 1] is used to make a mathematical prediction as to what path a given physical system will take over time following a set of known initial conditions. Solving this equation gives the position and the momentum of the physical system as a function of the external force [math]\displaystyle{ \mathbf{F} }[/math] on the system. Those two parameters are sufficient to describe its state at each time instant. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation.

经典力学中,牛顿第二定律(F = ma)[note 2] 用于数学预测给定物理系统在一组已知初始条件下随时间的变化路径。解这个方程就得到了物理系统的位置和动量,它是系统上外力的函数。这两个参数足以描述它在每个时刻的状态。在量子力学中,牛顿定律的对应物是薛定谔方程。

The concept of a wave function is a fundamental postulate of quantum mechanics; the wave function defines the state of the system at each spatial position and time. Using these postulates, Schrödinger's equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by the exponential of a self-adjoint operator, which is the quantum Hamiltonian. This derivation is explained below. 波函数的概念是一个基本的量子力学的假设;波函数定义了系统在每个空间位置和时间的状态。利用这些假设,薛定谔方程可以从时间演化算符必须是幺正的事实导出,因此必须由自伴算符的指数生成,即量子哈密顿量。这一推导解释如下。

The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation (TDSE), which gives a description of a system evolving with time:

薛定谔方程的形式取决于实际情况(特殊情况见下文)。最普遍的形式是时变薛定谔方程(TDSE) ,它描述了一个随时间演化的系统:


A [[wave function that satisfies the nonrelativistic Schrödinger equation with 0}}. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here.]]

[[满足非相对论薛定谔方程的[波函数}}。换句话说,这相当于一个粒子在真空中自由运动。波函数的实际部分在这里画出来了]]

In the Copenhagen interpretation of quantum mechanics, the wave function is the most complete description that can be given of a physical system. Solutions to Schrödinger's equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems, possibly even the whole universe.[4]:292ff

在量子力学的哥本哈根解释中,波函数是对一个物理系统最完整的描述。薛定谔方程的解不仅描述了分子原子ic和亚原子粒子系统,而且还描述了宏观系统,甚至可能描述了整个宇宙

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The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly by Richard Feynman. Paul Dirac incorporated matrix mechanics and the Schrödinger equation into a single formulation.

薛定谔方程并不是研究量子力学系统和进行预测的唯一方法。量子力学的其他公式包括Werner Heisenberg提出的matrix mechanics,以及主要由Richard Feynman提出的path integral formulationPaul Dirac将矩阵力学和薛定谔方程合并到一个公式中。

|title=Time-dependent Schrödinger equation (general)

| title = 依赖时间的薛定谔方程(一般)


|equation=[math]\displaystyle{ i \hbar \frac{d}{d t}\vert\Psi(t)\rangle = \hat H\vert\Psi(t)\rangle }[/math]

| equation = < math qid = q165498 > i hbar frac { d }{ d t } vert Psi (t) rangle = hat h vert Psi (t) rangle </math >

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Time-dependent equation时变方程

The form of the Schrödinger equation depends on the physical situation (see below for special cases). The most general form is the time-dependent Schrödinger equation (TDSE), which gives a description of a system evolving with time:

薛定谔方程的形式取决于物理情况(特殊情况见下文)。最普遍的形式是含时薛定谔方程(TDSE),它描述了系统随时间演化的过程:[5]:143

|equation=[math]\displaystyle{ i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t) }[/math]

文件:Wave packet (dispersion).gif
A wave function that satisfies the nonrelativistic Schrödinger equation with V = 0. In other words, this corresponds to a particle traveling freely through empty space. The real part of the wave function is plotted here.

[[文件:Wave packet(色散).gif | thumb | 200px | A波函数,满足具有V'=}0}的非相对论薛定谔方程。换句话说,这相当于一个粒子在空旷的空间中自由运动。这里绘制了波函数实部


The term "Schrödinger equation" can refer to both the general equation, or the specific nonrelativistic version. The general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in diverse expressions for the Hamiltonian. The specific nonrelativistic version is a strictly classical approximation to reality and yields accurate results in many situations, but only to a certain extent (see relativistic quantum mechanics and relativistic quantum field theory).

术语“薛定谔方程”既可以指一般方程,也可以指特定的非相对论方程。通过插入哈密顿量的不同表达式,一般的方程确实是相当普遍的,贯穿了整个量子力学,适用于从狄拉克方程到量子场论的所有事物。特定的非相对论版本是一个严格的经典近似,但只是在一定程度上,在许多情况下产生精确的结果(见相对论量子力学和相对论量子场理论)。

where [math]\displaystyle{ i }[/math] is the imaginary unit, [math]\displaystyle{ \hbar = \frac{h}{2 \pi} }[/math] is the reduced Planck constant having the dimension of action, 其中,[math]\displaystyle{ i }[/math]虚单位[math]\displaystyle{ \hbar=\frac{h}{2\pi} }[/math]是具有作用维数的约化普朗克常数引用错误:没有找到与</ref>对应的<ref>标签[6][note 3] [math]\displaystyle{ \Psi }[/math] (the Greek letter psi) is the state vector of the quantum system, [math]\displaystyle{ t }[/math] is time, and [math]\displaystyle{ \hat{H} }[/math] is the Hamiltonian operator. The position-space wave function of the quantum system is nothing but the components in the expansion of the state vector in terms of the position eigenvector [math]\displaystyle{ \vert\mathbf{r}\rangle }[/math]. It is a scalar function, expressed as [math]\displaystyle{ \Psi(\mathbf{r},t) = \langle \mathbf{r}\vert \Psi \rangle }[/math]. Similarly, the momentum-space wave function can be defined as [math]\displaystyle{ \tilde\Psi(\mathbf{p},t) = \langle \mathbf{p}\vert \Psi \rangle }[/math], where [math]\displaystyle{ \vert\mathbf{p}\rangle }[/math] is the momentum eigenvector.

行动的维度是能量乘以时间,而不是每次的能量,这是力量的维度。国际单位制的作用单位是焦耳秒,而国际单位制的功率单位是焦耳每秒(瓦特)。</ref>[math]\displaystyle{ \Psi }[/math](希腊字母 Psi)是量子系统的状态向量,[math]\displaystyle{ t }[/math]是时间,[math]\displaystyle{ \hat{H} }[/math] Hamiltonian operator。量子系统的“位置空间波函数”只不过是状态向量在位置特征向量方面展开的分量。它是一个标量函数,表示为[math]\displaystyle{ \Psi(\mathbf{r},t)=\langle\mathbf{r}\vert\Psi\rangle }[/math]。类似地,'动量空间波函数可以定义为[math]\displaystyle{ \tilde\Psi(\mathbf{p},t)=\langle\mathbf{p}\vert\Psi\rangle }[/math],其中[math]\displaystyle{ \vert\mathbf{p}\rangle }[/math]是动量特征向量。

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Each of these three rows is a wave function which satisfies the time-dependent Schrödinger equation for a harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability distribution of finding the particle with this wave function at a given position. The top two rows are examples of stationary states, which correspond to standing waves. The bottom row is an example of a state which is not a stationary state. The right column illustrates why stationary states are called "stationary".

[[文件:stationasanimation.gif|300px |拇指|右|这三行中的每一行都是满足谐振子的含时薛定谔方程的波函数。左:波函数的实部(蓝色)和虚部(红色)。右:在给定位置用这个波函数找到粒子的概率分布。上面两行是“'稳态s”的示例,对应于驻波s。下面一行是“不是”稳态的状态示例。右栏说明了为什么静止状态被称为“静止”。]]

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The most famous example is the nonrelativistic Schrödinger equation for the wave function in position space [math]\displaystyle{ \Psi(\mathbf{r},t) }[/math] of a single particle subject to a potential [math]\displaystyle{ V(\mathbf{r},t) }[/math], such as that due to an electric field.[7][note 4]

最著名的例子是非相对论薛定谔方程,它描述了单个粒子在位置空间中的波函数[math]\displaystyle{ \Psi(\mathbf{r},t) }[/math]的势[math]\displaystyle{ V(\mathbf{r},t) }[/math],例如由于电场[8][note 5]

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where [math]\displaystyle{ E }[/math] is a constant equal to the energy level of the system. This is only used when the Hamiltonian itself is not dependent on time explicitly. However, even in this case the total wave function still has a time dependency.

其中,[math]\displaystyle{ E }[/math]是一个与系统能量等值的常数。这只有在哈密顿量本身不明确依赖于时间的情况下才使用。然而,即使在这种情况下,总波函数仍然具有时间依赖性。

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|title=Time-dependent Schrödinger equation in position basis
(single nonrelativistic particle)

In the language of linear algebra, this equation is an eigenvalue equation. Therefore, the wave function is an eigenfunction of the Hamiltonian operator with corresponding eigenvalue(s) [math]\displaystyle{ E }[/math].

用线性代数的语言来说,这个方程是一个本征方程。因此,波函数是哈密顿算符的本征函数,其本征值与哈密顿算符的本征值相对应。

|equation=[math]\displaystyle{ i\hbar\frac{\partial}{\partial t} \Psi(\mathbf{r},t) = \left [ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r},t)\right ] \Psi(\mathbf{r},t) }[/math]

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As before, the most common manifestation is the nonrelativistic Schrödinger equation for a single particle moving in an electric field (but not a magnetic field):

和以前一样,最常见的表现形式是非相对论性的薛定谔方程在电场(但不是磁场)中运动的粒子:

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|title=Time-independent Schrödinger equation (single nonrelativistic particle)

| title = 与时间无关的薛定谔方程(单个非相对论粒子)

where [math]\displaystyle{ m }[/math] is the particle's mass, and [math]\displaystyle{ \nabla^2 }[/math] is the Laplacian.

其中,[math]\displaystyle{ m }[/math]是粒子的质量,[math]\displaystyle{ \nabla^2 }[/math]Laplacian

|equation=[math]\displaystyle{ \left[ \frac{-\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \right] \Psi(\mathbf{r}) = E \Psi(\mathbf{r}) }[/math]

| equation = < math > left [ frac {-hbar ^ 2}{2m } nabla ^ 2 + v (mathbf { r }) right ] Psi (mathbf { r }) = e Psi (mathbf { r }) </math >


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This is a diffusion equation, with an imaginary constant present in the transient term.

这是一个扩散方程,瞬变项中存在一个虚常数。

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The term "Schrödinger equation" can refer to both the general equation, or the specific nonrelativistic version. The general equation is indeed quite general, used throughout quantum mechanics, for everything from the Dirac equation to quantum field theory, by plugging in diverse expressions for the Hamiltonian. The specific nonrelativistic version is a strictly classical approximation to reality and yields accurate results in many situations, but only to a certain extent (see relativistic quantum mechanics and relativistic quantum field theory).

术语“薛定谔方程”既可以指一般方程,也可以指特定的非相对论形式。一般方程确实是非常普遍的,在整个量子力学中,从狄拉克方程量子场论的所有东西,都是通过插入不同的哈密顿量表达式。具体的非相对论版本是对现实的严格经典近似,但仅在一定程度上,在许多情况下产生准确的结果,(见相对论量子力学相对论量子场论)。

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To apply the Schrödinger equation, write down the Hamiltonian for the system, accounting for the kinetic and potential energies of the particles constituting the system, then insert it into the Schrödinger equation. The resulting partial differential equation is solved for the wave function, which contains information about the system.

为了应用薛定谔方程,写下系统的哈密顿量,计算组成系统的粒子的动能和势能,然后把它插入薛定谔方程。由此产生的偏微分方程可求解包含有关系统信息的波函数。

with definitions as above. Here, the form of the Hamiltonian operator comes from classical mechanics, where the Hamiltonian function is the sum of the kinetic and potential energies. That is, [math]\displaystyle{ H = T + V = \frac{\|\mathbf{p}\|^2}{2m} + V(x, y, z) }[/math] for a single particle in the non-relativistic limit.

定义同上。这里,哈密顿算符的形式来自经典力学,其中哈密顿函数是动能和势能之和。也就是说,对于非相对论极限下的单个粒子,[math]\displaystyle{ H=T+V=\frac{p}\^2}{2m}+V(x,y,z) }[/math]


模板:AnchorTime-independent equation{{锚定|时间无关方程}时间无关方程

The time-independent Schrödinger equation is discussed further below.

与时间无关的薛定谔方程将在下面进一步讨论。

The time-dependent Schrödinger equation described above predicts that wave functions can form standing waves, called stationary states.[note 6] These states are particularly important as their individual study later simplifies the task of solving the time-dependent Schrödinger equation for any state. Stationary states can also be described by a simpler form of the Schrödinger equation, the time-independent Schrödinger equation (TISE).

上述含时薛定谔方程预测波函数可以形成驻波s,在化学中称为稳态s.引用错误:没有找到与</ref>对应的<ref>标签

Related to diffraction, particles also display superposition and interference.

与衍射相关的粒子也表现出叠加和干涉现象。

Implications启示

The superposition property allows the particle to be in a quantum superposition of two or more quantum states at the same time. However, a "quantum state" in quantum mechanics means the probability that a system will be, for example at a position , not that the system will actually be at position . It does not imply that the particle itself may be in two classical states at once. Indeed, quantum mechanics is generally unable to assign values for properties prior to measurement at all.

叠加特性允许粒子同时处于两个或更多量子态的态叠加原理。然而,量子力学中的“量子态”意味着系统处于某个位置的概率,而不是系统实际处于某个位置的概率。这并不意味着粒子本身可能同时处于两个经典状态。事实上,量子力学通常根本无法在测量之前为物业指定价值。

Energy能量

The Hamiltonian is constructed in the same manner as in classical mechanics. However, in classical mechanics, the Hamiltonian is a scalar-valued function, whereas in quantum mechanics, it is an operator on a space of functions. It is not surprising that the eigenvalues of [math]\displaystyle{ \hat{H} }[/math] are the energy levels of the system.

哈密顿量的构造方式与经典力学中的相同。然而,在经典力学中,哈密顿量是一个标量值函数,而在量子力学中,哈密顿量是一个函数空间上的算符。毫不奇怪,[math]\displaystyle{ \hat{H} }[/math]特征值是系统的能级。

Quantization量子化

In classical mechanics, a particle has, at every moment, an exact position and an exact momentum. These values change deterministically as the particle moves according to Newton's laws. Under the Copenhagen interpretation of quantum mechanics, particles do not have exactly determined properties, and when they are measured, the result is randomly drawn from a probability distribution. The Schrödinger equation predicts what the probability distributions are, but fundamentally cannot predict the exact result of each measurement.

在经典力学中,一个粒子每时每刻都有一个精确的位置和一个精确的动量。当粒子按照牛顿定律运动时,这些值会发生决定性的变化。根据哥本哈根诠释量子力学,粒子并没有确切的性质,当他们被测量,结果是随机抽取的概率分布。美国薛定谔方程预测概率分布是什么,但从根本上不能预测每个测量的精确结果。

The Schrödinger equation predicts that if certain properties of a system are measured, the result may be quantized, meaning that only specific discrete values can occur. One example is energy quantization: the energy of an electron in an atom is always one of the quantized energy levels, a fact discovered via atomic spectroscopy. (Energy quantization is discussed below.) Another example is quantization of angular momentum. This was an assumption in the earlier Bohr model of the atom, but it is a prediction of the Schrödinger equation.

