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第209行： − 其中“ m”是粒子的质量，“ math”是拉普拉斯算符。+ 第233行：
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第307行： − 其中，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。+ 第381行：
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第385行： − 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求解波函数，其中包含有关系统的信息。+ 第469行：
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第616行： − + + + + + − 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.+ 第639行：
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第681行： + − + − 在马克斯 · 普朗克量子化光之后，Albert Einstein 将普朗克量子解释为光子，光的粒子，并提出光子的能量与其频率成正比，这是波粒二象性的第一个黑体辐射。由于能量和动量与狭义相对论的频率和波数有着相同的关系，因此可以推断，光子的动量与它的波长成反比，或者与它的波数成正比:+ − + − 如果你不喜欢这个，你可以试试这个 − + 第703行：
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→Time-dependent equation时变方程
{{Equation box 1
{{Equation box 1
|publisher=Kluwer Academic/Plenum Publishers
|publisher=Kluwer Academic/Plenum Publishers
|equation=<math>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>
|equation=<math>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>
[[File:Wave packet (dispersion).gif|thumb|200px|A [[wave function]] that satisfies the nonrelativistic Schrödinger equation with {{math|''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.]]
[[File:Wave packet (dispersion).gif|thumb|200px|A [[wave function]] that satisfies the nonrelativistic Schrödinger equation with {{math|''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.]]
|border
|border
|indent=:
|indent=:
where <math>m</math> is the particle's mass, and <math>\nabla^2</math> is the Laplacian.
where <math>m</math> is the particle's mass, and <math>\nabla^2</math> is the Laplacian.
其中<math>m</math>是粒子的质量，<math>\nabla^2</math> 是拉普拉斯算符。
|border
|border
where <math>i</math> is the [[imaginary unit]], <math>\hbar = \frac{h}{2 \pi}</math> is the reduced [[Planck constant]] having the dimension of action,<ref>{{cite journal |author1=P. R. Bunker |author2=I. M. Mills |author3=Per Jensen
where <math>i</math> is the [[imaginary unit]], <math>\hbar = \frac{h}{2 \pi}</math> is the reduced [[Planck constant]] having the dimension of action,<ref>{{cite journal |author1=P. R. Bunker |author2=I. M. Mills |author3=Per Jensen
其中，<math>i</math>是[[虚单位]]，<math>\hbar=\frac{h}{2\pi}</math>是具有作用维数的约化[[Planck常数]]，<ref>{{cite journal
其中，<math>i</math>是[[虚单位]]，<math>\hbar=\frac{h}{2\pi}</math>是具有作用维数的约化[[普朗克常数]]，<ref>{{cite journal
|author1=P. R. Bunker |author2=I. M. Mills |author3=Per Jensen
|author1=P. R. Bunker |author2=I. M. Mills |author3=Per Jensen
{{Equation box 1
{{Equation box 1
| title = The Planck constant of action <math>h</math><sub>A</sub>
| title = The Planck constant of action <math>h</math><sub>A</sub>
where <math>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.
where <math>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>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。
|indent=:
|indent=:
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]]).
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]]到[[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，并在许多情况下产生准确的结果，但仅在一定程度上（见[[相对论量子力学]]和[[相对论量子场论]]）。
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]]到[[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]]和[[相对论量子场论]]）。
|background colour = rgb(0,115,207,10%)}}
|background colour = rgb(0,115,207,10%)}}
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.
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>H = T + V = \frac{\|\mathbf{p}\|^2}{2m} + V(x, y, z)</math> for a single particle in the non-relativistic limit.
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>H = T + V = \frac{\|\mathbf{p}\|^2}{2m} + V(x, y, z)</math> for a single particle in the non-relativistic limit.
<math>i \hbar \frac{d\psi}{dt} = \hat{\mathcal{H}}\psi,</math>
<math>i \hbar \frac{d\psi}{dt} = \hat{\mathcal{H}}\psi,</math>
As before, the most common manifestation is the [[relativistic quantum mechanics|nonrelativistic]] Schrödinger equation for a single particle moving in an electric field (but not a magnetic field):
As before, the most common manifestation is the [[relativistic quantum mechanics|nonrelativistic]] Schrödinger equation for a single particle moving in an electric field (but not a magnetic field):
where the ordinary definition for the derivative was used. The operator <math>\hat{\mathcal{H}}</math> used here denotes an arbitrary Hermitian operator. Using the correspondence principle it is possible to show that in the classical limit, using appropriate units, the expectation value of <math>\hat{\mathcal{H}}</math> corresponds to the Hamiltonian of the system.
where the ordinary definition for the derivative was used. The operator <math>\hat{\mathcal{H}}</math> used here denotes an arbitrary Hermitian operator. Using the correspondence principle it is possible to show that in the classical limit, using appropriate units, the expectation value of <math>\hat{\mathcal{H}}</math> corresponds to the Hamiltonian of the system.
