更改

In [[classical mechanics]], [[Newton's second law]] ({{math|'''F''' {{=}} ''m'''''a'''}})<ref group="note">While this is the most famous form of Newton's second law, it is not the most general, being valid only for objects of constant mass. Newton's second law reads <math> \mathbf{F} = \frac{d}{dt}(m \mathbf{v})</math>, the net force acting on a body is equal to the total time derivative of the total momentum of that body—which is equivalent to the given form when mass is constant with time.</ref> is used to make a mathematical prediction as to what path a given physical system will take over time following a set of known initial conditions. Solving this equation gives the position and the momentum of the physical system as a function of the external force <math>\mathbf{F}</math> on the system. Those two parameters are sufficient to describe its state at each time instant. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation.

In [[classical mechanics]], [[Newton's second law]] ({{math|'''F''' {{=}} ''m'''''a'''}})<ref group="note">While this is the most famous form of Newton's second law, it is not the most general, being valid only for objects of constant mass. Newton's second law reads <math> \mathbf{F} = \frac{d}{dt}(m \mathbf{v})</math>, the net force acting on a body is equal to the total time derivative of the total momentum of that body—which is equivalent to the given form when mass is constant with time.</ref> is used to make a mathematical prediction as to what path a given physical system will take over time following a set of known initial conditions. Solving this equation gives the position and the momentum of the physical system as a function of the external force <math>\mathbf{F}</math> on the system. Those two parameters are sufficient to describe its state at each time instant. In quantum mechanics, the analogue of Newton's law is Schrödinger's equation.

在[[经典力学]]中，[[牛顿第二定律]]({{math|'''F''' {{=}} ''m'''''a'''}})<ref group="note">While this is the most famous form of Newton's second law, it is not the most general, being valid only for objects of constant mass. Newton's second law reads <math> \mathbf{F} = \frac{d}{dt}(m \mathbf{v})</math>, the net force acting on a body is equal to the total time derivative of the total momentum of that body—which is equivalent to the given form when mass is constant with time.</ref> 用于数学预测给定物理系统在一组已知初始条件下随时间的变化路径。解这个方程就得到了物理系统的位置和动量，它是系统上外力的函数。这两个参数足以描述它在每个时刻的状态。在量子力学中，牛顿定律的类似物是薛定谔方程。

在[[经典力学]]中，[[牛顿第二定律]]({{math|'''F''' {{=}} ''m'''''a'''}})<ref group="note">While this is the most famous form of Newton's second law, it is not the most general, being valid only for objects of constant mass. Newton's second law reads <math> \mathbf{F} = \frac{d}{dt}(m \mathbf{v})</math>, the net force acting on a body is equal to the total time derivative of the total momentum of that body—which is equivalent to the given form when mass is constant with time.</ref> 用于数学预测给定物理系统在一组已知初始条件下随时间的变化路径。解这个方程就得到了物理系统的位置和动量，它是系统上外力的函数。这两个参数足以描述它在每个时刻的状态。在量子力学中，牛顿定律的对应物是薛定谔方程。

The concept of a wave function is a fundamental [[Mathematical formulation of quantum mechanics|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 operator|unitary]], and must therefore be generated by the exponential of a [[self-adjoint operator]], which is the quantum [[Hamiltonian (quantum mechanics)|Hamiltonian]]. This derivation is explained below.

The concept of a wave function is a fundamental [[Mathematical formulation of quantum mechanics|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 operator|unitary]], and must therefore be generated by the exponential of a [[self-adjoint operator]], which is the quantum [[Hamiltonian (quantum mechanics)|Hamiltonian]]. This derivation is explained below.

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.]]

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

[[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 [[molecule|molecular]], [[atom]]ic, and [[subatomic particle|subatomic]] systems, but also [[macroscopic scale|macroscopic system]]s, possibly even the whole [[universe]].<ref name=Laloe>{{citation|last=Laloe|first=Franck|title=Do We Really Understand Quantum Mechanics| publisher=Cambridge University Press|year=2012| isbn = 978-1-107-02501-1}}</ref>{{rp|292ff}}

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 [[molecule|molecular]], [[atom]]ic, and [[subatomic particle|subatomic]] systems, but also [[macroscopic scale|macroscopic system]]s, possibly even the whole [[universe]].<ref name=Laloe>{{citation|last=Laloe|first=Franck|title=Do We Really Understand Quantum Mechanics| publisher=Cambridge University Press|year=2012| isbn = 978-1-107-02501-1}}</ref>{{rp|292ff}}

{{Equation box 1

{{Equation box 1

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.

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&nbsp;(general)

|title=Time-dependent Schrödinger equation&nbsp;(general)