薛定谔方程预测,如果测量系统的某些特性,结果可能是“量子化”的,这意味着只能出现特定的离散值。一个例子是“能量量子化”:原子中电子的能量总是量子化能级之一,这是通过原子光谱发现的事实。(能量量子化在下面讨论[#与时间无关的|下面]])另一个例子是角动量量子化。这是早期原子玻尔模型中的“假设”,但它是薛定谔方程的“预测”。

The Heisenberg uncertainty principle is one statement of the inherent measurement uncertainty in quantum mechanics. It states that the more precisely a particle's position is known, the less precisely its momentum is known, and vice versa.

海森堡不确定度原理是量子力学固有测量不确定度的一种表述。它指出粒子的位置知道得越精确,其动量知道得越少,反之亦然。

Another result of the Schrödinger equation is that not every measurement gives a quantized result in quantum mechanics. For example, position, momentum, time, and (in some situations) energy can have any value across a continuous range.[9]:165–167

薛定谔方程的另一个结果是,在量子力学中,并不是每个测量都给出一个量子化的结果。例如,位置、动量、时间和(在某些情况下)能量可以在连续范围内具有任何值。[9]:165–167

The Schrödinger equation describes the (deterministic) evolution of the wave function of a particle. However, even if the wave function is known exactly, the result of a specific measurement on the wave function is uncertain.

薛定谔方程描述了粒子波函数的确定性演化。然而,即使波函数已经精确地知道,对波函数的具体测量结果仍然是不确定的。

Quantum tunneling量子隧道

文件:TunnelEffektKling1.png
Quantum tunneling through a barrier. A particle coming from the left does not have enough energy to climb the barrier. However, it can sometimes "tunnel" to the other side.

[[文件:TunnelEffektKling1.png|宽度=300px |拇指|量子隧穿穿过势垒。从左边来的粒子没有足够的能量爬过障碍物。然而,它有时可以“隧穿”到另一边。]]

In classical physics, when a ball is rolled slowly up a large hill, it will come to a stop and roll back, because it doesn't have enough energy to get over the top of the hill to the other side. However, the Schrödinger equation predicts that there is a small probability that the ball will get to the other side of the hill, even if it has too little energy to reach the top. This is called quantum tunneling. It is related to the distribution of energy: although the ball's assumed position seems to be on one side of the hill, there is a chance of finding it on the other side.


在经典物理学中,当一个球慢慢地滚上一座大山时,它会停下来并向后滚,因为它没有足够的能量从山顶滚到另一边。然而,薛定谔方程预测球仍有很小的可能性到达山的另一边,即使它的能量太小而无法到达山顶。这被称为量子隧道。它与能量的分配有关:虽然球的假定位置似乎在山的一侧,但有可能在另一侧找到它。

The Schrödinger equation provides a way to calculate the wave function of a system and how it changes dynamically in time. However, the Schrödinger equation does not directly say what, exactly, the wave function is. Interpretations of quantum mechanics address questions such as what the relation is between the wave function, the underlying reality, and the results of experimental measurements.

薛定谔方程提供了一种方法来计算系统的波函数,以及它是如何随时间动态变化的。然而,薛定谔方程并没有直接说明波函数到底是什么。量子力学诠释解决了诸如波函数、基本现实和实验测量结果之间的关系这样的问题。

Particles as waves作为波的粒子

An important aspect is the relationship between the Schrödinger equation and wave function collapse. In the oldest Copenhagen interpretation, particles follow the Schrödinger equation except during wave function collapse, during which they behave entirely differently. The advent of quantum decoherence theory allowed alternative approaches (such as the Everett many-worlds interpretation and consistent histories), wherein the Schrödinger equation is always satisfied, and wave function collapse should be explained as a consequence of the Schrödinger equation.

一个重要的方面是薛定谔方程和波函数崩塌之间的关系。在最古老的哥本哈根诠释中,粒子遵循薛定谔方程定律,除了在波函数崩塌期间,粒子的行为完全不同。量子退相干理论的出现允许其他方法(如埃弗里特多世界诠释和一致性历史) ,其中薛定谔方程总是满足的,波函数崩溃应该被解释为薛定谔方程的结果。

{{主物质波}波粒二象性}双缝实验}}

In 1952, Erwin Schrödinger gave a lecture during which he commented,

1952年,埃尔温·薛定谔在一次演讲中评论道:

文件:Double-slit experiment results Tanamura 2.jpg
A double slit experiment showing the accumulation of electrons on a screen as time passes.

175px | thumb |一个双缝实验显示了电子在屏幕上随着时间的推移而累积。

Nearly every result [a quantum theorist] pronounces is about the probability of this or that or that ... happening—with usually a great many alternatives. The idea that they be not alternatives but all really happen simultaneously seems lunatic to him, just impossible.

几乎每一个(一个量子理论家)宣称的结果都有这个、那个或那个的概率... ... 发生ーー通常有很多可能性。认为它们不是交替发生,而是同时发生的想法在他看来简直是疯了,根本不可能。


The nonrelativistic Schrödinger equation is a type of partial differential equation called a wave equation. Therefore, it is often said particles can exhibit behavior usually attributed to waves. In some modern interpretations this description is reversed – the quantum state, i.e. wave, is the only genuine physical reality, and under the appropriate conditions it can show features of particle-like behavior. However, Ballentine[10]:Chapter 4, p.99 shows that such an interpretation has problems. Ballentine points out that whilst it is arguable to associate a physical wave with a single particle, there is still only one Schrödinger wave equation for many particles. He points out:

非相对论薛定谔方程是一种称为波动方程的偏微分方程。因此,人们常说粒子可以表现出通常归因于波的行为。在一些现代的解释中,这种描述是相反的——量子态,即波,是唯一真实的物理实在,在适当的条件下,它可以表现出类似粒子的行为特征。然而,巴伦丁[10]:Chapter 4, p.99 说明这样的解释有问题。巴伦丁指出,虽然将物理波与单个粒子联系起来是有争议的,但对于许多粒子来说,仍然只有“一个”薛定谔波动方程。他指出:

David Deutsch regarded this as the earliest known reference to a many-worlds interpretation of quantum mechanics, an interpretation generally credited to Hugh Everett III, while Jeffrey A. Barrett took the more modest position that it indicates a "similarity in ... general views" between Schrödinger and Everett.

大卫 · 多伊奇认为这是已知的最早提及多世界诠释量子力学的文献,这一解释通常归功于休·艾弗雷特三世,而杰弗里 · a · 巴雷特则持更为温和的立场,认为这表明薛定谔和埃弗雷特之间的“一般观点相似”。

"If a physical wave field were associated with a particle, or if a particle were identified with a wave packet, then corresponding to N interacting particles there should be N interacting waves in ordinary three-dimensional space. But according to (4.6) that is not the case; instead there is one "wave" function in an abstract 3N-dimensional configuration space. The misinterpretation of psi as a physical wave in ordinary space is possible only because the most common applications of quantum mechanics are to one-particle states, for which configuration space and ordinary space are isomorphic."

如果一个物理波场与一个粒子相关联,或者一个粒子被一个波包识别,那么对应于N个相互作用的粒子,在普通的三维空间中应该有N个相互作用的波。但根据(4.6),情况并非如此;相反,在抽象的3N维构型空间中有一个“波”函数。将psi误解为普通空间中的物理波是可能的,因为量子力学最常见的应用是单粒子态,组态空间和普通空间是同构的


Two-slit diffraction is a famous example of the strange behaviors that waves regularly display, that are not intuitively associated with particles. The overlapping waves from the two slits cancel each other out in some locations, and reinforce each other in other locations, causing a complex pattern to emerge. Intuitively, one would not expect this pattern from firing a single particle at the slits, because the particle should pass through one slit or the other, not a complex overlap of both.

双缝衍射是一个著名的例子,说明了波有规律地表现出奇怪的行为,而这些行为并不是直观地与粒子联系在一起的。来自两个狭缝的重叠波在某些位置相互抵消,在其他位置相互加强,从而形成一个复杂的模式。直观地说,人们不会期望这种模式在狭缝处发射单个粒子,因为粒子应该通过一个或另一个狭缝,而不是两者的复杂重叠。

Erwin Schrödinger

Erwin Schrödinger

埃尔文·薛定谔


However, since the Schrödinger equation is a wave equation, a single particle fired through a double-slit does show this same pattern (figure on right). The experiment must be repeated many times for the complex pattern to emerge. Although this is counterintuitive, the prediction is correct; in particular, electron diffraction and neutron diffraction are well understood and widely used in science and engineering.

然而,由于薛定谔方程是一个波动方程,通过双缝发射的单个粒子“确实”显示出相同的模式(右图)。这个实验必须重复多次才能出现复杂的模式。虽然这是违反直觉的,但预测是正确的;特别是,电子衍射中子衍射在科学和工程中得到了很好的理解和广泛的应用。

Following Max Planck's quantization of light (see black-body radiation), Albert Einstein interpreted Planck's quanta to be photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, one of the first signs of wave–particle duality. Since energy and momentum are related in the same way as frequency and wave number in special relativity, it followed that the momentum [math]\displaystyle{ p }[/math] of a photon is inversely proportional to its wavelength [math]\displaystyle{ \lambda }[/math], or proportional to its wave number [math]\displaystyle{ k }[/math]:

继马克斯·普朗克对光的量子化(见黑体辐射)之后,阿尔伯特·爱因斯坦将普朗克量子解释为光子,即光的粒子,并提出光子的能量与其频率成正比,这是波粒二象性的最初迹象之一。由于能量和动量与狭义相对论中的频率和波数的关系相同,因此光子的动量p与其波长λ成反比,或与其波数k成反比:

Related to diffraction, particles also display superposition and interference.

衍射有关,粒子也显示叠加干涉

[math]\displaystyle{ p = \frac{h}{\lambda} = \hbar k, }[/math]


The superposition property allows the particle to be in a quantum superposition of two or more quantum states at the same time. However, a "quantum state" in quantum mechanics means the probability that a system will be, for example at a position x, not that the system will actually be at position x. It does not imply that the particle itself may be in two classical states at once. Indeed, quantum mechanics is generally unable to assign values for properties prior to measurement at all.

叠加特性允许粒子同时处于两个或多个量子态的量子叠加。然而,量子力学中的“量子态”是指系统处于 x位置的“概率”,而不是系统实际处于 x位置。这并不意味着粒子本身可能同时处于两种经典状态。事实上,量子力学通常无法在测量之前为属性赋值。

where [math]\displaystyle{ h }[/math] is Planck's constant and [math]\displaystyle{ \hbar = {h}/{2\pi} }[/math] is the reduced Planck constant of action

普朗克常数是什么? 普朗克常数是不是行动的普朗克常数

Measurement and uncertainty测量和不确定度

These quantized orbits correspond to discrete energy levels, and de Broglie reproduced the Bohr model formula for the energy levels. The Bohr model was based on the assumed quantization of angular momentum [math]\displaystyle{ L }[/math] according to:

这些量子化的轨道对应于离散的能级,德布罗意再现了能级的玻尔模型公式。玻尔模型是基于假设量子化的角动量:

{{量子力学中的主要测量}海森堡测不准原理}量子力学的解释}}

[math]\displaystyle{ L = n{h \over 2\pi} = n\hbar. }[/math]

[ math > l = n { h/2 pi } = n hbar

In classical mechanics, a particle has, at every moment, an exact position and an exact momentum. These values change deterministically as the particle moves according to Newton's laws. Under the Copenhagen interpretation of quantum mechanics, particles do not have exactly determined properties, and when they are measured, the result is randomly drawn from a probability distribution. The Schrödinger equation predicts what the probability distributions are, but fundamentally cannot predict the exact result of each measurement.

在经典力学中,粒子在每一时刻都有一个精确的位置和一个精确的动量。当粒子按照牛顿定律运动时,这些值会改变决定论。在量子力学的哥本哈根解释下,粒子没有精确确定的性质,当对它们进行测量时,结果是从概率分布中随机抽取的。薛定谔方程预测了概率分布是什么,但从根本上说不能预测每次测量的确切结果。

According to de Broglie the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:

根据德布罗意的理论,电子是由一个波描述的,并且沿着电子轨道的周长必须有一整套波长:


The Heisenberg uncertainty principle is one statement of the inherent measurement uncertainty in quantum mechanics. It states that the more precisely a particle's position is known, the less precisely its momentum is known, and vice versa.

海森堡测不准原理是量子力学中固有测量不确定度的一种表述。它指出,粒子的位置知道得越精确,其动量知道得就越不精确,反之亦然。

[math]\displaystyle{ n \lambda = 2 \pi r.\, }[/math]

“ n lambda = 2 pi r,”


The Schrödinger equation describes the (deterministic) evolution of the wave function of a particle. However, even if the wave function is known exactly, the result of a specific measurement on the wave function is uncertain.

薛定谔方程描述了粒子波函数的(确定性)演化。然而,即使精确地知道波函数,对波函数的具体测量结果也是不确定的。

This approach essentially confined the electron wave in one dimension, along a circular orbit of radius [math]\displaystyle{ r }[/math].

这种方法实质上限制了电子波在一维空间中,沿着半径为 r </math > 的圆形轨道运动。

Interpretation of the wave function波函数的解释

In 1921, prior to de Broglie, Arthur C. Lunn at the University of Chicago had used the same argument based on the completion of the relativistic energy–momentum 4-vector to derive what we now call the de Broglie relation. Unlike de Broglie, Lunn went on to formulate the differential equation now known as the Schrödinger equation, and solve for its energy eigenvalues for the hydrogen atom. Unfortunately the paper was rejected by the Physical Review, as recounted by Kamen.

