# 薛定谔方程

{{简述}线性偏微分方程，其解描述了量子力学系统。}}

Schrödinger's equation inscribed on the gravestone of Annemarie and Erwin Schrödinger. (Newton's dot notation for the time derivative is used.)

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

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

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

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

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

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

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

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

{{Equation box 1

{方程式方框1

|indent=:

2012年10月22日:

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

|title=Time-dependent Schrödinger equation (general)

| title = 依赖时间的薛定谔方程(一般)

|equation=$\displaystyle{ i \hbar \frac{d}{d t}\vert\Psi(t)\rangle = \hat H\vert\Psi(t)\rangle }$

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

|border

## Equation

|border colour = rgb(80,200,120)

| border color = rgb (80,200,120)

|background colour = rgb(80,200,120,10%)}}

| background color = rgb (80,200,120,10%)}

### Time-dependent equation时变方程

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

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

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

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

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

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

|border

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

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

|border colour = rgb(80,200,120)

| border color = rgb (80,200,120)

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

|background colour = rgb(80,200,120,10%)}}

| background color = rgb (80,200,120,10%)}

{{Equation box 1

where $\displaystyle{ E }$ 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.

|indent=:

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

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

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

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

|border

|border colour = rgb(0,115,207)

{{Equation box 1

{方程式方框1

|background colour = rgb(0,115,207,10%)}}

|indent=:

2012年10月22日:

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

| title = 与时间无关的薛定谔方程(单个非相对论粒子)

where $\displaystyle{ m }$ is the particle's mass, and $\displaystyle{ \nabla^2 }$ is the Laplacian.

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

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

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

|border

|border colour = rgb(0,115,207)

| border colour = rgb (0,115,207)

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).

|background colour = rgb(0,115,207,10%)}}

| background color = 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.

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, $\displaystyle{ H = T + V = \frac{\|\mathbf{p}\|^2}{2m} + V(x, y, z) }$ for a single particle in the non-relativistic limit.

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

The time-independent Schrödinger equation is discussed further below.

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

Related to diffraction, particles also display superposition and interference.

## 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.

### 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 $\displaystyle{ \hat{H} }$ are the energy levels of the system.

### Quantization量子化

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

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

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.[9]:165–167

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

### Quantum tunneling量子隧道

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.

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.

{{主物质波}波粒二象性}双缝实验}}

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

1952年，埃尔温·薛定谔在一次演讲中评论道:

A double slit experiment showing the accumulation of electrons on a screen as time passes.

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[10]:Chapter 4, p.99 shows that such an interpretation has problems. Ballentine points out that whilst it is arguable to associate a physical wave with a single particle, there is still only one Schrödinger wave equation for many particles. He points out:

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."

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.

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.

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 $\displaystyle{ p }$ of a photon is inversely proportional to its wavelength $\displaystyle{ \lambda }$, or proportional to its wave number $\displaystyle{ k }$:

Related to diffraction, particles also display superposition and interference.

$\displaystyle{ p = \frac{h}{\lambda} = \hbar k, }$

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

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

#### 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 $\displaystyle{ L }$ according to:

{{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}}

$\displaystyle{ L = n{h \over 2\pi} = n\hbar. }$

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

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

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.

$\displaystyle{ n \lambda = 2 \pi r.\, }$

“ n lambda = 2 pi r，”

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

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

## Interpretation of the wave function波函数的解释

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

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

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

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

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

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

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

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

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

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

## Historical background and development历史背景与发展

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

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

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

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

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

$\displaystyle{ p = \frac{h}{\lambda} = \hbar k, }$

While at the cabin, Schrödinger decided that his earlier nonrelativistic calculations were novel enough to publish, and decided to leave off the problem of relativistic corrections for the future. Despite the difficulties in solving the differential equation for hydrogen (he had sought help from his friend the mathematician Hermann Weyl In the equation, Schrödinger computed the hydrogen spectral series by treating a hydrogen atom's electron as a wave $\displaystyle{ \Psi (\mathbf{x}, t) }$, moving in a potential well $\displaystyle{ V }$, 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:

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

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

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

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

$\displaystyle{ L = n{h \over 2\pi} = n\hbar. }$

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

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

$\displaystyle{ \sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\! }$

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

$\displaystyle{ n \lambda = 2 \pi r.\, }$

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

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

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 $\displaystyle{ V }$, 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 4-vector to derive what we now call the de Broglie relation.[18][19] Unlike de Broglie, Lunn went on to formulate the differential equation now known as the Schrödinger equation, and solve for its energy eigenvalues for the hydrogen atom. Unfortunately the paper was rejected by the Physical Review, as recounted by Kamen.[20]

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

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

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

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.

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

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

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

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

- \frac{m\omega x^2}{2 \hbar}} \cdot \mathcal{H}_n\left(\sqrt{\frac{m\omega}{\hbar}} x \right), [/itex]

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

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

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

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

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

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

The eigenvalues are

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

The Hamiltonian for one particle in three dimensions is:

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

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.

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

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

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

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$\displaystyle{ \Psi }$解释为概率振幅，其模平方等于概率密度[34]:220薛定谔，总是反对统计或概率方法，及其相关的不连续——很像爱因斯坦，他认为量子力学是一个统计近似的基础确定性理论——从来没有调和哥本哈根解释[35]

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

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

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

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

The Schrödinger equation is:

## The wave equation for particles粒子的波动方程

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

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

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

with stationary state solutions:

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

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

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

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,[37] and can be constructed as shown informally in the following sections.[38] For a more rigorous description of Schrödinger's equation, see also Resnick et al.[39]

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

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

### Consistency with energy conservation 与节能的一致性

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

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

$\displaystyle{ E = T + V =H \,\! }$

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

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

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

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.

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

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

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

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

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

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

$\displaystyle{ \psi(r,\theta,\varphi) = R(r)Y_\ell^m(\theta, \varphi) = R(r)\Theta(\theta)\Phi(\varphi), }$

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

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

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

### Linearity 线性

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

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

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 >

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

where:

where the A is the amplitude, k the wave vector, and $\displaystyle{ \omega }$ 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.

For discrete $\displaystyle{ \mathbf{k} }$ the sum is a superposition of plane waves:

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

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

1,2,3，点

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

0,1,2，dots，n-1

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

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

M & =-ell，dots，ell

\end{align}

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

[/itex]

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

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

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

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

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

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

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

where

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

$\displaystyle{ \rho=|\Psi|^2=\Psi^*(\mathbf{r},t)\Psi(\mathbf{r},t)\,\! }$

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

$\displaystyle{ E = h\nu = \hbar \omega \,\! }$

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

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

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

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

$\displaystyle{ p = \frac{h}{\lambda} = \hbar k\;, }$

is the probability current (flow per unit area).

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

$\displaystyle{ \mathbf{p} = \hbar \mathbf{k}\,,\quad |\mathbf{k}| = \frac{2\pi}{\lambda} \,. }$

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

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 $\displaystyle{ \hbar = 1 }$ 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.

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] late in 1925, was to express the phase of a plane wave as a complex phase factor using these relations:

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

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

$\displaystyle{ \langle \psi |\hat{A}|\psi \rangle }$

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

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

over all which are normalized.

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

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

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

Taking partial derivatives with respect to time gives

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

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

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

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

$\displaystyle{ \hat{E} \Psi = i\hbar\dfrac{\partial}{\partial t}\Psi = E\Psi }$

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

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

$\displaystyle{ E^2 = (pc)^2 + (m_0c^2)^2 \, , }$

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

$\displaystyle{ \hat{\mathbf{p}} \Psi = -i\hbar\nabla \Psi = \mathbf{p} \Psi }$

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

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

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

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

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

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:

$\displaystyle{ E= \dfrac{\mathbf{p}\cdot\mathbf{p}}{2m}+V \quad \rightarrow \quad \hat{E} = \dfrac{\hat{\mathbf{p}}\cdot\hat{\mathbf{p}}}{2m} + V }$

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

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

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

$\displaystyle{ \mathbf{p}\cdot\mathbf{p} \propto \mathbf{k}\cdot\mathbf{k} \propto T \propto \dfrac{1}{\lambda^2} }$

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

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:

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.