其中使用了导数的普通定义。这里使用的操作符<math>\hat{\mathcal{H}}</math>表示一个任意的Hermitian操作符。利用对应原理，可以证明在经典极限下，用适当的单位，hat{\mathcal{H}}</math>的期望值对应于系统的哈密顿量。
其中使用了导数的普通定义。这里使用的操作符<math>\hat{\mathcal{H}}</math>表示一个任意的哈密顿Hermitian操作符。利用对应原理，可以证明在经典极限下，用适当的单位，hat{\mathcal{H}}</math>的期望值对应于系统的哈密顿量。
{{Equation box 1
{{Equation box 1
[[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. ]]
[[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. ]]
[[量子穿隧效应通过障碍物。从左边来的粒子没有足够的能量爬过障碍。然而，它有时候会“隧道”到另一边。]]
[[量子穿隧效应通过障碍物。从左边来的粒子没有足够的能量爬过障碍。然而，它有时候会“隧穿”到另一边。]]
==Derivation推导==
==Derivation推导==
One may derive the Schrödinger equation starting from the [[Dirac-von Neumann axioms]]. Suppose the [[wave function]] <math>\psi(t_0)</math> represents a unit vector defined on a complex [[Hilbert space]] at some initial time <math>t_0</math>. The [[Unitarity (physics)|unitarity principle]] requires that there must exist a linear operator, <math>\hat{U}(t)</math>, such that for any time <math>t > t_0</math>,
One may derive the Schrödinger equation starting from the [[Dirac-von Neumann axioms]]. Suppose the [[wave function]] <math>\psi(t_0)</math> represents a unit vector defined on a complex [[Hilbert space]] at some initial time <math>t_0</math>. The [[Unitarity (physics)|unitarity principle]] requires that there must exist a linear operator, <math>\hat{U}(t)</math>, such that for any time <math>t > t_0</math>,
我们可以从[[狄拉克冯诺依曼公理]]出发导出薛定谔方程。假设[[wave function]<math>\psi（t_0）</math>表示在某个初始时间<math>t_0</math>在复[[Hilbert space]]上定义的单位向量。[[Unitarity（physics）| Unitarity principle]]要求必须存在一个线性算子，<math>\hat{U}（t）</math>，这样在任何时间<math>t > t_0</math>，
我们可以从[[狄拉克冯诺依曼公理]]出发导出薛定谔方程。假设[[波函数]]<math>\psi(t_0)</math>表示在某个初始时间<math>t_0</math>在复[[希尔伯特空间Hilbert space]]上定义的单位向量。[[统一性（物理学）|统一性原理]]要求必须存在一个线性算子，<math>\hat{U}（t）</math>，这样在任何时间<math>t > t_0</math>，
{{NumBlk|:|<math>\psi(t) = \hat{U}(t)\psi(t_0).</math>|{{EquationRef|1}}}}
{{NumBlk|:|<math>\psi(t) = \hat{U}(t)\psi(t_0).</math>|{{EquationRef|1}}}}
==Implications==
==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.
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。
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===
===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>\hat{H}</math> are the energy levels of the system.
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>\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>\hat{H}</math>的[[特征值]]是系统的能级。
===Quantization量子化===
===Quantization===
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.
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.
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 [[Energy level|quantized energy levels]], a fact discovered via [[atomic spectroscopy]]. (Energy quantization is discussed [[#Time-independent|below]].) Another example is [[angular momentum operator|quantization of angular momentum]]. This was an ''assumption'' in the earlier [[Bohr model|Bohr model of the atom]], but it is a ''prediction'' of the Schrödinger equation.
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 [[Energy level|quantized energy levels]], a fact discovered via [[atomic spectroscopy]]. (Energy quantization is discussed [[#Time-independent|below]].) Another example is [[angular momentum operator|quantization of angular momentum]]. This was an ''assumption'' in the earlier [[Bohr model|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[[能级|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[[角动量算符|角动量量子化]]。这是早期[[玻尔模型|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.
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.
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.<ref name="Zettili2009">{{cite book|author=Nouredine Zettili|title=Quantum Mechanics: Concepts and Applications|date=17 February 2009|publisher=John Wiley & Sons|isbn=978-0-470-02678-6}}</ref>{{rp|165–167}}
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.<ref name="Zettili2009">{{cite book|author=Nouredine Zettili|title=Quantum Mechanics: Concepts and Applications|date=17 February 2009|publisher=John Wiley & Sons|isbn=978-0-470-02678-6}}</ref>{{rp|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。<ref name="Zettili2009">{{cite book|author=Nouredine Zettili|title=Quantum Mechanics: Concepts and Applications|date=17 February 2009|publisher=John Wiley & Sons|isbn=978-0-470-02678-6}}</ref>{{rp|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.
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。
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===
===Quantum tunneling量子隧道===
{{main|Quantum tunneling}}
{{main|Quantum tunneling}}
{{main |量子隧道}}
[[File:TunnelEffektKling1.png|width = 300px|thumb|[[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. ]]
[[File:TunnelEffektKling1.png|width = 300px|thumb|[[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，薛定谔方程预测球到达山的另一边的可能性很小，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.
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.
===Particles as waves===
===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.
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.
{{main|Matter wave|Wave–particle duality|Double-slit experiment}}
{{main|Matter wave|Wave–particle duality|Double-slit experiment}}
{{主物质波}波粒二象性}双缝实验}}
In 1952, Erwin Schrödinger gave a lecture during which he commented,
In 1952, Erwin Schrödinger gave a lecture during which he commented,
[[File:Double-slit experiment results Tanamura 2.jpg|175px|thumb|A double slit experiment showing the accumulation of electrons on a screen as time passes.]]
[[File:Double-slit experiment results Tanamura 2.jpg|175px|thumb|A double slit experiment showing the accumulation of electrons on a screen as time passes.]]
[[文件：双缝实验结果Tanamura 2.jpg | 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.
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.
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<ref name=Ballentine>{{citation|last=Ballentine|first=Leslie|title=Quantum Mechanics: A Modern Development | publisher=World Scientific Publishing Co.|year=1998| isbn = 978-9810241056}}</ref>{{rp|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:
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<ref name=Ballentine>{{citation|last=Ballentine|first=Leslie|title=Quantum Mechanics: A Modern Development | publisher=World Scientific Publishing Co.|year=1998| isbn = 978-9810241056}}</ref>{{rp|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<ref name=Ballentine>{{citation|last=Ballentine|first=Leslie|title=Quantum Mechanics: A Modern Development | publisher=World Scientific Publishing Co.|year=1998| isbn = 978-9810241056}}</ref>{{rp|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.
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.
:''"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."''
:''"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
[[Double-slit experiment|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.
[[Double-slit experiment|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.
[[双缝实验|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]]
[[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.
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]]和[[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>p</math> of a photon is inversely proportional to its wavelength <math>\lambda</math>, or proportional to its wave number <math>k</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>p</math> of a photon is inversely proportional to its wavelength <math>\lambda</math>, or proportional to its wave number <math>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 principle#Application to waves|superposition]] and [[Interference (wave propagation)|interference]].
Related to [[diffraction]], particles also display [[Superposition principle#Application to waves|superposition]] and [[Interference (wave propagation)|interference]].
与[[衍射]]有关，粒子也显示[[叠加原理#应用于波|叠加]]和[[干涉（波传播）|干涉]]。
<math>p = \frac{h}{\lambda} = \hbar k,</math>
<math>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 {{math|''x''}}, not that the system will actually be at position {{math|''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.