1921年,在德布罗意之前,芝加哥大学的阿瑟 · c · 伦恩用相对论能量-动量4向量完成的理论,推导出了我们现在所说的德布罗意关系。与德布罗意不同的是,Lunn 继续构造了现在被称为微分方程的薛定谔方程,并解出了氢原子的能量本征值。不幸的是,这篇论文被《物理评论》拒绝了,正如卡门所述。

模板:量子力学的主要解释

Following up on de Broglie's ideas, physicist Peter Debye made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, Schrödinger decided to find a proper 3-dimensional wave equation for the electron. He was guided by William R. Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system—the trajectories of light rays become sharp tracks that obey Fermat's principle, an analog of the principle of least action. A modern version of his reasoning is reproduced below. The equation he found is:

下面是他推理的现代版本。他发现的方程式是:

Nearly every result [a quantum theorist] pronounces is about the probability of this or that or that ... happening—with usually a great many alternatives. The idea that they be not alternatives but all really happen simultaneously seems lunatic to him, just impossible.[11]

:几乎每个[量子理论家]宣称的结果都会,以这个或那个或那个的概率发生。。。通常有很多选择。他们不是交替发生,而是“所有”同时发生的想法在他看来是疯狂的,只是“不可能”[12]

[math]\displaystyle{ i\hbar \frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=-\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},\,t) + V(\mathbf{r})\Psi(\mathbf{r},\,t). }[/math]

2 m } nabla ^ 2(mathbf { r } ,,t) + v (mathbf { r }) Psi (mathbf { r } ,t) . </math >

David Deutsch regarded this as the earliest known reference to a many-worlds interpretation of quantum mechanics, an interpretation generally credited to Hugh Everett III,[13] while Jeffrey A. Barrett took the more modest position that it indicates a "similarity in ... general views" between Schrödinger and Everett.[14]

David Deutsch认为这是已知的最早的关于量子力学的多世界解释的引用,这种解释通常归功于Hugh Everett III[15] while Jeffrey A. Barrett took the more modest position that it indicates a "similarity in ... general views" between Schrödinger and Everett.[16]

However, by that time, Arnold Sommerfeld had refined the Bohr model with relativistic corrections. Schrödinger used the relativistic energy momentum relation to find what is now known as the Klein–Gordon equation in a Coulomb potential (in natural units):

然而,到那个时候,阿诺·索末菲已经用相对论修正改进了玻尔模型。薛定谔利用相对论能量动量关系,发现了库仑势(以自然单位计)中的克莱因-戈登方程:

Historical background and development历史背景与发展

[math]\displaystyle{ \left(E + {e^2\over r} \right)^2 \psi(x) = - \nabla^2\psi(x) + m^2 \psi(x). }[/math]

< math > left (e + { e ^ 2 over r } right) ^ 2 psi (x) =-nabla ^ 2 psi (x) + m ^ 2 psi (x) . </math >

模板:薛定谔方程的主要理论和实验证明


< ! -- < math > frac {1}{ c ^ 2} left (e + { e ^ 2 over 4 pi varepsilon _ 0 r } right) ^ 2 psi (x) =-hbar ^ 2 nabla ^ 2 psi (x) + frac { m ^ 2 c ^ 2}{ hbar ^ 2} psi (x) . </math > < < ? -- >


Following Max Planck's quantization of light (see black-body radiation), Albert Einstein interpreted Planck's quanta to be photons, particles of light, and proposed that the energy of a photon is proportional to its frequency, one of the first signs of wave–particle duality. Since energy and momentum are related in the same way as frequency and wave number in special relativity, it followed that the momentum [math]\displaystyle{ p }[/math] of a photon is inversely proportional to its wavelength [math]\displaystyle{ \lambda }[/math], or proportional to its wave number [math]\displaystyle{ k }[/math]:

Max Planck对光的量子化(见black body radiation)之后,Albert Einstein将Planck的 quanta解释为photons, particles of light,并提出光子的[[Planck关系|能量与其频率成正比],波粒二象性的最初迹象之一。由于能量和动量的关系与狭义相对论中的频率波数相同,因此光子的动量p与其波长[math]\displaystyle{ \lambda }[/math]成反比,或与其波数k</math>成反比:

He found the standing waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself with a mistress in a mountain cabin in December 1925.

他发现了这个相对论方程的驻波,但是相对论修正与索末菲公式不一致。1925年12月,气馁的他放下计算,与一个情妇一起躲在山间小屋里。


[math]\displaystyle{ p = \frac{h}{\lambda} = \hbar k, }[/math]

While at the cabin, Schrödinger decided that his earlier nonrelativistic calculations were novel enough to publish, and decided to leave off the problem of relativistic corrections for the future. Despite the difficulties in solving the differential equation for hydrogen (he had sought help from his friend the mathematician Hermann Weyl In the equation, Schrödinger computed the hydrogen spectral series by treating a hydrogen atom's electron as a wave [math]\displaystyle{ \Psi (\mathbf{x}, t) }[/math], moving in a potential well [math]\displaystyle{ V }[/math], created by the proton. This computation accurately reproduced the energy levels of the Bohr model. In a paper, Schrödinger himself explained this equation as follows:

在船舱里,薛定谔认为他早期的非相对论性计算足够新颖,可以发表,并决定为未来放弃相对论性修正的问题。尽管在解决氢的微分方程时遇到了困难(他向他的朋友,数学家 Hermann Weyl 寻求帮助)在这个方程式中,薛定谔把氢原子的电子当作一个由质子产生的势阱来计算氢原子光谱。这个计算精确地再现了玻尔模型的能级。在一篇论文中,薛定谔自己对这个等式解释如下:


where [math]\displaystyle{ h }[/math] is Planck's constant and [math]\displaystyle{ \hbar = {h}/{2\pi} }[/math] is the reduced Planck constant of action[6] (or the Dirac constant). Louis de Broglie hypothesized that this is true for all particles, even particles which have mass such as electrons. He showed that, assuming that the matter waves propagate along with their particle counterparts, electrons form standing waves, meaning that only certain discrete rotational frequencies about the nucleus of an atom are allowed.

其中,[math]\displaystyle{ h }[/math]Planck's constant,而[math]\displaystyle{ \hbar={h}/{2\pi} }[/math]是作用的约化普朗克常数[6](或狄拉克常数)。Louis de Broglie假设所有粒子都是这样,即使是有质量的粒子,比如电子。他指出,假设物质波和粒子波一起传播,电子形成驻波s,这意味着只有原子核周围的某些离散旋转频率是允许的。[17]

[[File:Wavefunction values.svg|300px|thumb|Diagrammatic summary of the quantities related to the wave function, as used in De broglie's hypothesis and development of the Schrödinger equation. Great care is required in how that limit is taken, and in what cases.

[文件: Wavefunction values.svg | 300px | thumb | diagramatic 汇总与波函数有关的量,用于德布罗意的假设和薛定谔方程的发展。在如何以及在什么情况下使用这一限制时,需要非常谨慎。

These quantized orbits correspond to discrete energy levels, and de Broglie reproduced the Bohr model formula for the energy levels. The Bohr model was based on the assumed quantization of angular momentum [math]\displaystyle{ L }[/math] according to:

这些量子化轨道对应于离散的能级s,德布罗意再现了玻尔模型能级公式。玻尔模型基于角动量[math]\displaystyle{ L }[/math]的假设量子化,根据:

[math]\displaystyle{ L = n{h \over 2\pi} = n\hbar. }[/math]

The limiting short-wavelength is equivalent to [math]\displaystyle{ \hbar }[/math] tending to zero because this is limiting case of increasing the wave packet localization to the definite position of the particle (see images right). Using the Heisenberg uncertainty principle for position and momentum, the products of uncertainty in position and momentum become zero as [math]\displaystyle{ \hbar \longrightarrow 0 }[/math]:

极限短波长相当于 < math > hbar </math > 趋于零,因为这是将波包局部化增加到粒子确定位置的极限情况(见右图)。利用海森堡位置和动量的不确定性原理,位置和动量的不确定性的乘积变成了0:

According to de Broglie the electron is described by a wave and a whole number of wavelengths must fit along the circumference of the electron's orbit:

根据德布罗意的理论,电子是由一个波来描述的,在电子轨道的周长上必须有许多波长:

[math]\displaystyle{ \sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\! }[/math]

(2)方向右行四边形 sigma (x) sigma (p _ x) geqslan0,

[math]\displaystyle{ n \lambda = 2 \pi r.\, }[/math]


where denotes the (root mean square) measurement uncertainty in and (and similarly for the and directions) which implies the position and momentum can only be known to arbitrary precision in this limit.

其中表示和(以及类似的和方向)的(平方平均数)测量不确定度,这意味着位置和动量只能在这个极限中被任意精确地知道。

This approach essentially confined the electron wave in one dimension, along a circular orbit of radius [math]\displaystyle{ r }[/math].

这种方法基本上把电子波限制在一维,沿着半径为[math]\displaystyle{ r }[/math]的圆形轨道。

One simple way to compare classical to quantum mechanics is to consider the time-evolution of the expected position and expected momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the Ehrenfest theorem. For a one-dimensional quantum particle moving in a potential [math]\displaystyle{ V }[/math], the Ehrenfest theorem says

一个简单的方法来比较经典和量子力学是考虑预期位置和预期动量的时间演化,然后可以比较普通位置和动量在经典力学的时间演化。量子期望值满足埃伦费斯特定理。对于一维的量子粒子以潜在的 < math > v </math > 运动,埃伦费斯特定理说

In 1921, prior to de Broglie, Arthur C. Lunn at the University of Chicago had used the same argument based on the completion of the relativistic energy–momentum 4-vector to derive what we now call the de Broglie relation.[18][19] Unlike de Broglie, Lunn went on to formulate the differential equation now known as the Schrödinger equation, and solve for its energy eigenvalues for the hydrogen atom. Unfortunately the paper was rejected by the Physical Review, as recounted by Kamen.[20]

1921年,在德布罗意之前,芝加哥大学的阿瑟·C·伦恩(Arthur C.Lunn)基于相对论能量-动量 4-vector的完备性,使用了同样的论点,导出了我们现在所说的德布罗意关系。[21][22] 与德布罗意不同的是,伦恩接着建立了现在称为薛定谔方程的微分方程,并求解了氢原子的能量本征值。不幸的是,这篇论文被Kamen重新叙述的“物理评论”拒绝了。[23]

[math]\displaystyle{ m\frac{d}{dt}\langle x\rangle = \langle p\rangle;\quad \frac{d}{dt}\langle p\rangle = -\left\langle V'(X)\right\rangle . }[/math]

四边形的边长 =-左边长 v’(x)右边长。 </math >


Although the first of these equations is consistent with the classical behavior, the second is not: If the pair [math]\displaystyle{ (\langle X\rangle,\langle P\rangle) }[/math] were to satisfy Newton's second law, the right-hand side of the second equation would have to be

虽然第一个方程符合经典行为,但第二个方程不符合: 如果要满足牛顿第二定律,那么第二个方程的右边必须是

Following up on de Broglie's ideas, physicist Peter Debye made an offhand comment that if particles behaved as waves, they should satisfy some sort of wave equation. Inspired by Debye's remark, Schrödinger decided to find a proper 3-dimensional wave equation for the electron. He was guided by William R. Hamilton's analogy between mechanics and optics, encoded in the observation that the zero-wavelength limit of optics resembles a mechanical system—the trajectories of light rays become sharp tracks that obey Fermat's principle, an analog of the principle of least action.

继德布罗意的想法之后,物理学家Peter Debye即兴评论说,如果粒子表现为波,它们应该满足某种波动方程。受德拜的启发,薛定谔决定为电子找到一个合适的三维波动方程。他遵循威廉·R·汉密尔顿力学光学的类比,即光学的零波长极限类似于一个机械系统,光线的轨迹变成了服从费马原理的尖锐轨迹,这是最小动作原理的类似物。[24] A modern version of his reasoning is reproduced below. The equation he found is:

下面是他的推理的现代版本。他找到的方程式是:[25]

where is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and is a function of time only.

其中是构成系统的所有粒子的空间坐标的函数,仅是时间的函数。


[math]\displaystyle{ i\hbar \frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=-\frac{\hbar^2}{2m}\nabla^2\Psi(\mathbf{r},\,t) + V(\mathbf{r})\Psi(\mathbf{r},\,t). }[/math]

Substituting for into the Schrödinger equation for the relevant number of particles in the relevant number of dimensions, solving by separation of variables implies the general solution of the time-dependent equation has the form: the potential of the system only influences each particle separately, so the total potential energy is the sum of potential energies for each particle:

用薛定谔方程代替相关维数中相关粒子的数量,用分离变量法求解意味着时变方程的一般解有这样的形式: 系统的势只影响每个粒子的势能,所以总势能是每个粒子的势能之和:


However, by that time, Arnold Sommerfeld had refined the Bohr model with relativistic corrections.

然而,到那时,Arnold Sommerfeld已经改进了玻尔模型相对论修正[26][27] Schrödinger used the relativistic energy momentum relation to find what is now known as the Klein–Gordon equation in a Coulomb potential (in natural units):

薛定谔利用相对论能量-动量关系,在库仑势(以自然单位)中找到了现在所知的克莱因-戈登方程

- \frac{m\omega x^2}{2 \hbar}} \cdot \mathcal{H}_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), </math>

- frac { m omega x ^ 2}{2 hbar } cdot mathcal { h } _ n left (sqrt { frac { m omega }{ hbar } x right) ,</math >


[math]\displaystyle{ \left(E + {e^2\over r} \right)^2 \psi(x) = - \nabla^2\psi(x) + m^2 \psi(x). }[/math]

where [math]\displaystyle{ n \in \{0, 1, 2, \ldots \} }[/math], and the functions [math]\displaystyle{ \mathcal{H}_n }[/math] are the Hermite polynomials of order [math]\displaystyle{ n }[/math]. The solution set may be generated by

其中[math]\displaystyle{ n \in \{0, 1, 2, \ldots \} }[/math],而函数 [math]\displaystyle{ \mathcal{H}_n }[/math][math]\displaystyle{ n }[/math] 的埃尔米特多项式。解决方案集可以由



[math]\displaystyle{ \psi_n(x) = \frac{1}{\sqrt{n!}} \left( \sqrt{\frac{m \omega}{2 \hbar}} \right)^{n} \left( x - \frac{\hbar}{m \omega} \frac{d}{dx}\right)^n \left( \frac{m \omega}{\pi \hbar} \right)^{\frac{1}{4}} e^{\frac{-m \omega x^2}{2\hbar}}. }[/math]

1}{ sqrt { n! }}左(sqrt { m omega }{2 hbar }右) ^ { n } left (x-frac { hbar }{ m omega } frac { d }{ dx }右) ^ n left (frac { m omega }{ pi hbar }右) ^ { frac {1}{4}{ frac {-m omega x ^ 2}{2 hbar } </math >

He found the standing waves of this relativistic equation, but the relativistic corrections disagreed with Sommerfeld's formula. Discouraged, he put away his calculations and secluded himself with a mistress in a mountain cabin in December 1925.[28]

他发现了这个相对论方程的驻波,但是相对论修正与索末菲公式不一致。1925年12月,他灰心丧气地放下计算,与一位情妇隐居在山间小屋里。[29]

The eigenvalues are

本征值是

While at the cabin, Schrödinger decided that his earlier nonrelativistic calculations were novel enough to publish, and decided to leave off the problem of relativistic corrections for the future. Despite the difficulties in solving the differential equation for hydrogen (he had sought help from his friend the mathematician Hermann Weyl[30]:3) Schrödinger showed that his nonrelativistic version of the wave equation produced the correct spectral energies of hydrogen in a paper published in 1926.

在船舱里时,薛定谔认为他早期的非相对论计算足够新颖,可以发表,并决定在未来抛开相对论修正的问题。尽管求解氢的微分方程有困难(他曾向他的朋友数学家Hermann Weyl[31]{rp | 3}}寻求帮助),但在1926年发表的一篇论文中,薛定谔表明,他对波动方程的非相对论版本产生了正确的氢光谱能量。[30]:1[32] In the equation, Schrödinger computed the hydrogen spectral series by treating a hydrogen atom's electron as a wave [math]\displaystyle{ \Psi (\mathbf{x}, t) }[/math], moving in a potential well [math]\displaystyle{ V }[/math], created by the proton. This computation accurately reproduced the energy levels of the Bohr model. In a paper, Schrödinger himself explained this equation as follows:

在方程中,薛定谔通过将氢原子电子视为波[math]\displaystyle{ \Psi (\mathbf{x}, t) }[/math],在质子产生的势阱[math]\displaystyle{ V }[/math]中移动来计算氢光谱系列。这个计算准确地再现了玻尔模型的能级。在一篇论文中,薛定谔本人对这个等式的解释如下:

The Hamiltonian for one particle in three dimensions is:

三维空间中一个粒子的哈密顿量是:


模板:Cquote

模板:Cquote

[math]\displaystyle{ \hat{H} = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r}) \,, \quad \hat{\mathbf{p}} = -i\hbar \nabla }[/math]

2m } + v (mathbf { r }) ,,quad hat { mathbf { p }}} =-i hbar nabla </math >


This 1926 paper was enthusiastically endorsed by Einstein, who saw the matter-waves as an intuitive depiction of nature, as opposed to Heisenberg's matrix mechanics, which he considered overly formal.