$\displaystyle{ \hat{T} \Psi = \frac{-\hbar^2}{2m}\nabla\cdot\nabla \Psi \, \propto \, \nabla^2 \Psi \,. }$

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

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

### Wave and particle motion波与质点运动

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

This equation allows for the inclusion of spin in nonrelativistic quantum mechanics. Squaring the above equation yields the Schrödinger equation in 1D. The matrices $\displaystyle{ \eta }$ obey the following properties

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

\eta^2=0 \\

2 = 0

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

(eta ^ dagger) ^ 2 = 0

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

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

\end{align}

$\displaystyle{ \sigma(x) \sigma(p_x) \geqslant \frac{\hbar}{2} \quad \rightarrow \quad \sigma(x) \sigma(p_x) \geqslant 0 \,\! }$

[/itex]

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

The 3 dimensional version of the equation is given by

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 $\displaystyle{ V }$, the Ehrenfest theorem says[43]

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

\begin{align}

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

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

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

$\displaystyle{ -V'\left(\left\langle X\right\rangle\right) }$,

\end{align}

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

[/itex]

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

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

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

The Schrödinger equation in its general form

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

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

$\displaystyle{ -\frac{\partial}{\partial t} S(q_i,t) = H\left(q_i,\frac{\partial S}{\partial q_i},t \right) \,\! }$

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

Substituting

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

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

The implications are as follows:

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

## Nonrelativistic quantum mechanics非相对论量子力学

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

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

### Time-independent时间无关性

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

$\displaystyle{ \Psi(\text{space coords},t)=\psi(\text{space coords})\tau(t)\,. }$

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

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

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

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

$\displaystyle{ \hat{H} \psi = E \psi }$

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

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

### One-dimensional examples一维例子

For a particle in one dimension, the Hamiltonian is:

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

and substituting this into the general Schrödinger equation gives:

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

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

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

Category:Differential equations

For N particles in one dimension, the Hamiltonian is:

Category:Partial differential equations

Category:Wave mechanics

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

Category:Functions of space and time

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

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

$\displaystyle{ z=re^{i\phi}=x+iy \,\! }$

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

< math > operatorname { hat h }

7. "Schrodinger equation". Hyperphysics. Department of Physics and Astronomy, George State University.
8. "Schrodinger equation". Hyperphysics. Department of Physics and Astronomy, George State University.
9. Nouredine Zettili (17 February 2009). Quantum Mechanics: Concepts and Applications. John Wiley & Sons. ISBN 978-0-470-02678-6.
10. Ballentine, Leslie (1998), Quantum Mechanics: A Modern Development, World Scientific Publishing Co., ISBN 978-9810241056
11. Schrödinger, Erwin (1995). The interpretation of quantum mechanics: Dublin seminars (1949–1955) and other unpublished essays. Ox Bow Press. ISBN 9781881987086.
12. Schrödinger, Erwin (1995). The interpretation of quantum mechanics: Dublin seminars (1949–1955) and other unpublished essays. Ox Bow Press. ISBN 9781881987086.
13. David Deutsch, The Beginning of Infinity, page 310
14. Barrett, Jeffrey A. (1999). The Quantum Mechanics of Minds and Worlds. Oxford University Press. pp. 63. ISBN 9780191583254.
15. David Deutsch, The Beginning of Infinity, page 310
16. Barrett, Jeffrey A. (1999). The Quantum Mechanics of Minds and Worlds. Oxford University Press. pp. 63. ISBN 9780191583254.
17. de Broglie This 1926 paper was enthusiastically endorsed by Einstein, who saw the matter-waves as an intuitive depiction of nature, as opposed to Heisenberg's matrix mechanics, which he considered overly formal. 这篇1926年的论文得到了 Einstein 的热烈支持，他认为物质波是对自然的直观描述，而不是 Heisenberg 的矩阵力学，他认为这种描述过于正式。, L. (1925 The Schrödinger equation details the behavior of $\displaystyle{ \Psi }$ but says nothing of its nature. Schrödinger tried to interpret it as a charge density in his fourth paper, but he was unsuccessful. In 1926, just a few days after Schrödinger's fourth and final paper was published, Max Born successfully interpreted $\displaystyle{ \Psi }$ as the probability amplitude, whose modulus squared is equal to probability density. The Schrödinger equation is therefore a new concept in itself; as Feynman put it: 薛定谔方程卫生组织详细描述了超能力的行为，但对其本质却只字未提。薛定谔在他的第四篇论文中试图将其解释为电荷密度，但他没有成功。1926年，就在薛定谔的第四篇也是最后一篇论文发表的几天之后，Max Born 成功地将《 math > Psi </math > 》解释为机率幅，其模数的平方等于概率密度。因此，薛定谔方程本身就是一个新概念，正如 Feynman 所说:). [https://web.archive.org/web/20090509012910/http://www.ensmp.fr/aflb/LDB-oeuvres/De_Broglie_Kracklauer.pdf where $\displaystyle{ d^3 \mathbf{k} = dk_x \,dk_y 在哪里? d ^ 3 mathbf { k } = dk x，dk y |archivedate = 9 May 2009 \,dk_z }$is the differential volume element in -space, and the integrals are taken over all $\displaystyle{ \mathbf{k} }$-space. The momentum wave function $\displaystyle{ \Phi (\mathbf{k}) }$ arises in the integrand since the position and momentum space wave functions are Fourier transforms of each other. ，dk _ z </math > 是空间中的微分体积元，并且所有的积分都是在整个 < math > mathbf { k } </math >-space 中进行的。动量波函数 < math > Phi (mathbf { k }) </math > 在被积函数中出现，因为位置波函数和动量空间波函数是相互之间的傅里叶变换。 "Recherches sur la théorie des quanta"] [On the Theory of Quanta]. Annales de Physique The foundation of the equation is structured to be a linear differential equation based on classical energy conservation, and consistent with the De Broglie relations. The solution is the wave function , which contains all the information that can be known about the system. In the Copenhagen interpretation, the modulus of is related to the probability the particles are in some spatial configuration at some instant of time. Solving the equation for can be used to predict how the particles will behave under the influence of the specified potential and with each other. 这个方程的基础是一个基于经典能量守恒的线性微分方程，与德布罗意关系是一致的。解决方案是波函数，它包含了所有可以知道的关于系统的信息。在哥本哈根诠释中，粒子的模数与粒子在某一时刻处于某种空间位置的概率有关。求解方程可以用来预测粒子在指定势能的影响下以及相互影响下的行为。. 10 (3 The Schrödinger equation was developed principally from the De Broglie hypothesis, a wave equation that would describe particles, and can be constructed as shown informally in the following sections. For a more rigorous description of Schrödinger's equation, see also Resnick et al. 薛定谔方程理论主要是从德布罗意假说发展而来的,德布罗意假说是一个描述粒子的波动方程,可以在下面的章节中非正式地构造出来。关于薛定谔方程更严格的描述,参见雷斯尼克等人的文章。): 22–128. Bibcode:1925AnPh...10...22D. doi:[//doi.org/10.1051%2Fanphys%2F192510030022%0A%0A%7F%27%22%60UNIQ--math-00000178-QINU%60%22%27%7F%0A%0A%3C%20math%20%3E%20Psi%20%28mathbf%20%7B%20r%20%7D%20%EF%BC%8Ct%29%20%3D%20frac%20%7B1%7D%7B%20left%20%28sqrt%20%7B2%20pi%20%7D%20%EF%BC%8Cright%29%20%5E%203%7D%20int%20Phi%20%28mathbf%20%7B%20k%20%7D%29%20e%20%5E%20%7B%20i%20%28mathbf%20%7B%20k%20%7D%20cdot%20mathbf%20%7B%20r%20%7D-omega%20t%29%7D%20d%20%5E%203%20mathbf%20%7B%20k%20%7D%20%EF%BC%8C%21%20%3C%2Fmath%20%3E 10.1051/anphys/192510030022 '"UNIQ--math-00000178-QINU"' < math > Psi (mathbf { r } ，t) = frac {1}{ left (sqrt {2 pi } ，right) ^ 3} int Phi (mathbf { k }) e ^ { i (mathbf { k } cdot mathbf { r }-omega t)} d ^ 3 mathbf { k } ，! </math >]. {{cite journal}}: |archive-url= requires |archive-date= (help); Check |archiveurl= value (help); Check |doi= value (help); Check date values in: |year= (help); line feed character in |archiveurl= at position 106 (help); line feed character in |doi= at position 28 (help); line feed character in |issue= at position 2 (help); line feed character in |journal= at position 24 (help); line feed character in |last= at position 11 (help); line feed character in |year= at position 5 (help)CS1 maint: extra punctuation (link) CS1 maint: multiple names: authors list (link) .
18. Weissman, M.B.; V. V. Iliev; I. Gutman (2008). "A pioneer remembered: biographical notes about Arthur Constant Lunn". Communications in Mathematical and in Computer Chemistry. 59 (3): 687–708.
19. Samuel I. Weissman; Michael Weissman (1997). "Alan Sokal's Hoax and A. Lunn's Theory of Quantum Mechanics". Physics Today. 50, 6 (6): 15. Bibcode:1997PhT....50f..15W. doi:10.1063/1.881789.
20. Kamen, Martin D. (1985). Radiant Science, Dark Politics. Berkeley and Los Angeles, California: University of California Press. pp. 29–32. ISBN 978-0-520-04929-1.
21. Weissman, M.B.; V. V. Iliev; I. Gutman (2008). "A pioneer remembered: biographical notes about Arthur Constant Lunn". Communications in Mathematical and in Computer Chemistry. 59 (3): 687–708.
22. Samuel I. Weissman; Michael Weissman (1997). "Alan Sokal's Hoax and A. Lunn's Theory of Quantum Mechanics". Physics Today. 50, 6 (6): 15. Bibcode:1997PhT....50f..15W. doi:10.1063/1.881789.
23. Kamen, Martin D. (1985). Radiant Science, Dark Politics. Berkeley and Los Angeles, California: University of California Press. pp. 29–32. ISBN 978-0-520-04929-1.