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 {{math|''x''}}, not that the system will actually be at position {{math|''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 {{math|''x''}}位置的“概率”，而不是系统实际处于 {{math|''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>h</math> is Planck's constant and <math>\hbar = {h}/{2\pi}</math> is the reduced Planck constant of action
where <math>h</math> is Planck's constant and <math>\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
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====
====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>L</math> according to:
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>L</math> according to:
{{main|Measurement in quantum mechanics|Heisenberg uncertainty principle|Interpretations of quantum mechanics}}
{{main|Measurement in quantum mechanics|Heisenberg uncertainty principle|Interpretations of quantum mechanics}}
{{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> L = n{h \over 2\pi} = n\hbar.</math>
<math> L = n{h \over 2\pi} = n\hbar.</math>
In classical mechanics, a particle has, at every moment, an exact position and an exact momentum. These values change [[determinism|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.
In classical mechanics, a particle has, at every moment, an exact position and an exact momentum. These values change [[determinism|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|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:
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:
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.
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>n \lambda = 2 \pi r.\,</math>
<math>n \lambda = 2 \pi r.\,</math>
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.
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>r</math>.
This approach essentially confined the electron wave in one dimension, along a circular orbit of radius <math>r</math>.
==Interpretation of the wave function==
==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.
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.
{{main|Interpretations of quantum mechanics}}
{{main|Interpretations of quantum mechanics}}
{{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.<!--
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.<!--
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.
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计算系统的波函数及其随时间动态变化的方法。然而，薛定谔方程并没有直接说明波函数是什么。[[量子力学的解释]]解决诸如波函数、潜在的真实性和实验测量结果之间的关系等问题。
--- Is the non-formulation by Hamilton really relevant?
--- Is the non-formulation by Hamilton really relevant?
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|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.
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|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[[波函数崩溃]]之间的关系。在最古老的[[哥本哈根解释]]中，粒子遵循薛定谔方程“除了”在波函数崩溃期间，在此期间它们的行为完全不同。[[quantum decoherence | quantum decoherence theory]]的出现允许使用替代方法（例如[[Everett多世界解释]]和[[consistent histories]]），其中薛定谔方程“总是”满足，波函数崩溃应解释为薛定谔方程的结果。
Hamilton believed that mechanics was the zero-wavelength limit of wave propagation, but did not formulate an equation for those waves.
Hamilton believed that mechanics was the zero-wavelength limit of wave propagation, but did not formulate an equation for those waves.
哈密顿相信力学是波的传播的零波长极限，但是没有给出这些波的方程式。
In 1952, [[Erwin Schrödinger]] gave a lecture during which he commented,
In 1952, [[Erwin Schrödinger]] gave a lecture during which he commented,
1952年，薛定谔[[Erwin Schrödinger]]作了一次演讲，他在演讲中发表了评论，
----> A modern version of his reasoning is reproduced below. The equation he found is:
----> A modern version of his reasoning is reproduced below. The equation he found is:
: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.''<ref>{{cite book|last=Schrödinger |first=Erwin |authorlink=Erwin Schrödinger|title=The interpretation of quantum mechanics: Dublin seminars (1949–1955) and other unpublished essays |year=1995 |publisher=Ox Bow Press |isbn=9781881987086}}</ref>
: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.''<ref>{{cite book|last=Schrödinger |first=Erwin |authorlink=Erwin Schrödinger|title=The interpretation of quantum mechanics: Dublin seminars (1949–1955) and other unpublished essays |year=1995 |publisher=Ox Bow Press |isbn=9781881987086}}</ref>
：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”<ref>{{cite book|last=Schrödinger |first=Erwin |authorlink=Erwin Schrödinger|title=The interpretation of quantum mechanics: Dublin seminars (1949–1955) and other unpublished essays |year=1995 |publisher=Ox Bow Press |isbn=9781881987086}}</ref>
<math>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>
<math>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>
[[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]],<ref>David Deutsch, ''The Beginning of Infinity'', page 310</ref> while [[Jeffrey A. Barrett]] took the more modest position that it indicates a "similarity in ... general views" between Schrödinger and Everett.<ref>{{cite book|last=Barrett |first=Jeffrey A. |authorlink=Jeffrey A. Barrett |title=The Quantum Mechanics of Minds and Worlds |pages=63 |publisher=[[Oxford University Press]] |year=1999 |isbn=9780191583254}}</ref>
[[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]],<ref>David Deutsch, ''The Beginning of Infinity'', page 310</ref> while [[Jeffrey A. Barrett]] took the more modest position that it indicates a "similarity in ... general views" between Schrödinger and Everett.<ref>{{cite book|last=Barrett |first=Jeffrey A. |authorlink=Jeffrey A. Barrett |title=The Quantum Mechanics of Minds and Worlds |pages=63 |publisher=[[Oxford University Press]] |year=1999 |isbn=9780191583254}}</ref>
[[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]]，<ref>David Deutsch, ''The Beginning of Infinity'', page 310</ref> while [[Jeffrey A. Barrett]] took the more modest position that it indicates a "similarity in ... general views" between Schrödinger and Everett.<ref>{{cite book|last=Barrett |first=Jeffrey A. |authorlink=Jeffrey A. Barrett |title=The Quantum Mechanics of Minds and Worlds |pages=63 |publisher=[[Oxford University Press]] |year=1999 |isbn=9780191583254}}</ref>
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):
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:
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==
==Historical background and development历史背景与发展==
[[File:Erwin Schrodinger2.jpg|right|thumb|[[Erwin Schrödinger]]]]
[[File:Erwin Schrodinger2.jpg|right|thumb|[[Erwin Schrödinger]]]]
{{Main|Theoretical and experimental justification for the Schrödinger equation}}
{{Main|Theoretical and experimental justification for the Schrödinger equation}}
{{薛定谔方程的主要理论和实验证明}}
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Following [[Max Planck]]'s quantization of light (see [[black-body radiation]]), [[Albert Einstein]] interpreted Planck's [[quantum|quanta]] to be [[photon]]s, [[corpuscular theory of light|particles of light]], and proposed that the [[Planck relation|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 [[wavenumber|wave number]] in [[special relativity]], it followed that the momentum <math>p</math> of a photon is inversely proportional to its [[wavelength]] <math>\lambda</math>, or proportional to its wave number <math>k</math>:
Following [[Max Planck]]'s quantization of light (see [[black-body radiation]]), [[Albert Einstein]] interpreted Planck's [[quantum|quanta]] to be [[photon]]s, [[corpuscular theory of light|particles of light]], and proposed that the [[Planck relation|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 [[wavenumber|wave number]] in [[special relativity]], it followed that the momentum <math>p</math> of a photon is inversely proportional to its [[wavelength]] <math>\lambda</math>, or proportional to its wave number <math>k</math>:
继[[Max Planck]]对光的量子化（见[[black body radiation]]）之后，[[Albert Einstein]]将Planck的[[quantum | quanta]]解释为[[photon]]s，[[Photocular theory of light | 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]]和[[波数|波数]]相同，因此光子的动量p与其[[波长]]<math>\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.