这篇1926年的论文得到了爱因斯坦的热情支持,他认为物质波是对自然的直观描述,而海森堡的矩阵力学,他认为过于正式。[33]

[math]\displaystyle{ \Psi(\mathbf{r},t) = \psi(\mathbf{r}) e^{-iEt/\hbar}, }[/math]

< math > Psi (mathbf { r } ,t) = Psi (mathbf { r }) e ^ {-iEt/hbar } ,</math >


The Schrödinger equation details the behavior of [math]\displaystyle{ \Psi }[/math] but says nothing of its nature. Schrödinger tried to interpret it as a charge density in his fourth paper, but he was unsuccessful.

薛定谔方程详细描述了Psi的行为,但没有提到它的“本质”。薛定谔在他的第四篇论文中试图将其解释为电荷密度,但没有成功。[34]:219 In 1926, just a few days after Schrödinger's fourth and final paper was published, Max Born successfully interpreted [math]\displaystyle{ \Psi }[/math] as the probability amplitude, whose modulus squared is equal to probability density.[34]:220 Schrödinger, though, always opposed a statistical or probabilistic approach, with its associated discontinuities—much like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying deterministic theory—and never reconciled with the Copenhagen interpretation.{{#tag:ref |It is clear that even in his last year of life, as shown in a letter to Max Born, that Schrödinger never accepted the Copenhagen interpretation.

1926年,就在薛定谔的第四篇也是最后一篇论文发表几天后,Max Born成功地将[math]\displaystyle{ \Psi }[/math]解释为概率振幅,其模平方等于概率密度[34]:220薛定谔,总是反对统计或概率方法,及其相关的不连续——很像爱因斯坦,他认为量子力学是一个统计近似的基础确定性理论——从来没有调和哥本哈根解释[35]

[math]\displaystyle{ \nabla_n = \mathbf{e}_x \frac{\partial}{\partial x_n} + \mathbf{e}_y\frac{\partial}{\partial y_n} + \mathbf{e}_z\frac{\partial}{\partial z_n}\,,\quad \nabla_n^2 = \nabla_n\cdot\nabla_n = \frac{\partial^2}{{\partial x_n}^2} + \frac{\partial^2}{{\partial y_n}^2} + \frac{\partial^2}{{\partial z_n}^2} }[/math]

,4 nabla n ^ 2 = nabla n cndot nabla n = frac { partial ^ 2}{ partial x _ n } ^ 2} + frac { partial y _ n } ^ 2}{ partial y _ n } ^ 2} + frac { partial ^ 2}{{ partial z _ n }模板:Partial z n{{ partial z _ n } ^ 2} ^ 2} </math >

Louis de Broglie in his later years proposed a real valued wave function connected to the complex wave function by a proportionality constant and developed the De Broglie–Bohm theory.

Louis de Broglie在晚年提出了一个实值的波函数通过一个比例常数连接到复波函数,并发展了de Broglie–Bohm理论


The Schrödinger equation is:

薛定谔方程是:

The wave equation for particles粒子的波动方程

模板:主

[math]\displaystyle{ -\frac{\hbar^2}{2}\sum_{n=1}^{N}\frac{1}{m_n}\nabla_n^2\Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N) + V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N)\Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N) = E\Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N) }[/math]

< math >-frac { hbar ^ 2}{2} sum { n = 1}{ n } frac {1}{ n } nabla _ n ^ 2 Psi (mathbf { r } _ 1,mathbf { r } _ 2,ldots,mathbf { r } _ n) + v (mathbf { r } _ 1,bf { r } _ 2,mathbf { r } _ 2,mathbf { r } _ n)(mathbf { r } _ 1,lbf { r } _ 2,lbf,mathf { r } _ n) = e Psi ({ r } _ 1,mathf _ 2,mathf _ 2,mathf _ l } _ 2,mathf _ 2,lbf,r } _ n </math >


The Schrödinger equation is a variation on the diffusion equation where the diffusion constant is imaginary. A spike of heat will decay in amplitude and spread out; however, because the imaginary i is the generator of rotations in the complex plane, a spike in the amplitude of a matter wave will also rotate in the complex plane over time. The solutions are therefore functions which describe wave-like motions. Wave equations in physics can normally be derived from other physical laws – the wave equation for mechanical vibrations on strings and in matter can be derived from Newton's laws, where the wave function represents the displacement of matter, and electromagnetic waves from Maxwell's equations, where the wave functions are electric and magnetic fields. The basis for Schrödinger's equation, on the other hand, is the energy of the system and a separate postulate of quantum mechanics: the wave function is a description of the system.[36] The Schrödinger equation is therefore a new concept in itself; as Feynman put it:

薛定谔方程是扩散方程的一个变体,其中扩散常数是虚的。热的尖峰会在振幅上衰减并扩散;然而,由于虚i是复平面中旋转的发生器,物质波振幅的尖峰也会随时间在复平面中旋转。因此,解是描述波浪运动的函数。物理学中的波动方程通常可以从其他物理定律推导出来——弦上和物质中机械振动的波动方程可以从牛顿定律推导出来,其中波函数表示物质的位移,而电磁波则来自麦克斯韦方程,其中波函数是电场磁场场。另一方面,薛定谔方程的基础是系统的能量和一个单独的量子力学的假设:波函数是对系统的描述。[36] 因此,薛定谔方程本身就是一个新概念;正如费曼所说:

with stationary state solutions:

定态解决方案:


模板:Cquote

模板:Cquote

[math]\displaystyle{ \Psi(\mathbf{r}_1,\mathbf{r}_2,\ldots, \mathbf{r}_N,t) = e^{-iEt/\hbar}\psi(\mathbf{r}_1,\mathbf{r}_2,\ldots, \mathbf{r}_N) }[/math]

< math > Psi (mathbf { r } _ 1,mathbf { r } _ 2,ldots,mathbf { r } _ n,t) = e ^ {-iEt/hbar } Psi (mathbf { r } _ 1,mathbf { r } _ 2,ldots,mathbf { r } _ n) </math >


The foundation of the equation is structured to be a linear differential equation based on classical energy conservation, and consistent with the De Broglie relations. The solution is the wave function ψ, which contains all the information that can be known about the system. In the Copenhagen interpretation, the modulus of ψ is related to the probability the particles are in some spatial configuration at some instant of time. Solving the equation for ψ can be used to predict how the particles will behave under the influence of the specified potential and with each other.

该方程的基础被构造成基于经典能量守恒的线性微分方程,并与德布罗意关系一致。解是波函数模板:数学,它包含了关于系统的所有已知信息。在哥本哈根解释中,{数学|ψ}的模与概率有关,粒子在某一时刻处于某种空间构型。求解{数学|ψ}方程可以用来预测粒子在特定势的影响下以及相互之间的行为。

Again, for non-interacting distinguishable particles the potential is the sum of particle potentials

同样,对于无相互作用可区分的粒子,势就是粒子势的和


The Schrödinger equation was developed principally from the De Broglie hypothesis, a wave equation that would describe particles,[37] and can be constructed as shown informally in the following sections.[38] For a more rigorous description of Schrödinger's equation, see also Resnick et al.[39]

薛定谔方程主要是从德布罗意假设发展而来的,这是一个描述粒子的波动方程,[37] 可按照以下各节的非正式说明进行构建。[38] 有关薛定谔方程的更严格描述,请参见Resnick“等人”的文章。[39]

[math]\displaystyle{ V(\mathbf{r}_1,\mathbf{r}_2,\ldots, \mathbf{r}_N) = \sum_{n=1}^N V(\mathbf{r}_n) }[/math]

< math > v (mathbf { r } _ 1,mathbf { r } _ 2,ldots,mathbf { r } _ n) = sum _ { n = 1} ^ n v (mathbf { r } _ n) </math >


Consistency with energy conservation 与节能的一致性

and the wave function is a product of the particle wave functions

波函数是粒子波函数的乘积

The total energy E of a particle is the sum of kinetic energy [math]\displaystyle{ T }[/math] and potential energy [math]\displaystyle{ V }[/math], this sum is also the frequent expression for the Hamiltonian [math]\displaystyle{ H }[/math] in classical mechanics:

粒子的总能量{math |“E'}}是动能[math]\displaystyle{ T }[/math]和势能[math]\displaystyle{ V }[/math]的和,这个和也是经典力学中哈密顿[math]\displaystyle{ H }[/math]的常用表达式:

[math]\displaystyle{ E = T + V =H \,\! }[/math]

[math]\displaystyle{ \Psi(\mathbf{r}_1,\mathbf{r}_2 ,\ldots, \mathbf{r}_N,t) = e^{-i{E t/\hbar}} \prod_{n=1}^N\psi(\mathbf{r}_n) \, . }[/math]

< math > Psi (mathbf { r } _ 1,mathbf { r } _ 2,ldots,mathbf { r } _ n,t) = e ^ {-i { e t/hbar } prod _ n = 1} ^ n Psi (mathbf { r } _ n) ,.数学

Explicitly, for a particle in one dimension with position [math]\displaystyle{ x }[/math], mass [math]\displaystyle{ m }[/math] and momentum [math]\displaystyle{ p }[/math], and potential energy [math]\displaystyle{ V }[/math] which generally varies with position and time [math]\displaystyle{ t }[/math]:

明确地说,对于一维粒子,其位置[math]\displaystyle{ x }[/math]mass[math]\displaystyle{ m }[/math]momentum[math]\displaystyle{ p }[/math],势能[math]\displaystyle{ V }[/math]通常 variable with position和时间[math]\displaystyle{ t }[/math]

For non-interacting identical particles, the potential is a sum but the wave function is a sum over permutations of products. The previous two equations do not apply to interacting particles.

对于不相互作用的全同粒子,势是一个和,但波函数是产品排列的和。前面的两个方程不适用于相互作用的粒子。

[math]\displaystyle{ E = \frac{p^2}{2m}+V(x,t)=H. }[/math]


Following are examples where exact solutions are known. See the main articles for further details.

下面是已知精确解的例子。详情请参阅主要文章。

For three dimensions, the position vector r and momentum vector p must be used:

对于三维,必须使用[[位置矢量]r和动量矢量p

[math]\displaystyle{ E = \frac{\mathbf{p}\cdot\mathbf{p}}{2m}+V(\mathbf{r},t)=H }[/math]


The Schrödinger equation for the hydrogen atom (or a hydrogen-like atom) is In this case, spherical polar coordinates are the most convenient. Thus,

在这种情况下,球极坐标是最方便的薛定谔方程类氢原子。因此,

This formalism can be extended to any fixed number of particles: the total energy of the system is then the total kinetic energies of the particles, plus the total potential energy, again the Hamiltonian. However, there can be interactions between the particles (an N-body problem), so the potential energy V can change as the spatial configuration of particles changes, and possibly with time. The potential energy, in general, is not the sum of the separate potential energies for each particle, it is a function of all the spatial positions of the particles. Explicitly:

这种形式可以推广到任意数量的粒子:系统的总能量就是粒子的总动能,加上总势能,再加上哈密顿量。然而,粒子之间可能存在相互作用(一个{数学| N'}体问题),因此势能{数学| V'}可能会随着粒子的空间构型的变化而变化,也可能随着时间的变化而变化。一般来说,势能“不是”每个粒子的独立势能之和,它是粒子所有空间位置的函数。明确地:

[math]\displaystyle{ \psi(r,\theta,\varphi) = R(r)Y_\ell^m(\theta, \varphi) = R(r)\Theta(\theta)\Phi(\varphi), }[/math]

< math > psi (r,Theta,varphi) = r (r) y _ ell ^ m (Theta,varphi) = r (r) Theta (Theta) Phi (varphi) ,</math >

[math]\displaystyle{ E=\sum_{n=1}^N \frac{\mathbf{p}_n\cdot\mathbf{p}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N,t) = H \,\! }[/math]


where are radial functions and [math]\displaystyle{ Y^m_l (\theta, \varphi) }[/math] are spherical harmonics of degree [math]\displaystyle{ \ell }[/math] and order [math]\displaystyle{ m }[/math]. This is the only atom for which the Schrödinger equation has been solved for exactly. Multi-electron atoms require approximate methods. The family of solutions are:

径向函数在哪里? y ^ m l (theta,varphi) </math > 是球谐函数 < math > </math > 和 < math > m </math > 。这是唯一的一个原子,薛定谔方程已经完全解决了。多电子原子需要近似方法。解决方案包括:

Linearity 线性

The simplest wave function is a plane wave of the form:

最简单的波函数是以下形式的平面波

[math]\displaystyle{ \psi_{n\ell m}(r,\theta,\varphi) = \sqrt {\left ( \frac{2}{n a_0} \right )^3\frac{(n-\ell-1)!}{2n[(n+\ell)!]} } e^{- r/na_0} \left(\frac{2r}{na_0}\right)^\ell L_{n-\ell-1}^{2\ell+1}\left(\frac{2r}{na_0}\right) \cdot Y_{\ell}^m(\theta, \varphi ) }[/math]

2n [(n + ell) ! ]{ e ^ {-r/na _ 0} left (frac {2r }{ na _ 0} right) ^ ell l _ { n-ell-1} ^ {2 ell + 1} left (frac {2r }{ na _ 0} right) cdot y _ { ell } ^ m (theta,varphi) </math >


[math]\displaystyle{ \Psi(\mathbf{r},t) = A e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} }[/math]

where:

其中:


where the A is the amplitude, k the wave vector, and [math]\displaystyle{ \omega }[/math] the angular frequency, of the plane wave. In general, physical situations are not purely described by plane waves, so for generality the superposition principle is required; any wave can be made by superposition of sinusoidal plane waves. So if the equation is linear, a linear combination of plane waves is also an allowed solution. Hence a necessary and separate requirement is that the Schrödinger equation is a linear differential equation.

其中{math |A}}是振幅,{math |k}是波矢量,[math]\displaystyle{ \omega }[/math]是平面波的角频率。一般来说,物理情况不完全用平面波来描述,因此一般需要[[叠加原理];任何波都可以由正弦平面波叠加而成。所以如果方程是线性的,平面波的线性组合也是允许的解。因此,一个必要的和单独的要求是薛定谔方程是一个线性微分方程

For discrete [math]\displaystyle{ \mathbf{k} }[/math] the sum is a superposition of plane waves:

对于离散的[math]\displaystyle{ \mathbf{k} }[/math]来说,和是平面波的叠加

[math]\displaystyle{ 《数学》 \begin{align} 开始{ align } :\lt math\gt \Psi(\mathbf{r},t) = \sum_{n=1}^\infty A_n e^{i(\mathbf{k}_n\cdot\mathbf{r}-\omega_n t)} \,\! }[/math]

n & = 1,2,3, \dots \\

1,2,3,点


\ell & = 0,1,2, \dots, n-1 \\

0,1,2,dots,n-1

for some real amplitude coefficients [math]\displaystyle{ A_n }[/math], and for continuous [math]\displaystyle{ \mathbf{k} }[/math] the sum becomes an integral, the Fourier transform of a momentum space wave function:[40]

对于一些实振幅系数,对于连续的,动量空间波函数的傅里叶变换[40]

m & = -\ell,\dots,\ell \\

M & =-ell,dots,ell


\end{align}

结束{ align }

[math]\displaystyle{ \Psi(\mathbf{r},t) = \frac{1}{\left(\sqrt{2\pi}\,\right)^3}\int\Phi(\mathbf{k})e^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)}d^3\mathbf{k} \,\! }[/math]

</math>

数学


where [math]\displaystyle{ d^3 \mathbf{k} = dk_x \,dk_y 其中 \lt math\gt d^3 \mathbf{k} = dk_x \,dk_y The generalized Laguerre polynomials are defined differently by different authors. See main article on them and the hydrogen atom. 不同的作者对广义拉盖尔多项式有不同的定义。参见关于它们和氢原子的主要文章。 \,dk_z }[/math]is the differential volume element in k-space, and the integrals are taken over all [math]\displaystyle{ \mathbf{k} }[/math]-space. The momentum wave function [math]\displaystyle{ \Phi (\mathbf{k}) }[/math] arises in the integrand since the position and momentum space wave functions are Fourier transforms of each other.