24. $\displaystyle{ -V'\left(\left\langle X\right\rangle\right) }$, [ math ]-v’ left (left langle x right rangle) {{Cite book which is typically not the same as $\displaystyle{ -\left\langle V'(X)\right\rangle }$. In the case of the quantum harmonic oscillator, however, $\displaystyle{ V' }$ is linear and this distinction disappears, so that in this very special case, the expected position and expected momentum do exactly follow the classical trajectories. 这通常不同于 < math >-left langle v’(x) right rangle </math > 。然而，在量子谐振子的例子中，< math > v’ </math > 是线性的，这种区别消失了，所以在这个特殊的例子中，预期的位置和预期的动量确实遵循了经典的轨迹。 |last=Schrödinger |first=E. |year=1984 For general systems, the best we can hope for is that the expected position and momentum will approximately follow the classical trajectories. If the wave function is highly concentrated around a point $\displaystyle{ x_0 }$, then $\displaystyle{ V'\left(\left\langle X\right\rangle\right) }$ and $\displaystyle{ \left\langle V'(X)\right\rangle }$ will be almost the same, since both will be approximately equal to $\displaystyle{ V'(x_0) }$. In that case, the expected position and expected momentum will remain very close to the classical trajectories, at least for as long as the wave function remains highly localized in position. When Planck's constant is small, it is possible to have a state that is well localized in both position and momentum. The small uncertainty in momentum ensures that the particle remains well localized in position for a long time, so that expected position and momentum continue to closely track the classical trajectories. 对于一般系统，我们所能期望的最好结果是，预期的位置和动量将近似遵循传统的轨迹。如果波函数高度集中在一个点附近，那么“ math”“ v”左(左小角 x 右小角)和“ math”“左小角 v’(x)右小角 v’(x)右小角 </math > 几乎是一样的，因为两者大致相等于 math < v’(x _ 0) </math > 。在这种情况下，预期的位置和预期的动量将保持非常接近传统的轨迹，至少在波函数仍然高度局部化的位置。当普朗克常数很小时，就可能有一个位置和动量都很好定域的状态。动量的微小不确定性保证了粒子在很长一段时间内保持很好的局部位置，从而使预期的位置和动量继续紧密地跟踪经典的轨迹。 |title=Collected papers |publisher=Friedrich Vieweg und Sohn The Schrödinger equation in its general form 薛定谔方程的一般形式 |isbn=978-3-7001-0573-2 }} See introduction to first 1926 paper.
25. Encyclopaedia of Physics (2nd Edition), R. G. Lerner, G. L. Trigg, VHC publishers, 1991, (Verlagsgesellschaft) 3-527-26954-1, (VHC Inc.)
26. $\displaystyle{ V(x_1,x_2,\ldots, x_N) = \sum_{n=1}^N V(x_n) \, . }$ V (x _ 1，x _ 2，ldots，x _ n) = sum _ { n = 1} ^ n v (x _ n) ，. </math > Sommerfeld, A. and the wave function can be written as a product of the wave functions for each particle: 波函数可以写成每个粒子的波函数的乘积: (1919). Atombau und Spektrallinien $\displaystyle{ \Psi(x_1,x_2,\ldots, x_N,t) = e^{-i{E t/\hbar}}\prod_{n=1}^N\psi(x_n) \, , }$ < math > Psi (x _ 1，x _ 2，ldots，x _ n，t) = e ^ {-i { e t/hbar } prod _ { n = 1} ^ n (x _ n) ，</math >. Braunschweig For non-interacting identical particles, the potential is still a sum, but wave function is a bit more complicated – it is a sum over the permutations of products of the separate wave functions to account for particle exchange. In general for interacting particles, the above decompositions are not possible. 对于没有相互作用的全同粒子，势仍然是一个和，但波函数要复杂一些——它是单独波函数的置换乘积的和，用来解释粒子交换。一般来说，对于相互作用的粒子，上面的分解是不可能的。: Friedrich Vieweg und Sohn. ISBN 978-3-87144-484-5.
