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===Time-dependent equation时变方程===

===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:<ref name=Shankar1994>

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

薛定谔方程的形式取决于物理情况（特殊情况见下文）。最普遍的形式是含时薛定谔方程（TDSE），它描述了系统随时间演化的过程：<ref name=Shankar1994>

薛定谔方程的形式取决于物理情况（特殊情况见下文）。最普遍的形式是含时薛定谔方程（TDSE），它描述了系统随时间演化的过程：<ref name=Shankar1994>

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".]]

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”。]

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”。]]

  |year=1943

  |year=1943

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

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

最著名的例子是位置空间 < math > Psi (mathbf { r } ，t) </math > 中波函数的非相对论性薛定谔方程。

最著名的例子是位置空间<math>V(\mathbf{r},t)</math> 中波函数的非相对论性薛定谔方程。

  |title=Principles of Quantum Mechanics

  |title=Principles of Quantum Mechanics

[[File:Wave packet (dispersion).gif|thumb|200px|A [[wave function]] that satisfies the nonrelativistic Schrödinger equation with {{math|''V'' {{=}} 0}}. In other words, this corresponds to a particle traveling freely through empty space. The [[real part]] of the [[wave function]] is plotted here.]]

[[File:Wave packet (dispersion).gif|thumb|200px|A [[wave function]] that satisfies the nonrelativistic Schrödinger equation with {{math|''V'' {{=}} 0}}. In other words, this corresponds to a particle traveling freely through empty space. The [[real part]] of the [[wave function]] is plotted here.]]

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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.<ref name="Quanta 1974"/>

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.<ref name="Quanta 1974"/>

===Wave and particle motion===

===Wave and particle motion===