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.
where <math>h</math> is [[Planck's constant]] and <math>\hbar = {h}/{2\pi}</math> is the reduced Planck constant of action<ref name = BunkerJensen2020/> (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 wave]]s, meaning that only certain discrete rotational frequencies about the nucleus of an atom are allowed.<ref>
where <math>h</math> is [[Planck's constant]] and <math>\hbar = {h}/{2\pi}</math> is the reduced Planck constant of action<ref name = BunkerJensen2020/> (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 wave]]s, meaning that only certain discrete rotational frequencies about the nucleus of an atom are allowed.
其中，<math>h</math>是[[Planck's constant]]，而<math>\hbar={h}/{2\pi}</math>是作用的约化普朗克常数<ref name=BunkerJensen2020/>（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。<ref>
{{Cite journal
{{Cite journal
These quantized orbits correspond to discrete [[energy level]]s, 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>L</math> according to:
These quantized orbits correspond to discrete [[energy level]]s, 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>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>L</math>的假设量子化，根据：
:<math> L = n{h \over 2\pi} = n\hbar.</math>
:<math> L = n{h \over 2\pi} = n\hbar.</math>
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:
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> \sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\!</math>
<math> \sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\!</math>
This approach essentially confined the electron wave in one dimension, along a circular orbit of radius <math>r</math>.
This approach essentially confined the electron wave in one dimension, along a circular orbit of radius <math>r</math>.
这种方法基本上把电子波限制在一维，沿着半径为<math>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>V</math>, the Ehrenfest theorem says
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>V</math>, the Ehrenfest theorem says
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 [[Four-vector|4-vector]] to derive what we now call the de Broglie relation.<ref>{{cite journal|last=Weissman|first=M.B. |author2=V. V. Iliev |author3=I. Gutman|title=A pioneer remembered: biographical notes about Arthur Constant Lunn|journal=Communications in Mathematical and in Computer Chemistry|year=2008|volume=59|issue=3|pages=687–708}}</ref><ref>{{cite journal|title=Alan Sokal's Hoax and A. Lunn's Theory of Quantum Mechanics|author=Samuel I. Weissman|author2=Michael Weissman|year=1997|journal=Physics Today|volume=50,6|issue=6|page=15|doi=10.1063/1.881789|bibcode=1997PhT....50f..15W}}</ref> 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.<ref>{{cite book|last=Kamen|first=Martin D.|title=Radiant Science, Dark Politics|year=1985|publisher=University of California Press|location=Berkeley and Los Angeles, California|isbn=978-0-520-04929-1|pages=[https://archive.org/details/radiantscienceda00kame/page/29 29–32]|url=https://archive.org/details/radiantscienceda00kame/page/29}}</ref>
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 [[Four-vector|4-vector]] to derive what we now call the de Broglie relation.<ref>{{cite journal|last=Weissman|first=M.B. |author2=V. V. Iliev |author3=I. Gutman|title=A pioneer remembered: biographical notes about Arthur Constant Lunn|journal=Communications in Mathematical and in Computer Chemistry|year=2008|volume=59|issue=3|pages=687–708}}</ref><ref>{{cite journal|title=Alan Sokal's Hoax and A. Lunn's Theory of Quantum Mechanics|author=Samuel I. Weissman|author2=Michael Weissman|year=1997|journal=Physics Today|volume=50,6|issue=6|page=15|doi=10.1063/1.881789|bibcode=1997PhT....50f..15W}}</ref> 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.<ref>{{cite book|last=Kamen|first=Martin D.|title=Radiant Science, Dark Politics|year=1985|publisher=University of California Press|location=Berkeley and Los Angeles, California|isbn=978-0-520-04929-1|pages=[https://archive.org/details/radiantscienceda00kame/page/29 29–32]|url=https://archive.org/details/radiantscienceda00kame/page/29}}</ref>
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[[Four vector | 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。<ref>{{cite journal|last=Weissman|first=M.B. |author2=V. V. Iliev |author3=I. Gutman|title=A pioneer remembered: biographical notes about Arthur Constant Lunn|journal=Communications in Mathematical and in Computer Chemistry|year=2008|volume=59|issue=3|pages=687–708}}</ref><ref>{{cite journal|title=Alan Sokal's Hoax and A. Lunn's Theory of Quantum Mechanics|author=Samuel I. Weissman|author2=Michael Weissman|year=1997|journal=Physics Today|volume=50,6|issue=6|page=15|doi=10.1063/1.881789|bibcode=1997PhT....50f..15W}}</ref> 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。<ref>{{cite book|last=Kamen|first=Martin D.|title=Radiant Science, Dark Politics|year=1985|publisher=University of California Press|location=Berkeley and Los Angeles, California|isbn=978-0-520-04929-1|pages=[https://archive.org/details/radiantscienceda00kame/page/29 29–32]|url=https://archive.org/details/radiantscienceda00kame/page/29}}</ref>
<math>m\frac{d}{dt}\langle x\rangle = \langle p\rangle;\quad \frac{d}{dt}\langle p\rangle = -\left\langle V'(X)\right\rangle .</math>
<math>m\frac{d}{dt}\langle x\rangle = \langle p\rangle;\quad \frac{d}{dt}\langle p\rangle = -\left\langle V'(X)\right\rangle .</math>
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
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 Rowan Hamilton|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]].<ref>
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 Rowan Hamilton|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·罗文·汉密尔顿|威廉·R·汉密尔顿]]对[[力学]]和[[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。<ref>
<math>-V'\left(\left\langle X\right\rangle\right)</math>,
<math>-V'\left(\left\langle X\right\rangle\right)</math>,
}}</ref> 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]]):
}}</ref> 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>
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.<ref>{{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}}</ref>
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.<ref>{{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}}</ref>
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，他灰心丧气地放下算计，与一位情妇隐居在山间小屋里。<ref>{{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}}</ref>
The eigenvalues are
The eigenvalues are
本征值是
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]]<ref name="Schrödinger1982"/>{{rp|3}}) Schrödinger showed that his nonrelativistic version of the wave equation produced the correct spectral energies of hydrogen in a paper published in 1926.<ref name="Schrödinger1982">{{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}}</ref>{{rp|1}}<ref>{{cite journal
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]]<ref name="Schrödinger1982"/>{{rp|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]]<ref name=“Schrödinger1982”/>{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。<ref name="Schrödinger1982">{{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}}</ref>{{rp|1}}<ref>{{cite journal
|last=Schrödinger |first=E.