27. For an English source, see 有关英文来源，请参阅Haar, T. For no potential, 0. {{cite journal}}: Cite journal requires |journal= (help); Missing or empty |title= (help); line feed character in |first= at position 3 (help), so the particle is free and the equation reads: and atoms or ions in lattices, and approximating other potentials near equilibrium points. It is also the basis of perturbation methods in quantum mechanics. 因为没有势，0. {{cite journal}}: Cite journal requires |journal= (help); |author2= has generic name (help); External link in |author2= (help); Missing or empty |title= (help); line feed character in |author2= at position 12 (help)CS1 maint: multiple names: authors list (link)，所以粒子是自由的，方程是: 和晶格中的原子或离子，并且近似平衡点附近的其他势。这也是量子力学微扰法的基础。 ed.). pp. 361–377 The extension from one dimension to three dimensions is straightforward, all position and momentum operators are replaced by their three-dimensional expressions and the partial derivative with respect to space is replaced by the gradient operator. 从一维到三维的扩展是很直接的，所有的位置和动量算符都被它们的三维表达式所代替，关于空间的偏导数被梯度算符所代替。. Bibcode 1926AnP...384..361S. doi:10.1002/andp.19263840404. ISBN [[Special:BookSources/$\displaystyle{ \psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{ \lt math \gt psi _ n (x) = sqrt { frac {1}{2 ^ n，n! }Cdot left (frac { m omega }{ pi hbar } right) ^ {1/4} cdot e ^ { }}\lt /span\gt \lt /li\gt \lt li id="cite_note-34"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-34|↑]]\lt /span\gt \lt span class="reference-text"\gt {{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}\lt /span\gt \lt /li\gt \lt li id="cite_note-35"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-35|↑]]\lt /span\gt \lt span class="reference-text"\gt {{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}\lt /span\gt \lt /li\gt \lt li id="cite_note-Schr.C3.B6dinger1982-36"\gt \lt span class="mw-cite-backlink"\gt ↑ \lt sup\gt [[#cite_ref-Schr.C3.B6dinger1982_36-0|30.0]]\lt /sup\gt \lt sup\gt [[#cite_ref-Schr.C3.B6dinger1982_36-1|30.1]]\lt /sup\gt \lt /span\gt \lt span class="reference-text"\gt {{cite book|author=Erwin Schrödinger|title=Collected Papers on Wave Mechanics: Third Edition|year=1982|publisher=American Mathematical Soc.|isbn=978-0-8218-3524-1}}\lt /span\gt \lt /li\gt \lt li id="cite_note-.E2.80.9CSchr.C3.B6dinger1982.E2.80.9D-37"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-.E2.80.9CSchr.C3.B6dinger1982.E2.80.9D_37-0|↑]]\lt /span\gt \lt span class="error mw-ext-cite-error" lang="zh-Hans-CN" dir="ltr"\gt 引用错误：无效\lt code\gt <ref>\lt /code\gt 标签；未给name属性为\lt code\gt “Schrödinger1982”\lt /code\gt 的引用提供文字\lt /span\gt \lt /li\gt \lt li id="cite_note-38"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-38|↑]]\lt /span\gt \lt span class="reference-text"\gt {{cite journal |last=Schrödinger |first=E. \lt math\gt E_n = \left(n + \frac{1}{2} \right) \hbar \omega. }$ < math > e _ n = left (n + frac {1}{2} right) hbar omega.数学|$\displaystyle{ \psi_n(x) = \sqrt{\frac{1}{2^n\,n!}} \cdot \left(\frac{m\omega}{\pi \hbar}\right)^{1/4} \cdot e^{ \lt math \gt psi _ n (x) = sqrt { frac {1}{2 ^ n，n! }Cdot left (frac { m omega }{ pi hbar } right) ^ {1/4} cdot e ^ { }}\lt /span\gt \lt /li\gt \lt li id="cite_note-34"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-34|↑]]\lt /span\gt \lt span class="reference-text"\gt {{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}\lt /span\gt \lt /li\gt \lt li id="cite_note-35"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-35|↑]]\lt /span\gt \lt span class="reference-text"\gt {{Cite news|last=Teresi|first=Dick|date=1990-01-07|title=THE LONE RANGER OF QUANTUM MECHANICS (Published 1990)|language=en-US|work=The New York Times|url=https://www.nytimes.com/1990/01/07/books/the-lone-ranger-of-quantum-mechanics.html|access-date=2020-10-13|issn=0362-4331}}\lt /span\gt \lt /li\gt \lt li id="cite_note-Schr.C3.B6dinger1982-36"\gt \lt span class="mw-cite-backlink"\gt ↑ \lt sup\gt [[#cite_ref-Schr.C3.B6dinger1982_36-0|30.0]]\lt /sup\gt \lt sup\gt [[#cite_ref-Schr.C3.B6dinger1982_36-1|30.1]]\lt /sup\gt \lt /span\gt \lt span class="reference-text"\gt {{cite book|author=Erwin Schrödinger|title=Collected Papers on Wave Mechanics: Third Edition|year=1982|publisher=American Mathematical Soc.