\,dkëz</math>是k-空间中的微分体积元素,积分占据了所有的[math]\displaystyle{ \mathbf{k} }[/math]-空间。动量波函数Phi(\mathbf{k})</math>出现在被积函数中,因为位置和动量空间波函数是相互的傅里叶变换。

Consistency with the de Broglie relations 与德布罗意关系的一致性

The equation for any two-electron system, such as the neutral helium atom (He, [math]\displaystyle{ Z = 2 }[/math]), the negative hydrogen ion (H, [math]\displaystyle{ Z = 1 }[/math]), or the positive lithium ion (Li+, [math]\displaystyle{ Z = 3 }[/math]) is:

对于任意两电子系统,如中性氦原子(He,< math > z = 2 </math >) ,负氢离子(h < sup >- ,< math > z = 1 </math >) ,或正锂离子(Li < sup > + ,< math > z = 3 </math >)的方程式是:

文件:Wavefunction values.svg
Diagrammatic summary of the quantities related to the wave function, as used in De broglie's hypothesis and development of the Schrödinger equation.[37]

300px |拇指|与波函数有关的量的图解总结,用于德布罗意的假设和薛定谔方程的发展。[41]

[math]\displaystyle{ { \partial \over \partial t} \rho\left(\mathbf{r},t\right) + \nabla \cdot \mathbf{j} = 0, }[/math]

< math > { partial over partial t } rho left (mathbf { r } ,t right) + nabla cdot mathbf { j } = 0,</math >


where


Einstein's light quanta hypothesis (1905) states that the energy E of a quantum of light or photon is proportional to its frequency [math]\displaystyle{ \nu }[/math] (or angular frequency, [math]\displaystyle{ \omega = 2\pi \nu }[/math])

爱因斯坦的光量子假说(1905)指出光或光子量子的能量{数学| E'}与其频率 成正比[math]\displaystyle{ \nu }[/math] (或 角频率, [math]\displaystyle{ \omega = 2\pi \nu }[/math])

[math]\displaystyle{ \rho=|\Psi|^2=\Psi^*(\mathbf{r},t)\Psi(\mathbf{r},t)\,\! }[/math]

< math > rho = | Psi | ^ 2 = Psi ^ * (mathbf { r } ,t) Psi (mathbf { r } ,t) ,

[math]\displaystyle{ E = h\nu = \hbar \omega \,\! }[/math]

is the probability density (probability per unit volume, denotes complex conjugate), and

是概率密度(单位体积概率,表示共轭复数) ,以及

Likewise De Broglie's hypothesis (1924) states that any particle can be associated with a wave, and that the momentum [math]\displaystyle{ p }[/math] of the particle is inversely proportional to the wavelength [math]\displaystyle{ \lambda }[/math] of such a wave (or proportional to the wavenumber, [math]\displaystyle{ k = \frac{2\pi}{\lambda} }[/math]), in one dimension, by:

[math]\displaystyle{ \mathbf{j} = {1 \over 2m} \left( \Psi^*\hat{\mathbf{p}}\Psi - \Psi\hat{\mathbf{p}}\Psi^* \right)\,\! }[/math]

< math > mathbf { j } = {1/2m } left (Psi ^ * hat { mathbf { p } Psi-Psi hat { mathbf { p } Psi ^ * right) ,! </math >

[math]\displaystyle{ p = \frac{h}{\lambda} = \hbar k\;, }[/math]

is the probability current (flow per unit area).

是概率流(单位面积流量)。

while in three dimensions, wavelength λ is related to the magnitude of the wavevector k:

[math]\displaystyle{ \mathbf{p} = \hbar \mathbf{k}\,,\quad |\mathbf{k}| = \frac{2\pi}{\lambda} \,. }[/math]

Hence predictions from the Schrödinger equation do not violate probability conservation.

因此,来自薛定谔方程的预测并不违背概率守恒。


The Planck–Einstein and de Broglie relations illuminate the deep connections between energy with time, and space with momentum, and express wave–particle duality. In practice, natural units comprising [math]\displaystyle{ \hbar = 1 }[/math] are used, as the De Broglie equations reduce to identities: allowing momentum, wave number, energy and frequency to be used interchangeably, to prevent duplication of quantities, and reduce the number of dimensions of related quantities. For familiarity SI units are still used in this article.

普朗克-爱因斯坦和德布罗意的关系阐明了能量与时间、空间与动量之间的深层联系,并表达了波粒二象性。在实践中,使用由[math]\displaystyle{ \hbar=1 }[/math]组成的自然单位,因为德布罗意的“方程式”简化为“恒等式”:允许动量、波数、能量和频率互换使用,以防止量的重复,并减少相关量的维数。为便于熟悉,本文仍使用国际单位制。

If the potential is bounded from below, meaning there is a minimum value of potential energy, the eigenfunctions of the Schrödinger equation have energy which is also bounded from below. This can be seen most easily by using the variational principle, as follows. (See also below).

如果势是从下面有界的,这意味着存在势能的最小值,薛定谔方程的本征函数的能量也是从下面有界的。这一点可以通过使用变分原理数据库最容易看到,如下所示。(另见下文)。

Schrödinger's insight,[citation needed] late in 1925, was to express the phase of a plane wave as a complex phase factor using these relations:

薛定谔的见解是,1925年晚些时候,{引证需要|日期=2014年1月}将平面波相位表示为相位因子,使用以下关系:

For any linear operator bounded from below, the eigenvector with the smallest eigenvalue is the vector that minimizes the quantity

对于任何从下面有界的线性算子,具有最小特征值的特征向量是使量最小的向量

[math]\displaystyle{ \Psi = Ae^{i(\mathbf{k}\cdot\mathbf{r}-\omega t)} = Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} }[/math]


[math]\displaystyle{ \langle \psi |\hat{A}|\psi \rangle }[/math]

[ math > langle psi | hat { a } | psi rangle

and to realize that the first order partial derivatives with respect to space were

认识到关于空间的一阶偏导数是

over all which are normalized.

所有这些都是正常的。

[math]\displaystyle{ \nabla\Psi = \dfrac{i}{\hbar}\mathbf{p}Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} = \dfrac{i}{\hbar}\mathbf{p}\Psi. }[/math]


[math]\displaystyle{ {\partial \over \partial \tau} X(\mathbf{r},\tau) = \frac{\hbar}{2m} \nabla ^2 X(\mathbf{r},\tau) \, , \quad X(\mathbf{r},\tau) = \Psi(\mathbf{r},\tau/i) }[/math]

< math > { partial over tau } x (mathbf { r } ,tau) = frac { hbar }{2m } nabla ^ 2 x (mathbf { r } ,tau) ,,quad x (mathbf { r } ,tau) = Psi (mathbf { r } ,tau/i) </math >

Taking partial derivatives with respect to time gives

对时间求偏导数

which has the same form as the diffusion equation, with diffusion coefficient }}.

它的形式与扩散方程相同,但扩散系数为}。

[math]\displaystyle{ \dfrac{\partial \Psi}{\partial t} = -\dfrac{i E}{\hbar} Ae^{i(\mathbf{p}\cdot\mathbf{r}-Et)/\hbar} = -\dfrac{i E}{\hbar} \Psi. }[/math]


Another postulate of quantum mechanics is that all observables are represented by linear Hermitian operators which act on the wave function, and the eigenvalues of the operator are the values the observable takes. The previous derivatives are consistent with the energy operator (or Hamiltonian operator), corresponding to the time derivative,

量子力学的另一个假设是,所有可观测物都由作用于波函数的线性厄米算子表示,算子的本征值就是可观测物所取的值。前面的导数与能量算符(或哈密顿算符)一致,对应于时间导数,

On the space [math]\displaystyle{ L^2 }[/math] of square-integrable densities, the Schrödinger semigroup [math]\displaystyle{ e^{it\hat{H}} }[/math] is a unitary evolution, and therefore surjective. The flows satisfy the Schrödinger equation [math]\displaystyle{ i\partial_t u = \hat{H}u }[/math], where the derivative is taken in the distribution sense. However, since [math]\displaystyle{ \hat{H} }[/math] for most physically reasonable Hamiltonians (e.g., the Laplace operator, possibly modified by a potential) is unbounded in [math]\displaystyle{ L^2 }[/math], this shows that the semigroup flows lack Sobolev regularity in general. Instead, solutions of the Schrödinger equation satisfy a Strichartz estimate.

在平方可积密度空间 < math > l ^ 2 </math > 上,薛定谔半群 < math > e ^ { it { h } </math > 是幺正演化,因此是满射的。流满足薛定谔方程分布意义下的导数。然而,对于大多数物理上合理的汉密尔顿人来说(例如,拉普拉斯算子,可能被势修改过) ,在 < math > l ^ 2 </math > 中是无界的,这表明半群流通常缺乏 Sobolev 规则性。相反,薛定谔方程银行的解决方案满足了斯特里查兹的估计。


[math]\displaystyle{ \hat{E} \Psi = i\hbar\dfrac{\partial}{\partial t}\Psi = E\Psi }[/math]


Relativistic quantum mechanics is obtained where quantum mechanics and special relativity simultaneously apply. In general, one wishes to build relativistic wave equations from the relativistic energy–momentum relation

相对论量子力学是在量子力学和狭义相对论同时适用的情况下得到的。一般来说,人们希望从相对论能量-动量关系建立相对论波动方程

where E are the energy eigenvalues, and the momentum operator, corresponding to the spatial derivatives (the gradient [math]\displaystyle{ \nabla }[/math]),

其中E'是能量本征值s和动量算子,对应于空间导数( gradient[math]\displaystyle{ \nabla }[/math]),

[math]\displaystyle{ E^2 = (pc)^2 + (m_0c^2)^2 \, , }[/math]

2 = (pc) ^ 2 + (m0c ^ 2) ^ 2,,</math >

[math]\displaystyle{ \hat{\mathbf{p}} \Psi = -i\hbar\nabla \Psi = \mathbf{p} \Psi }[/math]


instead of classical energy equations. The Klein–Gordon equation and the Dirac equation are two such equations. The Klein–Gordon equation,

而不是经典的能量方程。克莱因-戈登方程和狄拉克方程就是这样两个方程。克莱因-戈登方程,

where p is a vector of the momentum eigenvalues. In the above, the "hats" ( ˆ ) indicate these observables are operators, not simply ordinary numbers or vectors. The energy and momentum operators are differential operators, while the potential energy operator [math]\displaystyle{ V }[/math] is just a multiplicative factor.

其中p是动量本征值的向量。在上面的例子中,“ hats”(ˆ)表示这些可观测值是运算符,而不仅仅是普通数或向量。能量和动量算符是“微分算符”,而势能算符[math]\displaystyle{ V }[/math]只是一个乘法因子。

[math]\displaystyle{ \frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0. }[/math],

1}{ c ^ 2} frac { partial ^ 2}{ partial t ^ 2} psi-nabla ^ 2 psi + frac { m ^ 2 c ^ 2}{ hbar ^ 2} psi = 0.数学,

Substituting the energy and momentum operators into the classical energy conservation equation obtains the operator:

将能量和动量算符代入经典能量守恒方程,得到算符:

was the first such equation to be obtained, even before the nonrelativistic one, and applies to massive spinless particles. The Dirac equation arose from taking the "square root" of the Klein–Gordon equation by factorizing the entire relativistic wave operator into a product of two operators – one of these is the operator for the entire Dirac equation. Entire Dirac equation:

是第一个得到的这样的方程,甚至在非相对论性方程之前,并且适用于大质量的无自旋粒子。狄拉克方程起源于将整个相对论性波算符因式分解为两个算符的乘积,从而得到 Klein-Gordon 方程的“平方根” ,其中一个算符是整个狄拉克方程的算符。整个狄拉克方程:

[math]\displaystyle{ E= \dfrac{\mathbf{p}\cdot\mathbf{p}}{2m}+V \quad \rightarrow \quad \hat{E} = \dfrac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V }[/math]

[math]\displaystyle{ \left(\beta mc^2 + c\left(\sum_{n \mathop =1}^{3}\alpha_n p_n\right)\right) \psi = i \hbar \frac{\partial\psi }{\partial t} }[/math]

左(beta mc ^ 2 + c left (sum { n mathop = 1} ^ {3} alpha _ n p _ n right) psi = i hbar frac { partial t } </math >


so in terms of derivatives with respect to time and space, acting this operator on the wave function Ψ immediately led Schrödinger to his equation:[citation needed]

因此,关于时间和空间的导数,把这个算符作用于波函数{{{1}}}与动量{mathp'}的平方有关。当粒子的动量增加时,动能增加得更快,但由于波数模板:数学k模板:!}}增加波长模板:数学减少。对于普通标量和向量量(不是运算符):

in which the (γ1, γ2, γ3)}} and are the Dirac gamma matrices related to the spin of the particle. The Dirac equation is true for all ]]}} particles, and the solutions to the equation are spinor fields with two components corresponding to the particle and the other two for the antiparticle.

其中(γ < sup > 1 ,γ < sup > 2 ,γ < sup > 3 )}和与粒子自旋有关的 Dirac gamma 矩阵。对于所有的]}粒子,狄拉克方程是正确的,方程的解是旋量场,其中两个分量对应于粒子,另外两个分量对应于反粒子。


[math]\displaystyle{ \mathbf{p}\cdot\mathbf{p} \propto \mathbf{k}\cdot\mathbf{k} \propto T \propto \dfrac{1}{\lambda^2} }[/math]

For the Klein–Gordon equation, the general form of the Schrödinger equation is inconvenient to use, and in practice the Hamiltonian is not expressed in an analogous way to the Dirac Hamiltonian. The equations for relativistic quantum fields can be obtained in other ways, such as starting from a Lagrangian density and using the Euler–Lagrange equations for fields, or use the representation theory of the Lorentz group in which certain representations can be used to fix the equation for a free particle of given spin (and mass).

对于 Klein-Gordon 方程来说,薛定谔方程的一般形式不便于使用,而且在实践中,哈密顿量并没有以类似于 Dirac 哈密顿量的方式来表示。相对论量子场的方程可以通过其他方式得到,比如从拉格朗日密度出发,利用场的 Euler-Lagrange 方程,或者利用洛伦兹群的表示论,在洛伦兹群中某些表示可以用来修正给定自旋(和质量)的自由粒子的方程。


The kinetic energy is also proportional to the second spatial derivatives, so it is also proportional to the magnitude of the curvature of the wave, in terms of operators:

动能也与空间二阶导数成正比,因此它也与波的‘曲率’的大小成正比,就算符而言:

In general, the Hamiltonian to be substituted in the general Schrödinger equation is not just a function of the position and momentum operators (and possibly time), but also of spin matrices. Also, the solutions to a relativistic wave equation, for a massive particle of spin , are complex-valued -component}} spinor fields.