|last=Schrödinger |first=E.
|bibcode = 1926AnP...384..361S }}</ref> In the equation, Schrödinger computed the [[hydrogen spectral series]] by treating a [[hydrogen atom]]'s [[electron]] as a wave <math>\Psi (\mathbf{x}, t)</math>, moving in a [[potential well]] <math>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:
|bibcode = 1926AnP...384..361S }}</ref> In the equation, Schrödinger computed the [[hydrogen spectral series]] by treating a [[hydrogen atom]]'s [[electron]] as a wave <math>\Psi (\mathbf{x}, t)</math>, moving in a [[potential well]] <math>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]]的[[电子]]视为波<math>\Psi（\mathbf{x}，t）</math>，在[[质子]]产生的[[势阱]]<math>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:
The Hamiltonian for one particle in three dimensions is:
{{cquote|The already ... mentioned psi-function.... is now the means for predicting probability of measurement results. In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog. |20px|20px|Erwin Schrödinger<ref>Erwin Schrödinger, "The Present situation in Quantum Mechanics", p. 9 of 22. The English version was translated by John D. Trimmer. The translation first appeared first in ''Proceedings of the American Philosophical Society'', 124, 323–38. It later appeared as Section I.11 of Part I of ''Quantum Theory and Measurement'' by J. A. Wheeler and W. H. Zurek, eds., Princeton University Press, New Jersey 1983.</ref>}}
{{cquote|The already ... mentioned psi-function.... is now the means for predicting probability of measurement results. In it is embodied the momentarily attained sum of theoretically based future expectation, somewhat as laid down in a catalog. |20px|20px|Erwin Schrödinger<ref>Erwin Schrödinger, "The Present situation in Quantum Mechanics", p. 9 of 22. The English version was translated by John D. Trimmer. The translation first appeared first in ''Proceedings of the American Philosophical Society'', 124, 323–38. It later appeared as Section I.11 of Part I of ''Quantum Theory and Measurement'' by J. A. Wheeler and W. H. Zurek, eds., Princeton University Press, New Jersey 1983.</ref>}}
{{cquote |已经。。。所述psi功能。。。。现在是预测测量结果概率的手段。它体现了理论上基于未来预期的瞬间达到的总和，有点像目录中规定的那样。|20px | 20px | Erwin Schrödinger公司<ref>Erwin Schrödinger, "The Present situation in Quantum Mechanics", p. 9 of 22. The English version was translated by John D. Trimmer. The translation first appeared first in ''Proceedings of the American Philosophical Society'', 124, 323–38. It later appeared as Section I.11 of Part I of ''Quantum Theory and Measurement'' by J. A. Wheeler and W. H. Zurek, eds., Princeton University Press, New Jersey 1983.</ref>}}
<math> \hat{H} = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V(\mathbf{r}) \,, \quad \hat{\mathbf{p}} = -i\hbar \nabla </math>
<math> \hat{H} = \frac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{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.<ref>
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。<ref>
generating the equation
generating the equation
The Schrödinger equation details the behavior of <math>\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.<ref name=Moore1992>
The Schrödinger equation details the behavior of <math>\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。<ref name=Moore1992>
where the position of the particle is <math> \mathbf{r} </math>.
where the position of the particle is <math> \mathbf{r} </math>.