|isbn=978-0-8218-3524-1}}\lt /span\gt \lt /li\gt \lt li id="cite_note-.E2.80.9CSchr.C3.B6dinger1982.E2.80.9D-37"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-.E2.80.9CSchr.C3.B6dinger1982.E2.80.9D_37-0|↑]]\lt /span\gt \lt span class="error mw-ext-cite-error" lang="zh-Hans-CN" dir="ltr"\gt 引用错误：无效\lt code\gt <ref>\lt /code\gt 标签；未给name属性为\lt code\gt “Schrödinger1982”\lt /code\gt 的引用提供文字\lt /span\gt \lt /li\gt \lt li id="cite_note-38"\gt \lt span class="mw-cite-backlink"\gt [[#cite_ref-38|↑]]\lt /span\gt \lt span class="reference-text"\gt {{cite journal |last=Schrödinger |first=E. \lt math\gt E_n = \left(n + \frac{1}{2} \right) \hbar \omega. }$ < math > e _ n = left (n + frac {1}{2} right) hbar omega.数学]].
28. generating the equation 生成方程式 Einstein, A. $\displaystyle{ \left[-\frac{\hbar^2}{2m}\nabla^2 + V(\mathbf{r}) \right]\psi(\mathbf{r}) = E\psi(\mathbf{r}) }$ < math > left [-frac { hbar ^ 2}{2m } nabla ^ 2 + v (mathbf { r }) right ] psi (mathbf { r }) = e psi (mathbf { r }) </math >; et al. "Letters on Wave Mechanics: Schrodinger–Planck–Einstein–Lorentz with stationary state solutions of the form 以定态溶液的形式". {{cite journal}}: Cite journal requires |journal= (help); line feed character in |first= at position 3 (help); line feed character in |title= at position 63 (help)
29. where the position of the particle is $\displaystyle{ \mathbf{r} }$. 其中粒子的位置是 < math > mathbf { r } </math > 。 Moore, W.J. For $\displaystyle{ N }$ particles in three dimensions, the Hamiltonian is 对于三维空间中的粒子，哈密顿函数是 (1992). Schrödinger: Life and Thought $\displaystyle{ \hat{H} = \sum_{n=1}^{N}\frac{\hat{\mathbf{p}}_n\cdot\hat{\mathbf{p}}_n}{2m_n} + V(\mathbf{r}_1,\mathbf{r}_2,\ldots,\mathbf{r}_N) \,,\quad \hat{\mathbf{p}}_n = -i\hbar \nabla_n }$ 2 m n } + v (mathbf { r } _ 1，mathbf { r } _ 2，ldots，mathbf { r } _ n) ，4 h { p } _ n =-i hbar _ n </math >. Cambridge University Press. ISBN [[Special:BookSources/978-0-521-43767-7 where the position of particle is and the gradient operators are partial derivatives with respect to the particle's position coordinates. In Cartesian coordinates, for particle , the position vector is (xn, yn, zn)|978-0-521-43767-7 where the position of particle is and the gradient operators are partial derivatives with respect to the particle's position coordinates. In Cartesian coordinates, for particle , the position vector is (xn, yn, zn)]].  while the gradient and Laplacian operator are respectively: 其中粒子的位置是和梯度算子是偏导数的粒子的位置坐标。在笛卡尔坐标系下，对于粒子，位置向量为(x < sub > n ，y < sub > n ，z < sub > n )} ，而梯度算子和拉普拉斯算子分别为: }}
30. 很明显即使在他生命的最后一年，如写给马克斯·伯恩的信所示，薛定谔从未接受哥本哈根的解释。[34]:220
31. Molecular Quantum Mechanics Parts I and II: An Introduction to Quantum Chemistry (Volume 1), P. W. Atkins, Oxford University Press, 1977,
32. Quanta: A handbook of concepts, P. W. Atkins, Oxford University Press, 1974,
33. Physics of Atoms and Molecules, B. H. Bransden, C. J. Joachain, Longman, 1983,
34. Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985,
35. Quantum Mechanics Demystified, D. McMahon, McGraw Hill (USA), 2006,
36. 引用错误：无效<ref>标签；未给name属性为“Quanta的引用提供文字
37. Analytical Mechanics, L. N. Hand, J. D. Finch, Cambridge University Press, 2008,
38. 脚本错误：没有“Footnotes”这个模块。 Section 3.7.5
39. 脚本错误：没有“Footnotes”这个模块。 Section 3.7.5
40. 脚本错误：没有“Footnotes”这个模块。 p. 78
41. {harvnb | Hall | 2013}p.78