一般来说,在一般薛定谔方程中被替换的哈密顿量不仅仅是位置和动量算符(可能还有时间)的函数,也是自旋矩阵的函数。此外,对于自旋的大质量粒子,相对论波动方程的解是复数分量}旋量场。


[math]\displaystyle{ \hat{T} \Psi = \frac{-\hbar^2}{2m}\nabla\cdot\nabla \Psi \, \propto \, \nabla^2 \Psi \,. }[/math]


The general equation is also valid and used in quantum field theory, both in relativistic and nonrelativistic situations. However, the solution is no longer interpreted as a "wave", but should be interpreted as an operator acting on states existing in a Fock space.

在相对论和非相对论情况下,广义方程在量子场论中也是有效的。然而,解不再被解释为“波” ,而应被解释为作用于存在于 Fock 空间中的状态的一个算符。

As the curvature increases, the amplitude of the wave alternates between positive and negative more rapidly, and also shortens the wavelength. So the inverse relation between momentum and wavelength is consistent with the energy the particle has, and so the energy of the particle has a connection to a wave, all in the same mathematical formulation.[37]

Wave and particle motion波与质点运动

The Schrödinger equation can also be derived from a first order form similar to the manner in which the Klein–Gordon equation can be derived from the Dirac equation. In 1D the first order equation is given by

薛定谔方程也可以从一阶形式导出,类似于从 Dirac 方程导出 Klein-Gordon 方程的方式。在一维情况下,一阶方程由

脚本错误:没有“Multiple image”这个模块。

[math]\displaystyle{ “数学显示屏” \begin{align} Schrödinger required that a [[wave packet]] solution near position \lt math\gt \mathbf{r} }[/math] with wave vector near [math]\displaystyle{ \mathbf{k} }[/math] will move along the trajectory determined by classical mechanics for times short enough for the spread in [math]\displaystyle{ \mathbf{k} }[/math] (and hence in velocity) not to substantially increase the spread in r. Since, for a given spread in k, the spread in velocity is proportional to Planck's constant [math]\displaystyle{ \hbar }[/math], it is sometimes said that in the limit as [math]\displaystyle{ \hbar }[/math] approaches zero, the equations of classical mechanics are restored from quantum mechanics.[42] Great care is required in how that limit is taken, and in what cases.

薛定谔要求一个靠近位置[math]\displaystyle{ \mathbf{r} }[/math]wave packet解,其波矢量在[math]\displaystyle{ \mathbf{k} }[/math]附近,将沿着经典力学确定的轨迹移动足够短的时间,以使[math]\displaystyle{ \mathbf{k} }[/math]中的传播(因此在速度上)不会实质性地增加[math]\displaystyle{ \mathbf{k} }[/math]中的传播模板:数学。因为,对于{math |k}中的给定扩散,速度扩散与普朗克常数成正比,所以有时有人说,当[math]\displaystyle{ \hbar }[/math]接近零时,经典力学的方程就从量子力学中恢复了。[42] 在如何确定极限以及在何种情况下,需要非常小心。

\eta^2=0 \\

2 = 0


(\eta^\dagger)^2=0 \\

(eta ^ dagger) ^ 2 = 0

The limiting short-wavelength is equivalent to [math]\displaystyle{ \hbar }[/math] tending to zero because this is limiting case of increasing the wave packet localization to the definite position of the particle (see images right). Using the Heisenberg uncertainty principle for position and momentum, the products of uncertainty in position and momentum become zero as [math]\displaystyle{ \hbar \longrightarrow 0 }[/math]:

极限短波长相当于趋向于零的[math]\displaystyle{ \hbar }[/math],因为这是将波包局部化增加到粒子的确定位置的极限情况(见右图)。使用Heisenberg测不准原理计算位置和动量,位置和动量的不确定度乘积变为零,如[math]\displaystyle{ \hbar\longrightarrow 0 }[/math]

\left\lbrace \eta, \eta^\dagger \right\rbrace= 2 I


\end{align}


[math]\displaystyle{ \sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\! }[/math]

</math>



where σ denotes the (root mean square) measurement uncertainty in x and px (and similarly for the y and z directions) which implies the position and momentum can only be known to arbitrary precision in this limit.

其中{math |σ'}表示{math |x'}和{math |px'}方向上的(均方根)测量不确定度,这意味着位置和动量只能在该极限下以任意精度已知。

The 3 dimensional version of the equation is given by

这个方程的三维形式是由


One simple way to compare classical to quantum mechanics is to consider the time-evolution of the expected position and expected momentum, which can then be compared to the time-evolution of the ordinary position and momentum in classical mechanics. The quantum expectation values satisfy the Ehrenfest theorem. For a one-dimensional quantum particle moving in a potential [math]\displaystyle{ V }[/math], the Ehrenfest theorem says[43]

比较经典力学和量子力学的一个简单方法是考虑“预期”位置和“预期”动量的时间演化,然后可以将其与经典力学中普通位置和动量的时间演化进行比较。量子期望值满足Ehrenfest定理。Ehrenfest定理说,对于一维量子粒子在势中运动[44]

[math]\displaystyle{ “数学显示屏” :\lt math\gt m\frac{d}{dt}\langle x\rangle = \langle p\rangle;\quad \frac{d}{dt}\langle p\rangle = -\left\langle V'(X)\right\rangle . }[/math]

\begin{align}


Although the first of these equations is consistent with the classical behavior, the second is not: If the pair [math]\displaystyle{ (\langle X\rangle,\langle P\rangle) }[/math] were to satisfy Newton's second law, the right-hand side of the second equation would have to be

尽管这些方程中的第一个与经典行为一致,但第二个却不一致:如果这对[math]\displaystyle{ (\langle X\rangle,\langle P\rangle) }[/math]要满足牛顿第二定律,第二个方程的右边就必须是

-i \gamma_i \partial_i \psi = (i \eta \partial_t + \eta^\dagger m) \psi

- i gamma _ i partial _ i psi = (i eta partial _ t + eta ^ dagger m) psi

[math]\displaystyle{ -V'\left(\left\langle X\right\rangle\right) }[/math],

\end{align}


which is typically not the same as [math]\displaystyle{ -\left\langle V'(X)\right\rangle }[/math]. In the case of the quantum harmonic oscillator, however, [math]\displaystyle{ V' }[/math] is linear and this distinction disappears, so that in this very special case, the expected position and expected momentum do exactly follow the classical trajectories.

通常与[math]\displaystyle{ -\left\langle V'(X)\right\rangle不同。然而,在量子谐振子的情况下,\lt math\gt V' }[/math]是线性的,这种区别消失了,所以在这个非常特殊的情况下,期望位置和期望动量确实遵循经典轨迹。

</math>


For general systems, the best we can hope for is that the expected position and momentum will approximately follow the classical trajectories. If the wave function is highly concentrated around a point [math]\displaystyle{ x_0 }[/math], then [math]\displaystyle{ V'\left(\left\langle X\right\rangle\right) }[/math] and [math]\displaystyle{ \left\langle V'(X)\right\rangle }[/math] will be almost the same, since both will be approximately equal to [math]\displaystyle{ V'(x_0) }[/math]. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least for as long as the wave function remains highly localized in position.[45] When Planck's constant is small, it is possible to have a state that is well localized in both position and momentum. The small uncertainty in momentum ensures that the particle remains well localized in position for a long time, so that expected position and momentum continue to closely track the classical trajectories.

对于一般系统,我们所能期望的最好结果是,预期的位置和动量将“近似”地遵循经典轨迹。如果波函数高度集中在一个点上,那么[math]\displaystyle{ V'\left(\left\langle x\right\rangle\right) }[/math][math]\displaystyle{ \left\langle V'(x)\right\rangle }[/math]将“几乎”相同,因为两者都将近似等于[math]\displaystyle{ V'(x\u 0) }[/math]。在这种情况下,预期位置和预期动量将保持非常接近经典轨迹,至少只要波函数在位置上保持高度局部化。[46]当普朗克常数很小时,有可能有一个状态在“位置和动量”都很好地局部化。动量的小不确定性确保了粒子在很长一段时间内“保持”良好的局部化位置,因此预期的位置和动量继续与经典轨迹密切相关。

Here [math]\displaystyle{ \eta=(\gamma_0+i \gamma_5)/\sqrt{2} }[/math] is a [math]\displaystyle{ 4 \times 4 }[/math] nilpotent matrix and [math]\displaystyle{ \gamma_i }[/math] are the Dirac gamma matrices ([math]\displaystyle{ i=1,2,3 }[/math]). The Schrödinger equation in 3D can be obtained by squaring the above equation.

这里 < math > eta = (gamma _ 0 + i gamma _ 5)/sqrt {2} </math > 是 < math > > 4 </math > 幂零矩阵和 < math > gamma </math > 是 Dirac gamma 矩阵(< math > i = 1,2,3 </math >)。3 d 中的薛定谔方程可以通过平方上面的方程得到。


In the nonrelativistic limit [math]\displaystyle{ E-m \simeq E' }[/math] and [math]\displaystyle{ E+m \simeq 2m }[/math], the above equation can be derived from the Dirac equation.

在非相对论性极限 < math > E-m simeq e’ </math > 和 < math > e + m simeq 2m </math > 中,上述方程可由 Dirac 方程导出。

The Schrödinger equation in its general form

薛定谔方程的一般形式

[math]\displaystyle{ i\hbar \frac{\partial}{\partial t} \Psi\left(\mathbf{r},t\right) = \hat{H} \Psi\left(\mathbf{r},t\right) \,\! }[/math]


is closely related to the Hamilton–Jacobi equation (HJE)

汉密尔顿-雅可比方程密切相关(HJE)

[math]\displaystyle{ -\frac{\partial}{\partial t} S(q_i,t) = H\left(q_i,\frac{\partial S}{\partial q_i},t \right) \,\! }[/math]


where [math]\displaystyle{ S }[/math] is the classical action and [math]\displaystyle{ H }[/math] is the Hamiltonian function (not operator). Here the generalized coordinates [math]\displaystyle{ q_i }[/math] for [math]\displaystyle{ i = 1, 2, 3 }[/math] (used in the context of the HJE) can be set to the position in Cartesian coordinates as [math]\displaystyle{ \mathbf{r} = (q_1, q_2, q_3) = (x, y, z) }[/math].[42]

其中,[math]\displaystyle{ S }[/math]是经典的作用,而[math]\displaystyle{ H }[/math]哈密顿函数(非算符)。这里,[math]\displaystyle{ i=1,2,3 }[/math]generalized coordinates[math]\displaystyle{ qu i }[/math](在HJE上下文中使用)可以设置为笛卡尔坐标中的位置,如[math]\displaystyle{ \mathbf{r}=(qu 1,qu 2,qu 3)=(x,y,z) }[/math][42]


Substituting

替代

[math]\displaystyle{ \Psi = \sqrt{\rho(\mathbf{r},t)} e^{iS(\mathbf{r},t)/\hbar}\,\! }[/math]


where [math]\displaystyle{ \rho }[/math] is the probability density, into the Schrödinger equation and then taking the limit [math]\displaystyle{ \hbar \longrightarrow 0 }[/math] in the resulting equation yield the Hamilton–Jacobi equation.

其中,[math]\displaystyle{ \rho }[/math]是概率密度,放入薛定谔方程,然后在得到的方程中取极限值[math]\displaystyle{ \hbar\longrightarrow 0 }[/math],得到汉密尔顿-雅可比方程。


The implications are as follows:

其含义如下:

  • The motion of a particle, described by a (short-wavelength) wave packet solution to the Schrödinger equation, is also described by the Hamilton–Jacobi equation of motion.
  • 粒子的运动由薛定谔方程的(短波长)波包解描述,也由哈密顿-雅可比运动方程描述。
  • The Schrödinger equation includes the wave function, so its wave packet solution implies the position of a (quantum) particle is fuzzily spread out in wave fronts. On the contrary, the Hamilton–Jacobi equation applies to a (classical) particle of definite position and momentum, instead the position and momentum at all times (the trajectory) are deterministic and can be simultaneously known.
  • 薛定谔方程包含了波函数,所以它的波包解意味着(量子)粒子的位置在波前是模糊分布的。相反,哈密顿-雅可比方程适用于具有确定位置和动量的(经典)粒子,相反,任何时候的位置和动量(轨迹)都是确定的,并且可以同时知道。

Nonrelativistic quantum mechanics非相对论量子力学

The quantum mechanics of particles without accounting for the effects of special relativity, for example particles propagating at speeds much less than light, is known as nonrelativistic quantum mechanics. Following are several forms of Schrödinger's equation in this context for different situations: time independence and dependence, one and three spatial dimensions, and one and N particles.

不考虑狭义相对论影响的粒子量子力学,例如以远低于光速的速度传播的粒子,被称为“非相对论量子力学”。以下是薛定谔方程在不同情况下的几种形式:时间独立性和依赖性,一个和三个空间维度,以及一个和{数学|'}粒子。


In actuality, the particles constituting the system do not have the numerical labels used in theory. The language of mathematics forces us to label the positions of particles one way or another, otherwise there would be confusion between symbols representing which variables are for which particle.[39]

实际上,组成系统的粒子没有理论上使用的数字标签。数学语言迫使我们以这样或那样的方式标注粒子的位置,否则在表示哪个变量是哪个粒子的符号之间会有混淆。[39]


Time-independent时间无关性

If the Hamiltonian is not an explicit function of time, the equation is separable into a product of spatial and temporal parts. In general, the wave function takes the form:

如果哈密顿量不是时间的显式函数,则方程是可分离空间和时间部分的乘积。一般而言,波函数的形式如下:

[math]\displaystyle{ \Psi(\text{space coords},t)=\psi(\text{space coords})\tau(t)\,. }[/math]


where ψ(space coords) is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and τ(t) is a function of time only.

其中{math |ψ(空间坐标)}}仅是构成系统的粒子的所有空间坐标的函数,{math |τ('t')}}仅是时间的函数。


Substituting for ψ into the Schrödinger equation for the relevant number of particles in the relevant number of dimensions, solving by separation of variables implies the general solution of the time-dependent equation has the form:[25]

将{数学|ψ}}代入薛定谔方程中,得到相关维数中的相关粒子数,通过变量分离求解意味着含时方程的通解具有以下形式:[25]

[math]\displaystyle{ \Psi(\text{space coords},t) = \psi(\text{space coords}) e^{-i{E t/\hbar}} \,. }[/math]


Since the time dependent phase factor is always the same, only the spatial part needs to be solved for in time independent problems. Additionally, the energy operator Ê = 模板:Sfrac can always be replaced by the energy eigenvalue E, thus the time independent Schrödinger equation is an eigenvalue equation for the Hamiltonian operator:[5]:143ff

由于与时间相关的相位因子总是相同的,所以对于与时间无关的问题,只需要求解空间部分。此外,能量算符Ê=iħ{sfrac }}总是可以被能量本征值{math | E'}替换,因此与时间无关的薛定谔方程是哈密顿算符的本征值方程:[5]:143ff

[math]\displaystyle{ \hat{H} \psi = E \psi }[/math]


This is true for any number of particles in any number of dimensions (in a time independent potential). This case describes the standing wave solutions of the time-dependent equation, which are the states with definite energy (instead of a probability distribution of different energies). In physics, these standing waves are called "stationary states" or "energy eigenstates"; in chemistry they are called "atomic orbitals" or "molecular orbitals". Superpositions of energy eigenstates change their properties according to the relative phases between the energy levels.