其中粒子的位置是 < math > mathbf { r } </math > 。
{{cite book
{{cite book
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 </sub > ，y < sub > n </sub > ，z < sub > n </sub >)} ，而梯度算子和拉普拉斯算子分别为:
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 </sub > ，y < sub > n </sub > ，z < sub > n </sub >)} ，而梯度算子和拉普拉斯算子分别为:
}}</ref>{{rp|219}} In 1926, just a few days after Schrödinger's fourth and final paper was published, [[Max Born]] successfully interpreted <math>\Psi</math> as the [[probability amplitude]], whose modulus squared is equal to [[Probability density function|probability density]].<ref name=Moore1992/>{{rp|220}} Schrödinger, though, always opposed a statistical or probabilistic approach, with its associated [[wave function collapse|discontinuities]]—much like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying [[determinism|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.<ref name=Moore1992/>{{rp|220}}}}
}}</ref>{{rp|219}} In 1926, just a few days after Schrödinger's fourth and final paper was published, [[Max Born]] successfully interpreted <math>\Psi</math> as the [[probability amplitude]], whose modulus squared is equal to [[Probability density function|probability density]].<ref name=Moore1992/>{{rp|220}} Schrödinger, though, always opposed a statistical or probabilistic approach, with its associated [[wave function collapse|discontinuities]]—much like Einstein, who believed that quantum mechanics was a statistical approximation to an underlying [[determinism|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>\Psi</math>解释为[[概率振幅]]，其模平方等于[[概率密度函数|概率密度]]。<ref name=Moore1992/>{{rp|220}}薛定谔，总是反对统计或概率方法，及其相关的[[波函数崩溃|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[[决定论|确定性理论]]——从来没有调和[[哥本哈根解释]]{{#tag:ref |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。<ref name=Moore1992/>{{rp|220}}}}
<math>\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>
<math>\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>
薛定谔方程是:
薛定谔方程是:
==The wave equation for particles==
==The wave equation for particles粒子的波动方程==
{{main|Wave–particle duality}}
{{main|Wave–particle duality}}
{{主|波粒二象性}}
<math> -\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}{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>
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 [[Vibration|mechanical vibrations]] on strings and in matter can be derived from [[Newton's laws]], where the wave function represents the [[Displacement (vector)|displacement]] of matter, and [[electromagnetic waves]] from [[Maxwell's equations]], where the wave functions are [[electric field|electric]] and [[magnetic field|magnetic]] fields. The basis for Schrödinger's equation, on the other hand, is the energy of the system and a separate [[Mathematical formulation of quantum mechanics#Postulates of quantum mechanics|postulate of quantum mechanics]]: the wave function is a description of the system.<ref name="Quantum Chemistry 1977">''Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry'' (Volume 1), P. W. Atkins, Oxford University Press, 1977, {{isbn|0-19-855129-0}}</ref> The Schrödinger equation is therefore a new concept in itself; as Feynman put it:
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 [[Vibration|mechanical vibrations]] on strings and in matter can be derived from [[Newton's laws]], where the wave function represents the [[Displacement (vector)|displacement]] of matter, and [[electromagnetic waves]] from [[Maxwell's equations]], where the wave functions are [[electric field|electric]] and [[magnetic field|magnetic]] fields. The basis for Schrödinger's equation, on the other hand, is the energy of the system and a separate [[Mathematical formulation of quantum mechanics#Postulates of quantum mechanics|postulate of quantum mechanics]]: the wave function is a description of the system.<ref name="Quantum Chemistry 1977">''Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry'' (Volume 1), P. W. Atkins, Oxford University Press, 1977, {{isbn|0-19-855129-0}}</ref> 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[[振动|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（矢量）|位移]]，而[[电磁波]]则来自[[麦克斯韦方程]]，其中波函数是[[电场|电场]]和[[磁场|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[[量子力学的数学公式ţ量子力学的假设|量子力学的假设]]：波函数是对系统的描述。<ref name="Quantum Chemistry 1977">''Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry'' (Volume 1), P. W. Atkins, Oxford University Press, 1977, {{isbn|0-19-855129-0}}</ref> 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:
with stationary state solutions:
{{cquote|Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger. |20px|20px|Richard Feynman<ref>''The New Quantum Universe'', T. Hey, P. Walters, Cambridge University Press, 2009, {{isbn|978-0-521-56457-1}}</ref>}}
{{cquote|Where did we get that (equation) from? Nowhere. It is not possible to derive it from anything you know. It came out of the mind of Schrödinger. |20px|20px|Richard Feynman<ref>''The New Quantum Universe'', T. Hey, P. Walters, Cambridge University Press, 2009, {{isbn|978-0-521-56457-1}}</ref>}}
{{cquote |我们从哪里得到的（方程式）？哪里也没有。不可能从你所知道的任何东西中得到它。这是薛定谔的想法。|20px|20px|Richard Feynman<ref>''The New Quantum Universe'', T. Hey, P. Walters, Cambridge University Press, 2009, {{isbn|978-0-521-56457-1}}</ref>}}
<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>
<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 {{math|''ψ''}}, which contains all the information that can be known about the system. In the [[Copenhagen interpretation]], the modulus of {{math|''ψ''}} is related to the [[probability]] the particles are in some spatial configuration at some instant of time. Solving the equation for {{math|''ψ''}} can be used to predict how the particles will behave under the influence of the specified potential and with each other.
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 {{math|''ψ''}}, which contains all the information that can be known about the system. In the [[Copenhagen interpretation]], the modulus of {{math|''ψ''}} is related to the [[probability]] the particles are in some spatial configuration at some instant of time. Solving the equation for {{math|''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{{数学|''ψ''}}，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
Again, for non-interacting distinguishable particles the potential is the sum of particle potentials
The Schrödinger equation was developed principally from the [[De Broglie hypothesis]], a wave equation that would describe particles,<ref name="Quanta 1974">''Quanta: A handbook of concepts'', P. W. Atkins, Oxford University Press, 1974, {{isbn|0-19-855493-1}}</ref> and can be constructed as shown informally in the following sections.<ref name="Molecules, B.H. Bransden 1983">''Physics of Atoms and Molecules'', B. H. Bransden, C. J. Joachain, Longman, 1983, {{isbn|0-582-44401-2}}</ref> For a more rigorous description of Schrödinger's equation, see also Resnick ''et al''.<ref name="Atoms, Molecules 1985"/>
The Schrödinger equation was developed principally from the [[De Broglie hypothesis]], a wave equation that would describe particles,<ref name="Quanta 1974">''Quanta: A handbook of concepts'', P. W. Atkins, Oxford University Press, 1974, {{isbn|0-19-855493-1}}</ref> and can be constructed as shown informally in the following sections.<ref name="Molecules, B.H. Bransden 1983">''Physics of Atoms and Molecules'', B. H. Bransden, C. J. Joachain, Longman, 1983, {{isbn|0-582-44401-2}}</ref> For a more rigorous description of Schrödinger's equation, see also Resnick ''et al''.<ref name="Atoms, Molecules 1985"/>
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，<ref name="Quanta 1974">''Quanta: A handbook of concepts'', P. W. Atkins, Oxford University Press, 1974, {{isbn|0-19-855493-1}}</ref> 可按照以下各节的非正式说明进行构建。<ref name="Molecules, B.H. Bransden 1983">''Physics of Atoms and Molecules'', B. H. Bransden, C. J. Joachain, Longman, 1983, {{isbn|0-582-44401-2}}</ref> 有关薛定谔方程的更严格描述，请参见Resnick“等人”的文章。<ref name="Atoms, Molecules 1985"/>
<math> 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 ===
=== Consistency with energy conservation 与节能的一致性===
and the wave function is a product of the particle wave functions
and the wave function is a product of the particle wave functions
The total energy {{math|''E''}} of a particle is the sum of kinetic energy <math>T</math> and potential energy <math>V</math>, this sum is also the frequent expression for the [[Hamiltonian mechanics#Basic physical interpretation|Hamiltonian]] <math>H</math> in classical mechanics:
The total energy {{math|''E''}} of a particle is the sum of kinetic energy <math>T</math> and potential energy <math>V</math>, this sum is also the frequent expression for the [[Hamiltonian mechanics#Basic physical interpretation|Hamiltonian]] <math>H</math> in classical mechanics:
粒子的总能量{math |“E'}}是动能<math>T</math>和势能<math>V</math>的和，这个和也是经典力学中[[哈密顿力学#基本物理解释|哈密顿]]<math>H</math>的常用表达式：
:<math>E = T + V =H \,\!</math>
:<math>E = T + V =H \,\!</math>
Explicitly, for a particle in one dimension with position <math>x</math>, [[mass]] <math>m</math> and [[momentum]] <math>p</math>, and potential energy <math>V</math> which generally [[Harmonic function|varies with position]] and time <math>t</math>:
Explicitly, for a particle in one dimension with position <math>x</math>, [[mass]] <math>m</math> and [[momentum]] <math>p</math>, and potential energy <math>V</math> which generally [[Harmonic function|varies with position]] and time <math>t</math>:
明确地说，对于一维粒子，其位置<math>x</math>，[[mass]]<math>m</math>和[[momentum]]<math>p</math>，势能<math>V</math>通常[[Harmonic function | variable with position]]和时间<math>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.
