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→Nonrelativistic quantum mechanics
*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。
*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==
==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 [[speed of light|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 {{math|''N''}} particles.
The quantum mechanics of particles without accounting for the effects of [[special relativity]], for example particles propagating at speeds much less than [[speed of light|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 {{math|''N''}} particles.
不考虑[[狭义相对论]]影响的粒子量子力学，例如以远低于[[光速|光速]]的速度传播的粒子，被称为“非相对论量子力学”。以下是薛定谔方程在不同情况下的几种形式：时间独立性和依赖性，一个和三个空间维度，以及一个和{数学|''''}粒子。
In actuality, the particles constituting the system do not have the numerical labels used in theory. The language of mathematics forces us to label the positions of particles one way or another, otherwise there would be confusion between symbols representing which variables are for which particle.<ref name="Atoms, Molecules 1985">''Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles'' (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, {{isbn|978-0-471-87373-0}}</ref>
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.<ref name="Atoms, Molecules 1985">''Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles'' (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, {{isbn|978-0-471-87373-0}}</ref>
实际上，组成系统的粒子没有理论上使用的数字标签。数学语言迫使我们以这样或那样的方式标注粒子的位置，否则在表示哪个变量是哪个粒子的符号之间会有混淆。<ref name="Atoms, Molecules 1985">''Quantum Physics of Atoms, Molecules, Solids, Nuclei and Particles'' (2nd Edition), R. Resnick, R. Eisberg, John Wiley & Sons, 1985, {{isbn|978-0-471-87373-0}}</ref>
===Time-independent===
===Time-independent时间无关性===
If the Hamiltonian is not an explicit function of time, the equation is [[Separation of variables|separable]] into a product of spatial and temporal parts. In general, the wave function takes the form:
If the Hamiltonian is not an explicit function of time, the equation is [[Separation of variables|separable]] into a product of spatial and temporal parts. In general, the wave function takes the form:
如果哈密顿量不是时间的显式函数，则方程是[[变量分离|可分离]]空间和时间部分的乘积。一般而言，波函数的形式如下：
:<math>\Psi(\text{space coords},t)=\psi(\text{space coords})\tau(t)\,.</math>
:<math>\Psi(\text{space coords},t)=\psi(\text{space coords})\tau(t)\,.</math>
where {{math|''ψ''(space coords)}} is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and {{math|''τ''(''t'')}} is a function of time only.
where {{math|''ψ''(space coords)}} is a function of all the spatial coordinate(s) of the particle(s) constituting the system only, and {{math|''τ''(''t'')}} is a function of time only.
其中{math |''ψ''（空间坐标）}}仅是构成系统的粒子的所有空间坐标的函数，{math |''τ''（'t'）}}仅是时间的函数。
Substituting for {{math|''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:<ref name="verlagsgesellschaft1991"/>
Substituting for {{math|''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:<ref name="verlagsgesellschaft1991"/>
将{数学|''ψ''}}代入薛定谔方程中，得到相关维数中的相关粒子数，通过[[变量分离]]求解意味着含时方程的通解具有以下形式：<ref name="verlagsgesellschaft1991"/>
:<math> \Psi(\text{space coords},t) = \psi(\text{space coords}) e^{-i{E t/\hbar}} \,.</math>
:<math> \Psi(\text{space coords},t) = \psi(\text{space coords}) e^{-i{E t/\hbar}} \,.</math>
Since the time dependent phase factor is always the same, only the spatial part needs to be solved for in time independent problems. Additionally, the energy operator {{math|''Ê'' {{=}} ''iħ''{{sfrac|∂|∂''t''}}}} can always be replaced by the energy eigenvalue {{math|''E''}}, thus the time independent Schrödinger equation is an [[eigenvalue]] equation for the Hamiltonian operator:<ref name=Shankar1994/>{{rp|143ff}}
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 {{math|''Ê'' {{=}} ''iħ''{{sfrac|∂|∂''t''}}}} can always be replaced by the energy eigenvalue {{math|''E''}}, thus the time independent Schrödinger equation is an [[eigenvalue]] equation for the Hamiltonian operator:<ref name=Shankar1994/>{{rp|143ff}}
由于与时间相关的相位因子总是相同的，所以对于与时间无关的问题，只需要求解空间部分。此外，能量算符{{math |''Ê{{=}}iħ{sfrac |∂Ӟ∂t'}}}}总是可以被能量本征值{math | E'}替换，因此与时间无关的薛定谔方程是哈密顿算符的[[本征值]]方程：<ref name=Shankar1994/>{{rp|143ff}}
:<math>\hat{H} \psi = E \psi </math>
:<math>\hat{H} \psi = E \psi </math>
This is true for any number of particles in any number of dimensions (in a time independent potential). This case describes the [[standing wave]] solutions of the time-dependent equation, which are the states with definite energy (instead of a probability distribution of different energies). In physics, these standing waves are called "[[stationary state]]s" or "[[energy eigenstate]]s"; in chemistry they are called "[[atomic orbital]]s" or "[[molecular orbital]]s". Superpositions of energy eigenstates change their properties according to the relative phases between the energy levels.