这是真实的任何数量的粒子在任何数量的维度(在一个时间无关的潜力)。这种情况描述了含时方程的驻波解,即具有确定能量的状态(而不是不同能量的概率分布)。在物理学中,这些驻波被称为“稳态s”或“能量本征态s”;在化学中,它们被称为“原子轨道s”或“分子轨道s”。能量本征态的叠加根据能级间的相对相位而改变其性质。

The energy eigenvalues from this equation form a discrete spectrum of values, so mathematically energy must be quantized. More specifically, the energy eigenstates form a basis – any wave function may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure. This is the spectral theorem in mathematics, and in a finite state space it is just a statement of the completeness of the eigenvectors of a Hermitian matrix.

这个方程的能量特征值形成了一个离散的值,因此在数学上能量必须量子化。更具体地说,能量本征态构成了一个基础——任何波函数都可以写为离散能量态的和或连续能量态的积分,或者更一般地写为测度的积分。这就是数学中的谱定理,在有限状态空间中,它只是厄米矩阵特征向量完备性的一种表述。

One-dimensional examples一维例子

For a particle in one dimension, the Hamiltonian is:

对于一维粒子,哈密顿量为:

[math]\displaystyle{ \hat{H} = \frac{\hat{p}^2}{2m} + V(x) \,, \quad \hat{p} = -i\hbar \frac{d}{d x} }[/math]


and substituting this into the general Schrödinger equation gives:

把它代入一般的薛定谔方程得到:

[math]\displaystyle{ \left[-\frac{\hbar^2}{2m}\frac{d^2}{d x^2} + V(x) \right]\psi(x) = E\psi(x) }[/math]


This is the only case the Schrödinger equation is an ordinary differential equation, rather than a partial differential equation. The general solutions are always of the form:

这是薛定谔方程是一个微分方程,而不是一个微分方程的唯一情况。一般解决方案的形式总是:

[math]\displaystyle{ \Psi(x,t)=\psi(x) e^{-iEt/\hbar} \, . }[/math]


Category:Differential equations

类别: 微分方程

For N particles in one dimension, the Hamiltonian is:

Category:Partial differential equations

类别: 偏微分方程


Category:Wave mechanics

类别: 波动力学

[math]\displaystyle{ \hat{H} = \sum_{n=1}^{N}\frac{\hat{p}_n^2}{2m_n} + V(x_1,x_2,\ldots, x_N) \,,\quad \hat{p}_n = -i\hbar \frac{\partial}{\partial x_n} }[/math]

Category:Functions of space and time

类别: 空间和时间的作用


This page was moved from wikipedia:en:Schrödinger equation. Its edit history can be viewed at 薛定谔方程/edithistory

此页摘自维基百科:英文:薛定谔方程。其编辑历史记录可在薛定谔方程/编辑历史查阅

  1. Griffiths, David J. (2004), Introduction to Quantum Mechanics (2nd ed.), Prentice Hall, ISBN 978-0-13-111892-8
  2. "Physicist Erwin Schrödinger's Google doodle marks quantum mechanics work". The Guardian. 13 August 2013. Retrieved 25 August 2013.
  3. The Schrödinger equation is a linear partial differential equation that describes the wave function or state function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of the subject. The equation is named after Erwin Schrödinger, who postulated the equation in 1925, and published it in 1926, forming the basis for the work that resulted in his Nobel Prize in Physics in 1933. 薛定谔方程Schrödinger equation是一个描述量子力学系统的波函数或态函数的线性偏微分方程。这是21量子力学的一个关键成果,它的发现也是这门学科发展史上的一个重要里程碑。这个方程是以埃尔温·薛定谔的名字命名的,他在1925年假设了这个方程,并在1926年发表了这个方程,为他的工作奠定了基础,并最终在1933年获得了诺贝尔物理学奖。 Schrödinger, E. In classical mechanics, Newton's second law ( ma. line feed character in |first= at position 3 (help); Cite journal requires |journal= (help); Missing or empty |title= (help)) is used to make a mathematical prediction as to what path a given physical system will take over time following a set of known initial conditions. Solving this equation gives the position and the momentum of the physical system as a function of the external force [math]\displaystyle{ \mathbf{F} }[/math] on the system. Those two parameters are sufficient to describe its state at each time instant. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation. 在经典力学,牛顿第二定律(ma })被用来做一个数学预测,来预测一个给定的物理系统在遵循一系列已知的初始条件的情况下会经历什么样的路径。解这个方程给出了物理系统的位置和动量作为系统外力的函数。这两个参数足以描述它在每个时刻的状态。在20世纪量子力学,牛顿定律的类似物是薛定谔方程。 | title = An Undulatory Theory of the Mechanics of Atoms and Molecules | url = http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf The concept of a wave function is a fundamental postulate of quantum mechanics; the wave function defines the state of the system at each spatial position and time. Using these postulates, Schrödinger's equation can be derived from the fact that the time-evolution operator must be unitary, and must therefore be generated by the exponential of a self-adjoint operator, which is the quantum Hamiltonian. This derivation is explained below. 波函数的概念是量子力学的基本假设; 波函数定义了系统在每个空间位置和时间的状态。根据这些假设, 薛定谔方程可以从时间演化算符必须是幺正的这一事实中推导出来,因此必须是由量子哈密顿自共轭算符self-adjoint operator的指数生成的。下面将解释这个推导过程。 | archiveurl = https://web.archive.org/web/20081217040121/http://home.tiscali.nl/physis/HistoricPaper/Schroedinger/Schroedinger1926c.pdf | archivedate = 17 December 2008 In the Copenhagen interpretation of quantum mechanics, the wave function is the most complete description that can be given of a physical system. Solutions to Schrödinger's equation describe not only molecular, atomic, and subatomic systems, but also macroscopic systems, possibly even the whole universe. 在量子力学的哥本哈根解释中,波函数是对物理系统最完整的描述。薛定谔方程的解不仅描述分子、原子和亚原子系统,而且描述宏观系统,甚至可能描述整个宇宙。 | journal = Physical Review | volume = 28 | issue = 6 | pages = 1049–1070 The Schrödinger equation is not the only way to study quantum mechanical systems and make predictions. The other formulations of quantum mechanics include matrix mechanics, introduced by Werner Heisenberg, and the path integral formulation, developed chiefly by Richard Feynman. Paul Dirac incorporated matrix mechanics and the Schrödinger equation into a single formulation. 薛定谔方程并不是研究量子力学系统和进行预测的唯一方法。量子力学的其他公式包括维尔纳·海森堡提出的矩阵力学和理查德·费曼提出的路径积分公式。保罗狄拉克将矩阵力学和薛定谔方程合并到一个公式中。 | year = 1926 | doi = 10.1103/PhysRev.28.1049 |bibcode = 1926PhRv...28.1049S }}
  4. Laloe, Franck (2012), Do We Really Understand Quantum Mechanics, Cambridge University Press, ISBN 978-1-107-02501-1
  5. 5.0 5.1 5.2 where [math]\displaystyle{ i }[/math] is the imaginary unit, [math]\displaystyle{ \hbar = \frac{h}{2 \pi} }[/math] is the reduced Planck constant having the dimension of action, [math]\displaystyle{ \Psi }[/math] (the Greek letter psi) is the state vector of the quantum system, [math]\displaystyle{ t }[/math] is time, and [math]\displaystyle{ \hat{H} }[/math] is the Hamiltonian operator. The position-space wave function of the quantum system is nothing but the components in the expansion of the state vector in terms of the position eigenvector [math]\displaystyle{ \vert\mathbf{r}\rangle }[/math]. It is a scalar function, expressed as [math]\displaystyle{ \Psi(\mathbf{r},t) = \langle \mathbf{r}\vert \Psi \rangle }[/math]. Similarly, the momentum-space wave function can be defined as [math]\displaystyle{ \tilde\Psi(\mathbf{p},t) = \langle \mathbf{p}\vert \Psi \rangle }[/math], where [math]\displaystyle{ \vert\mathbf{p}\rangle }[/math] is the momentum eigenvector. 这里 < math > i </math > 是虚单位,< math > hbar = frac { h }{2 pi } </math > 是具有作用维度的减少的普朗克常数,< math > Psi </math > (希腊字母 Psi)是量子系统的状态向量,< math > t </math > 是时间,< math > hat { h } </math > 是 Hamiltonian 算符。量子系统的位置空间波函数不过是状态矢量展开式中的位置本征矢量的分量。它是一个标量函数,表示为 < math > Psi (mathbf { r } ,t) = langle mathbf { r } vert Psi rangle </math > 。类似地,动量空间波函数可以定义为 < math > tilde Psi (mathbf { p } ,t) = langle mathbf { p } vert Psi rangle </math > ,其中 < math > vert mathbf { p } rangle </math > 是动量本征向量。 Shankar, R. harmonic oscillator. Left: The real part (blue) and imaginary part (red) of the wave function. Right: The probability distribution of finding the particle with this wave function at a given position. The top two rows are examples of stationary states, which correspond to standing waves. The bottom row is an example of a state which is not a stationary state. The right column illustrates why stationary states are called "stationary".]] 谐振子。左图: 波函数的实部(蓝色)和虚部(红色)。右图: 在给定位置找到具有这个波函数的粒子的概率分布。上面的两排是定态的例子,它们对应于驻波。下面一行是一个不是定态的州的例子。右边一栏说明了为什么定态被称为“定态”。]] (1926). [http://gallica.bnf.fr/ark:/12148/bpt6k153811.image.langFR.f373.pagination Quantisierung als Eigenwertproblem; von Erwin Schrödinger The case [math]\displaystyle{ n = 0 }[/math] is called the ground state, its energy is called the zero-point energy, and the wave function is a Gaussian. 这种情况下,[math]\displaystyle{ n = 0 }[/math] 被称为基态,它的能量被称为零点能量,波函数是高斯函数。]. 384 (2nd
    位置基础上随时间变化的薛定谔方程(单个非相对论粒子)

    [math]\displaystyle{ z=re^{i\phi}=x+iy \,\! }[/math]