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.
For three dimensions, the [[position vector]] {{math|'''r'''}} and momentum vector {{math|'''p'''}} must be used:
For three dimensions, the [[position vector]] {{math|'''r'''}} and momentum vector {{math|'''p'''}} must be used:
对于三维，必须使用[[位置矢量]{{math |''r''}}和动量矢量{{math |''p''}}：
:<math>E = \frac{\mathbf{p}\cdot\mathbf{p}}{2m}+V(\mathbf{r},t)=H</math>
:<math>E = \frac{\mathbf{p}\cdot\mathbf{p}}{2m}+V(\mathbf{r},t)=H</math>
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#Physics|interactions]] between the particles (an [[many body problem|{{math|''N''}}-body problem]]), so the potential energy {{math|''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:
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#Physics|interactions]] between the particles (an [[many body problem|{{math|''N''}}-body problem]]), so the potential energy {{math|''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#物理|相互作用]]（一个[[多体问题|{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> \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>
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:
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 ===
=== Linearity 线性===
The simplest wave function is a [[plane wave]] of the form:
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> \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>
<math> \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>
where:
where:
其中：
where the {{math|''A''}} is the amplitude, {{math|'''k'''}} the wave vector, and <math>\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]].
where the {{math|''A''}} is the amplitude, {{math|'''k'''}} the wave vector, and <math>\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>\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>\mathbf{k}</math> the sum is a [[superposition principle#Application to waves|superposition]] of plane waves:
对于离散的<math>\mathbf{k}</math>来说，和是平面波的[[叠加原理#应用于波|叠加]]：
<math>
<math>
for some real amplitude coefficients <math>A_n</math>, and for continuous <math>\mathbf{k}</math> the sum becomes an integral, the [[Fourier transform]] of a momentum space wave function:<ref name="Quantum Mechanics Demystified 2006">''Quantum Mechanics Demystified'', D. McMahon, McGraw Hill (USA), 2006, {{isbn|0-07-145546-9}}</ref>
for some real amplitude coefficients <math>A_n</math>, and for continuous <math>\mathbf{k}</math> the sum becomes an integral, the [[Fourier transform]] of a momentum space wave function:<ref name="Quantum Mechanics Demystified 2006">''Quantum Mechanics Demystified'', D. McMahon, McGraw Hill (USA), 2006, {{isbn|0-07-145546-9}}</ref>
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]]：<ref name="Quantum Mechanics Demystified 2006">''Quantum Mechanics Demystified'', D. McMahon, McGraw Hill (USA), 2006, {{isbn|0-07-145546-9}}</ref>
m & = -\ell,\dots,\ell \\
m & = -\ell,\dots,\ell \\
where <math>d^3 \mathbf{k} = dk_x \,dk_y
where <math>d^3 \mathbf{k} = dk_x \,dk_y
其中 <math>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.
The generalized Laguerre polynomials are defined differently by different authors. See main article on them and the hydrogen atom.
\,dk_z</math>is the differential volume element in [[momentum space|{{math|'''k'''}}-space]], and the integrals are taken over all <math>\mathbf{k}</math>-space. The momentum wave function <math>\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>is the differential volume element in [[momentum space|{{math|'''k'''}}-space]], and the integrals are taken over all <math>\mathbf{k}</math>-space. The momentum wave function <math>\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[[动量空间|{{math |''k''}}-空间]]中的微分体积元素，积分占据了所有的<math>\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===
=== Consistency with the de Broglie relations ===
The equation for any two-electron system, such as the neutral helium atom (He, <math> Z = 2 </math>), the negative hydrogen ion (H<sup>−</sup>, <math> Z = 1 </math>), or the positive lithium ion (Li<sup>+</sup>, <math> Z = 3 </math>) is:
The equation for any two-electron system, such as the neutral helium atom (He, <math> Z = 2 </math>), the negative hydrogen ion (H<sup>−</sup>, <math> Z = 1 </math>), or the positive lithium ion (Li<sup>+</sup>, <math> Z = 3 </math>) is:
[[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.<ref name="Quanta 1974"/>]]
[[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.<ref name="Quanta 1974"/>]]
[[文件：Wavefunction values.svg|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。<ref name=“Quanta 1974”/>]]
<math>{ \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
where
[[Photoelectric effect|Einstein's light quanta hypothesis]] (1905) states that the energy {{math|''E''}} of a quantum of light or photon is proportional to its [[frequency]] <math>\nu</math> (or [[angular frequency]], <math>\omega = 2\pi \nu</math>)
[[Photoelectric effect|Einstein's light quanta hypothesis]] (1905) states that the energy {{math|''E''}} of a quantum of light or photon is proportional to its [[frequency]] <math>\nu</math> (or [[angular frequency]], <math>\omega = 2\pi \nu</math>)
[[光电效应|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>\nu</math> (或 [[角频率]], <math>\omega = 2\pi \nu</math>)
<math>\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>
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>\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.