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 state]]s" or "[[energy eigenstate]]s"; in chemistry they are called "[[atomic orbital]]s" or "[[molecular orbital]]s". Superpositions of energy eigenstates change their properties according to the relative phases between the energy levels.
这是真实的任何数量的粒子在任何数量的维度（在一个时间无关的潜力）。这种情况描述了含时方程的[[驻波]]解，即具有确定能量的状态（而不是不同能量的概率分布）。在物理学中，这些驻波被称为“[[稳态]]s”或“[[能量本征态]]s”；在化学中，它们被称为“[[原子轨道]]s”或“[[分子轨道]]s”。能量本征态的叠加根据能级间的相对相位而改变其性质。
The energy eigenvalues from this equation form a discrete [[Spectrum (functional analysis)|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]].
The energy eigenvalues from this equation form a discrete [[Spectrum (functional analysis)|spectrum]] of values, so mathematically energy must be quantized. More specifically, the energy eigenstates form a basis – any wave function may be written as a sum over the discrete energy states or an integral over continuous energy states, or more generally as an integral over a measure. This is the [[spectral theorem]] in mathematics, and in a finite state space it is just a statement of the completeness of the eigenvectors of a [[Hermitian matrix]].
这个方程的能量特征值形成了一个离散的[[谱（泛函分析）|谱]]值，因此在数学上能量必须量子化。更具体地说，能量本征态构成了一个基础——任何波函数都可以写为离散能量态的和或连续能量态的积分，或者更一般地写为测度的积分。这就是数学中的[[谱定理]]，在有限状态空间中，它只是[[厄米矩阵]]特征向量完备性的一种表述。
===One-dimensional examples一维例子===
===One-dimensional examples===
For a particle in one dimension, the Hamiltonian is:
For a particle in one dimension, the Hamiltonian is:
对于一维粒子，哈密顿量为：
:<math> \hat{H} = \frac{\hat{p}^2}{2m} + V(x) \,, \quad \hat{p} = -i\hbar \frac{d}{d x} </math>
:<math> \hat{H} = \frac{\hat{p}^2}{2m} + V(x) \,, \quad \hat{p} = -i\hbar \frac{d}{d x} </math>
and substituting this into the general Schrödinger equation gives:
and substituting this into the general Schrödinger equation gives:
把它代入一般的薛定谔方程得到：
:<math> \left[-\frac{\hbar^2}{2m}\frac{d^2}{d x^2} + V(x) \right]\psi(x) = E\psi(x) </math>
:<math> \left[-\frac{\hbar^2}{2m}\frac{d^2}{d x^2} + V(x) \right]\psi(x) = E\psi(x) </math>
This is the only case the Schrödinger equation is an [[ordinary derivative|ordinary]] differential equation, rather than a [[partial derivative|partial]] differential equation. The general solutions are always of the form:
This is the only case the Schrödinger equation is an [[ordinary derivative|ordinary]] differential equation, rather than a [[partial derivative|partial]] differential equation. The general solutions are always of the form:
这是薛定谔方程是一个[[常导数|常]]微分方程，而不是一个[[偏导数|偏]]微分方程的唯一情况。一般解决方案的形式总是：
:<math> \Psi(x,t)=\psi(x) e^{-iEt/\hbar} \, . </math>
:<math> \Psi(x,t)=\psi(x) e^{-iEt/\hbar} \, . </math>
<small>This page was moved from [[wikipedia:en:Schrödinger equation]]. Its edit history can be viewed at [[薛定谔方程/edithistory]]</small></noinclude>
<small>This page was moved from [[wikipedia:en:Schrödinger equation]]. Its edit history can be viewed at [[薛定谔方程/edithistory]]</small></noinclude>
<small>此页摘自[[维基百科：英文：薛定谔方程]]。其编辑历史记录可在[[薛定谔方程/编辑历史]]查阅</small></noinclude>
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