  6. 6.0 6.1 6.2 P. R. Bunker; Per Jensen
    与时间无关的薛定谔方程

    < math > operatorname { hat h }

  7. "Schrodinger equation". Hyperphysics. Department of Physics and Astronomy, George State University.
  8. "Schrodinger equation". Hyperphysics. Department of Physics and Astronomy, George State University.
  9. 9.0 9.1 Nouredine Zettili (17 February 2009). Quantum Mechanics: Concepts and Applications. John Wiley & Sons. ISBN 978-0-470-02678-6. 
  10. 10.0 10.1 Ballentine, Leslie (1998), Quantum Mechanics: A Modern Development, World Scientific Publishing Co., ISBN 978-9810241056
  11. Schrödinger, Erwin (1995). The interpretation of quantum mechanics: Dublin seminars (1949–1955) and other unpublished essays. Ox Bow Press. ISBN 9781881987086. 
  12. Schrödinger, Erwin (1995). The interpretation of quantum mechanics: Dublin seminars (1949–1955) and other unpublished essays. Ox Bow Press. ISBN 9781881987086. 
  13. David Deutsch, The Beginning of Infinity, page 310
  14. Barrett, Jeffrey A. (1999). The Quantum Mechanics of Minds and Worlds. Oxford University Press. pp. 63. ISBN 9780191583254. 
  15. David Deutsch, The Beginning of Infinity, page 310
  16. Barrett, Jeffrey A. (1999). The Quantum Mechanics of Minds and Worlds. Oxford University Press. pp. 63. ISBN 9780191583254. 
  17. de Broglie This 1926 paper was enthusiastically endorsed by Einstein, who saw the matter-waves as an intuitive depiction of nature, as opposed to Heisenberg's matrix mechanics, which he considered overly formal. 这篇1926年的论文得到了 Einstein 的热烈支持,他认为物质波是对自然的直观描述,而不是 Heisenberg 的矩阵力学,他认为这种描述过于正式。, L. (1925 The Schrödinger equation details the behavior of [math]\displaystyle{ \Psi }[/math] but says nothing of its nature. Schrödinger tried to interpret it as a charge density in his fourth paper, but he was unsuccessful. In 1926, just a few days after Schrödinger's fourth and final paper was published, Max Born successfully interpreted [math]\displaystyle{ \Psi }[/math] as the probability amplitude, whose modulus squared is equal to probability density. The Schrödinger equation is therefore a new concept in itself; as Feynman put it: 薛定谔方程卫生组织详细描述了超能力的行为,但对其本质却只字未提。薛定谔在他的第四篇论文中试图将其解释为电荷密度,但他没有成功。1926年,就在薛定谔的第四篇也是最后一篇论文发表的几天之后,Max Born 成功地将《 math > Psi </math > 》解释为机率幅,其模数的平方等于概率密度。因此,薛定谔方程本身就是一个新概念,正如 Feynman 所说:). [https://web.archive.org/web/20090509012910/http://www.ensmp.fr/aflb/LDB-oeuvres/De_Broglie_Kracklauer.pdf where [math]\displaystyle{ d^3 \mathbf{k} = dk_x \,dk_y 在哪里? d ^ 3 mathbf { k } = dk x,dk y |archivedate = 9 May 2009 \,dk_z }[/math]is the differential volume element in -space, and the integrals are taken over all [math]\displaystyle{ \mathbf{k} }[/math]-space. The momentum wave function [math]\displaystyle{ \Phi (\mathbf{k}) }[/math] arises in the integrand since the position and momentum space wave functions are Fourier transforms of each other. ,dk _ z </math > 是空间中的微分体积元,并且所有的积分都是在整个 < math > mathbf { k } </math >-space 中进行的。动量波函数 < math > Phi (mathbf { k }) </math > 在被积函数中出现,因为位置波函数和动量空间波函数是相互之间的傅里叶变换。 "Recherches sur la théorie des quanta"] Check |archiveurl= value (help) [On the Theory of Quanta]. Annales de Physique The foundation of the equation is structured to be a linear differential equation based on classical energy conservation, and consistent with the De Broglie relations. The solution is the wave function , which contains all the information that can be known about the system. In the Copenhagen interpretation, the modulus of is related to the probability the particles are in some spatial configuration at some instant of time. Solving the equation for can be used to predict how the particles will behave under the influence of the specified potential and with each other. 这个方程的基础是一个基于经典能量守恒的线性微分方程,与德布罗意关系是一致的。解决方案是波函数,它包含了所有可以知道的关于系统的信息。在哥本哈根诠释中,粒子的模数与粒子在某一时刻处于某种空间位置的概率有关。求解方程可以用来预测粒子在指定势能的影响下以及相互影响下的行为。. 10 (3 The Schrödinger equation was developed principally from the De Broglie hypothesis, a wave equation that would describe particles, and can be constructed as shown informally in the following sections. For a more rigorous description of Schrödinger's equation, see also Resnick et al. 薛定谔方程理论主要是从德布罗意假说发展而来的,德布罗意假说是一个描述粒子的波动方程,可以在下面的章节中非正式地构造出来。关于薛定谔方程更严格的描述,参见雷斯尼克等人的文章。): 22–128. Bibcode:1925AnPh...10...22D. doi:[//doi.org/10.1051%2Fanphys%2F192510030022%0A%0A%7F%27%22%60UNIQ--math-0000016C-QINU%60%22%27%7F%0A%0A%3C+math+%3E+Psi+%28mathbf+%7B+r+%7D+%EF%BC%8Ct%29+%3D+frac+%7B1%7D%7B+left+%28sqrt+%7B2+pi+%7D+%EF%BC%8Cright%29+%5E+3%7D+int+Phi+%28mathbf+%7B+k+%7D%29+e+%5E+%7B+i+%28mathbf+%7B+k+%7D+cdot+mathbf+%7B+r+%7D-omega+t%29%7D+d+%5E+3+mathbf+%7B+k+%7D+%EF%BC%8C%21+%3C%2Fmath+%3E 10.1051/anphys/192510030022 '"`UNIQ--math-0000016C-QINU`"' < math > Psi (mathbf { r } ,t) = frac {1}{ left (sqrt {2 pi } ,right) ^ 3} int Phi (mathbf { k }) e ^ { i (mathbf { k } cdot mathbf { r }-omega t)} d ^ 3 mathbf { k } ,! </math >] Check |doi= value (help). Archived from the original (PDF) on |archive-url= requires |archive-date= (help). line feed character in |archiveurl= at position 106 (help); line feed character in |last= at position 11 (help); line feed character in |doi= at position 28 (help); line feed character in |journal= at position 24 (help); line feed character in |issue= at position 2 (help); line feed character in |year= at position 5 (help); Check date values in: |year= (help)CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link) .
  18. Weissman, M.B.; V. V. Iliev; I. Gutman (2008). "A pioneer remembered: biographical notes about Arthur Constant Lunn". Communications in Mathematical and in Computer Chemistry. 59 (3): 687–708.
  19. Samuel I. Weissman; Michael Weissman (1997). "Alan Sokal's Hoax and A. Lunn's Theory of Quantum Mechanics". Physics Today. 50, 6 (6): 15. Bibcode:1997PhT....50f..15W. doi:10.1063/1.881789.
  20. Kamen, Martin D. (1985). Radiant Science, Dark Politics. Berkeley and Los Angeles, California: University of California Press. pp. 29–32. ISBN 978-0-520-04929-1. https://archive.org/details/radiantscienceda00kame/page/29. 
  21. Weissman, M.B.; V. V. Iliev; I. Gutman (2008). "A pioneer remembered: biographical notes about Arthur Constant Lunn". Communications in Mathematical and in Computer Chemistry. 59 (3): 687–708.
  22. Samuel I. Weissman; Michael Weissman (1997). "Alan Sokal's Hoax and A. Lunn's Theory of Quantum Mechanics". Physics Today. 50, 6 (6): 15. Bibcode:1997PhT....50f..15W. doi:10.1063/1.881789.
  23. Kamen, Martin D. (1985). Radiant Science, Dark Politics. Berkeley and Los Angeles, California: University of California Press. pp. 29–32. ISBN 978-0-520-04929-1. https://archive.org/details/radiantscienceda00kame/page/29. 
  24. [math]\displaystyle{ -V'\left(\left\langle X\right\rangle\right) }[/math], [ math ]-v’ left (left langle x right rangle) {{Cite book which is typically not the same as [math]\displaystyle{ -\left\langle V'(X)\right\rangle }[/math]. In the case of the quantum harmonic oscillator, however, [math]\displaystyle{ V' }[/math] is linear and this distinction disappears, so that in this very special case, the expected position and expected momentum do exactly follow the classical trajectories. 这通常不同于 < math >-left langle v’(x) right rangle </math > 。然而,在量子谐振子的例子中,< math > v’ </math > 是线性的,这种区别消失了,所以在这个特殊的例子中,预期的位置和预期的动量确实遵循了经典的轨迹。 |last=Schrödinger |first=E. |year=1984 For general systems, the best we can hope for is that the expected position and momentum will approximately follow the classical trajectories. If the wave function is highly concentrated around a point [math]\displaystyle{ x_0 }[/math], then [math]\displaystyle{ V'\left(\left\langle X\right\rangle\right) }[/math] and [math]\displaystyle{ \left\langle V'(X)\right\rangle }[/math] will be almost the same, since both will be approximately equal to [math]\displaystyle{ V'(x_0) }[/math]. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least for as long as the wave function remains highly localized in position. When Planck's constant is small, it is possible to have a state that is well localized in both position and momentum. The small uncertainty in momentum ensures that the particle remains well localized in position for a long time, so that expected position and momentum continue to closely track the classical trajectories. 对于一般系统,我们所能期望的最好结果是,预期的位置和动量将近似遵循传统的轨迹。如果波函数高度集中在一个点附近,那么“ math”“ v”左(左小角 x 右小角)和“ math”“左小角 v’(x)右小角 v’(x)右小角 </math > 几乎是一样的,因为两者大致相等于 math < v’(x _ 0) </math > 。在这种情况下,预期的位置和预期的动量将保持非常接近传统的轨迹,至少在波函数仍然高度局部化的位置。当普朗克常数很小时,就可能有一个位置和动量都很好定域的状态。动量的微小不确定性保证了粒子在很长一段时间内保持很好的局部位置,从而使预期的位置和动量继续紧密地跟踪经典的轨迹。 |title=Collected papers |publisher=Friedrich Vieweg und Sohn The Schrödinger equation in its general form 薛定谔方程的一般形式 |isbn=978-3-7001-0573-2 }} See introduction to first 1926 paper.
  25. 25.0 25.1 25.2 Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.)
  26. [math]\displaystyle{ V(x_1,x_2,\ldots, x_N) = \sum_{n=1}^N V(x_n) \, . }[/math] V (x _ 1,x _ 2,ldots,x _ n) = sum _ { n = 1} ^ n v (x _ n) ,. </math > Sommerfeld, A. and the wave function can be written as a product of the wave functions for each particle: 波函数可以写成每个粒子的波函数的乘积: (1919). Atombau und Spektrallinien [math]\displaystyle{ \Psi(x_1,x_2,\ldots, x_N,t) = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(x_n) \, , }[/math] < math > Psi (x _ 1,x _ 2,ldots,x _ n,t) = e ^ {-i { e t/hbar } prod _ { n = 1} ^ n (x _ n) ,</math >. Braunschweig For non-interacting identical particles, the potential is still a sum, but wave function is a bit more complicated – it is a sum over the permutations of products of the separate wave functions to account for particle exchange. In general for interacting particles, the above decompositions are not possible. 对于没有相互作用的全同粒子,势仍然是一个和,但波函数要复杂一些——它是单独波函数的置换乘积的和,用来解释粒子交换。一般来说,对于相互作用的粒子,上面的分解是不可能的。: Friedrich Vieweg und Sohn. ISBN 978-3-87144-484-5. 
  27. For an English source, see 有关英文来源,请参阅Haar, T. For no potential, 0. line feed character in |first= at position 3 (help); Cite journal requires |journal= (help); Missing or empty |title= (help), so the particle is free and the equation reads: and atoms or ions in lattices, and approximating other potentials near equilibrium points. It is also the basis of perturbation methods in quantum mechanics. 因为没有势,0. line feed character in |author2= at position 12 (help); Cite journal requires |journal= (help); Missing or empty |title= (help)CS1 maint: multiple names: authors list (link),所以粒子是自由的,方程是: 和晶格中的原子或离子,并且近似平衡点附近的其他势。这也是量子力学微扰法的基础。 ed.). pp. 361–377 The extension from one dimension to three dimensions is straightforward, all position and momentum operators are replaced by their three-dimensional expressions and the partial derivative with respect to space is replaced by the gradient operator. 从一维到三维的扩展是很直接的,所有的位置和动量算符都被它们的三维表达式所代替,关于空间的偏导数被梯度算符所代替。. Bibcode 1926AnP...384..361S. doi:10.1002/andp.19263840404. ISBN [[Special:BookSources/[math]\displaystyle{ \psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{ \lt math \gt psi _ n (x) = sqrt { frac {1}{2 ^ n,n! }Cdot left (frac { m omega }{ pi hbar } right) ^ {1/4} cdot e ^ { }}\lt /span\gt \lt /li\gt \lt li id="cite_note-34"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-34|↑]]\lt /span\gt \lt span class="reference-text"\gt {{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}\lt /span\gt \lt /li\gt \lt li id="cite_note-35"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-35|↑]]\lt /span\gt \lt span class="reference-text"\gt {{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}\lt /span\gt \lt /li\gt \lt li id="cite_note-Schr.C3.B6dinger1982-36"\gt \lt span class="mw-cite-backlink"\gt ↑ \lt sup\gt [[#cite_ref-Schr.C3.B6dinger1982_36-0|30.0]]\lt /sup\gt \lt sup\gt [[#cite_ref-Schr.C3.B6dinger1982_36-1|30.1]]\lt /sup\gt \lt /span\gt \lt span class="reference-text"\gt {{cite book|author=Erwin Schrödinger|title=Collected Papers on Wave Mechanics: Third Edition|year=1982|publisher=American Mathematical Soc.|isbn=978-0-8218-3524-1}}\lt /span\gt \lt /li\gt \lt li id="cite_note-.E2.80.9CSchr.C3.B6dinger1982.E2.80.9D-37"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-.E2.80.9CSchr.C3.B6dinger1982.E2.80.9D_37-0|↑]]\lt /span\gt \lt span class="error mw-ext-cite-error" lang="zh-Hans-CN" dir="ltr"\gt 引用错误:无效\lt code\gt <ref>\lt /code\gt 标签;未给name属性为\lt code\gt “Schrödinger1982”\lt /code\gt 的引用提供文字\lt /span\gt \lt /li\gt \lt li id="cite_note-38"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-38|↑]]\lt /span\gt \lt span class="reference-text"\gt {{cite journal |last=Schrödinger |first=E. \lt math\gt E_n = \left(n + \frac{1}{2} \right) \hbar \omega. }[/math] < math > e _ n = left (n + frac {1}{2} right) hbar omega.数学|[math]\displaystyle{ \psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{ \lt math \gt psi _ n (x) = sqrt { frac {1}{2 ^ n,n! }Cdot left (frac { m omega }{ pi hbar } right) ^ {1/4} cdot e ^ { }}\lt /span\gt \lt /li\gt \lt li id="cite_note-34"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-34|↑]]\lt /span\gt \lt span class="reference-text"\gt {{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}\lt /span\gt \lt /li\gt \lt li id="cite_note-35"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-35|↑]]\lt /span\gt \lt span class="reference-text"\gt {{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}\lt /span\gt \lt /li\gt \lt li id="cite_note-Schr.C3.B6dinger1982-36"\gt \lt span class="mw-cite-backlink"\gt ↑ \lt sup\gt [[#cite_ref-Schr.C3.B6dinger1982_36-0|30.0]]\lt /sup\gt \lt sup\gt [[#cite_ref-Schr.C3.B6dinger1982_36-1|30.1]]\lt /sup\gt \lt /span\gt \lt span class="reference-text"\gt {{cite book|author=Erwin Schrödinger|title=Collected Papers on Wave Mechanics: Third Edition|year=1982|publisher=American Mathematical Soc.|isbn=978-0-8218-3524-1}}\lt /span\gt \lt /li\gt \lt li id="cite_note-.E2.80.9CSchr.C3.B6dinger1982.E2.80.9D-37"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-.E2.80.9CSchr.C3.B6dinger1982.E2.80.9D_37-0|↑]]\lt /span\gt \lt span class="error mw-ext-cite-error" lang="zh-Hans-CN" dir="ltr"\gt 引用错误:无效\lt code\gt <ref>\lt /code\gt 标签;未给name属性为\lt code\gt “Schrödinger1982”\lt /code\gt 的引用提供文字\lt /span\gt \lt /li\gt \lt li id="cite_note-38"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-38|↑]]\lt /span\gt \lt span class="reference-text"\gt {{cite journal |last=Schrödinger |first=E. \lt math\gt E_n = \left(n + \frac{1}{2} \right) \hbar \omega. }[/math] < math > e _ n = left (n + frac {1}{2} right) hbar omega.数学]]. http://gallica.bnf.fr/ark:/12148/bpt6k153811.image.langFR.f373.pagination. 
  28. generating the equation 生成方程式 Einstein, A. [math]\displaystyle{ \left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \right]\psi(\mathbf{r}) = E\psi(\mathbf{r}) }[/math] < math > left [-frac { hbar ^ 2}{2m } nabla ^ 2 + v (mathbf { r }) right ] psi (mathbf { r }) = e psi (mathbf { r }) </math >; et al. "Letters on Wave Mechanics: Schrodinger–Planck–Einstein–Lorentz with stationary state solutions of the form 以定态溶液的形式". line feed character in |title= at position 63 (help); line feed character in |first= at position 3 (help); Cite journal requires |journal= (help)
  29. 34.0 34.1 34.2 34.3 where the position of the particle is [math]\displaystyle{ \mathbf{r} }[/math]. 其中粒子的位置是 < math > mathbf { r } </math > 。 Moore, W.J. For [math]\displaystyle{ N }[/math] particles in three dimensions, the Hamiltonian is 对于三维空间中的粒子,哈密顿函数是 (1992). Schrödinger: Life and Thought [math]\displaystyle{ \hat{H} = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N) \,,\quad \hat{\mathbf{p}}_n = -i\hbar \nabla_n }[/math] 2 m n } + v (mathbf { r } _ 1,mathbf { r } _ 2,ldots,mathbf { r } _ n) ,4 h { p } _ n =-i hbar _ n </math >. Cambridge University Press. ISBN [[Special:BookSources/978-0-521-43767-7 where the position of particle is and the gradient operators are partial derivatives with respect to the particle's position coordinates. In Cartesian coordinates, for particle , the position vector is (xn, yn, zn)|978-0-521-43767-7 where the position of particle is and the gradient operators are partial derivatives with respect to the particle's position coordinates. In Cartesian coordinates, for particle , the position vector is (xn, yn, zn)]].  while the gradient and Laplacian operator are respectively: 其中粒子的位置是和梯度算子是偏导数的粒子的位置坐标。在笛卡尔坐标系下,对于粒子,位置向量为(x < sub > n ,y < sub > n ,z < sub > n )} ,而梯度算子和拉普拉斯算子分别为: }}
  30. 很明显即使在他生命的最后一年,如写给马克斯·伯恩的信所示,薛定谔从未接受哥本哈根的解释。[34]:220
  31. 36.0 36.1 Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry (Volume 1), P. W. Atkins, Oxford University Press, 1977,
  32. 37.0 37.1 37.2 37.3 Quanta: A handbook of concepts, P. W. Atkins, Oxford University Press, 1974,
  33. 38.0 38.1 Physics of Atoms and Molecules, B. H. Bransden, C. J. Joachain, Longman, 1983,
  34. 39.0 39.1 39.2 39.3 Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985,
  35. 40.0 40.1 Quantum Mechanics Demystified, D. McMahon, McGraw Hill (USA), 2006,
  36. 引用错误:无效<ref>标签;未给name属性为“Quanta的引用提供文字
  37. 42.0 42.1 42.2 42.3 Analytical Mechanics, L. N. Hand, J. D. Finch, Cambridge University Press, 2008,
  38. 脚本错误:没有“Footnotes”这个模块。 Section 3.7.5
  39. 脚本错误:没有“Footnotes”这个模块。 Section 3.7.5
  40. 脚本错误:没有“Footnotes”这个模块。 p. 78
  41. {harvnb | Hall | 2013}p.78


引用错误:组名为“note”的<ref>标签存在,但没有找到相应的<references group="note"/>标签