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>\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>\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).
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).
Schrödinger's insight,{{citation needed|date=January 2014}} late in 1925, was to express the [[Phase (waves)|phase]] of a [[plane wave]] as a [[complex number|complex]] [[phase factor]] using these relations:
Schrödinger's insight,{{citation needed|date=January 2014}} late in 1925, was to express the [[Phase (waves)|phase]] of a [[plane wave]] as a [[complex number|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]]的[[相位（波）|相位]]表示为[[复数|复]][[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
For any linear operator bounded from below, the eigenvector with the smallest eigenvalue is the vector that minimizes the quantity
and to realize that the first order [[partial derivatives]] with respect to space were
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.
over all which are normalized.
Taking partial derivatives with respect to time gives
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 }}.
which has the same form as the diffusion equation, with diffusion coefficient }}.
Another postulate of quantum mechanics is that all observables are represented by [[linear operator|linear]] [[Self-adjoint operator|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,
Another postulate of quantum mechanics is that all observables are represented by [[linear operator|linear]] [[Self-adjoint operator|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[[线性算子|线性]][[自伴算子|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>L^2</math> of square-integrable densities, the Schrödinger semigroup <math>e^{it\hat{H}}</math> is a unitary evolution, and therefore surjective. The flows satisfy the Schrödinger equation <math>i\partial_t u = \hat{H}u</math>, where the derivative is taken in the distribution sense. However, since <math>\hat{H}</math> for most physically reasonable Hamiltonians (e.g., the Laplace operator, possibly modified by a potential) is unbounded in <math>L^2</math>, this shows that the semigroup flows lack Sobolev regularity in general. Instead, solutions of the Schrödinger equation satisfy a Strichartz estimate.
On the space <math>L^2</math> of square-integrable densities, the Schrödinger semigroup <math>e^{it\hat{H}}</math> is a unitary evolution, and therefore surjective. The flows satisfy the Schrödinger equation <math>i\partial_t u = \hat{H}u</math>, where the derivative is taken in the distribution sense. However, since <math>\hat{H}</math> for most physically reasonable Hamiltonians (e.g., the Laplace operator, possibly modified by a potential) is unbounded in <math>L^2</math>, this shows that the semigroup flows lack Sobolev regularity in general. Instead, solutions of the Schrödinger equation satisfy a Strichartz estimate.
where {{math|''E''}} are the energy [[eigenvalue]]s, and the [[momentum operator]], corresponding to the spatial derivatives (the [[del|gradient]] <math>\nabla </math>),
where {{math|''E''}} are the energy [[eigenvalue]]s, and the [[momentum operator]], corresponding to the spatial derivatives (the [[del|gradient]] <math>\nabla </math>),
其中{{math |''E'}}是能量[[本征值]]s和[[动量算子]]，对应于空间导数（[[del | gradient]]<math>\nabla</math>），
<math>E^2 = (pc)^2 + (m_0c^2)^2 \, , </math>
<math>E^2 = (pc)^2 + (m_0c^2)^2 \, , </math>
where {{math|'''p'''}} is a vector of the momentum eigenvalues. In the above, the "[[circumflex|hats]]" ( {{math|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>V</math> is just a multiplicative factor.
where {{math|'''p'''}} is a vector of the momentum eigenvalues. In the above, the "[[circumflex|hats]]" ( {{math|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>V</math> is just a multiplicative factor.
其中{{math |''p''}}是动量本征值的向量。在上面的例子中，“[[扬抑符| hats]]”（{{math |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>V</math>只是一个乘法因子。
<math> \frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0. </math>,
<math> \frac {1}{c^2} \frac{\partial^2}{\partial t^2} \psi - \nabla^2 \psi + \frac {m^2 c^2}{\hbar^2} \psi = 0. </math>,
Substituting the energy and momentum operators into the classical energy conservation equation obtains the operator:
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:
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:
so in terms of derivatives with respect to time and space, acting this operator on the wave function {{math|''Ψ''}} immediately led Schrödinger to his equation:{{citation needed|date=January 2014}}
so in terms of derivatives with respect to time and space, acting this operator on the wave function {{math|''Ψ''}} immediately led Schrödinger to his equation:{{citation needed|date=January 2014}}
因此，关于时间和空间的导数，把这个算符作用于波函数{{math |''Ψ''}立即导致薛定谔的方程：{{citation needed|date=January 2014}}
The general form of the Schrödinger equation remains true in relativity, but the Hamiltonian is less obvious. For example, the Dirac Hamiltonian for a particle of mass and electric charge in an electromagnetic field (described by the electromagnetic potentials and ) is:
The general form of the Schrödinger equation remains true in relativity, but the Hamiltonian is less obvious. For example, the Dirac Hamiltonian for a particle of mass and electric charge in an electromagnetic field (described by the electromagnetic potentials and ) is:
Wave–particle duality can be assessed from these equations as follows. The kinetic energy {{math|''T''}} is related to the square of momentum {{math|'''p'''}}. As the particle's momentum increases, the kinetic energy increases more rapidly, but since the wave number {{math|{{!}}'''k'''{{!}}}} increases the wavelength {{math|''λ''}} decreases. In terms of ordinary scalar and vector quantities (not operators):
Wave–particle duality can be assessed from these equations as follows. The kinetic energy {{math|''T''}} is related to the square of momentum {{math|'''p'''}}. As the particle's momentum increases, the kinetic energy increases more rapidly, but since the wave number {{math|{{!}}'''k'''{{!}}}} increases the wavelength {{math|''λ''}} decreases. In terms of ordinary scalar and vector quantities (not operators):
波粒二象性可以从以下方程中评估。动能{math''T'}}与动量{math''p'}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{{数学|{！}}''k''{{！}}}}增加波长{{数学|''λ''}}减少。对于普通标量和向量量（不是运算符）：
in which the (γ<sup>1</sup>, γ<sup>2</sup>, γ<sup>3</sup>)}} 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.
in which the (γ<sup>1</sup>, γ<sup>2</sup>, γ<sup>3</sup>)}} 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.
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:
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.
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.