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=== Infinite-order PDEs in quantum mechanics ===

=== Infinite-order PDEs in quantum mechanics ===

In the [[phase space formulation]] of quantum mechanics,  one may consider the [[Method of quantum characteristics|quantum Hamilton's equations]] for trajectories of quantum particles. These equations are infinite-order PDEs. However, in the semiclassical expansion, one has a finite system of ODEs at any fixed order of [[Dirac constant|{{mvar|ħ}}]].  The evolution equation of the [[Wigner quasi-probability distribution|Wigner function]] is also an infinite-order PDE. The quantum trajectories are [[Method of quantum characteristics|quantum characteristics]], with the use of which one could calculate the evolution of the Wigner function.

In the [[phase space formulation]] of quantum mechanics,  one may consider the [[Method of quantum characteristics|quantum Hamilton's equations]] for trajectories of quantum particles. These equations are infinite-order PDEs. However, in the semiclassical expansion, one has a finite system of ODEs at any fixed order of [[Dirac constant|{{mvar|ħ}}]].  The evolution equation of the [[Wigner quasi-probability distribution|c]] is also an infinite-order PDE. The quantum trajectories are [[Method of quantum characteristics|quantum characteristics]], with the use of which one could calculate the evolution of the Wigner function.

In the phase space formulation of quantum mechanics,  one may consider the quantum Hamilton's equations for trajectories of quantum particles. These equations are infinite-order PDEs. However, in the semiclassical expansion, one has a finite system of ODEs at any fixed order of Dirac constant|.  The evolution equation of the Wigner function is also an infinite-order PDE. The quantum trajectories are quantum characteristics, with the use of which one could calculate the evolution of the Wigner function.

In the phase space formulation of quantum mechanics,  one may consider the quantum Hamilton's equations for trajectories of quantum particles. These equations are infinite-order PDEs. However, in the semiclassical expansion, one has a finite system of ODEs at any fixed order of Dirac constant|.  The evolution equation of the Wigner function is also an infinite-order PDE. The quantum trajectories are quantum characteristics, with the use of which one could calculate the evolution of the Wigner function.

在《相空间表述量子力学，我们可以考虑量子哈密顿的量子粒子轨迹方程。这些方程是无限阶偏微分方程。然而，在半经典展开中，我们有一个有限的狄拉克常数序列 | 的常微分方程组。维格纳函数的发展方程也是一个无限阶偏微分方程。量子轨道具有量子特性，利用量子轨道可以计算维格纳函数的演化。

在量子力学相空间表述下，我们可以考虑用于求解量子粒子的轨迹的量子哈密顿的方程。这些方程是无限阶偏微分方程。然而，在半经典展开中，我们在给定[[Dirac constant|{{mvar|ħ}}]]阶数下有一个有限的的常微分方程组。'''<font color="#ff8000">维格纳函数  Wigner Function</font>的演化方程也是一个无限阶偏微分方程。由于量子轨道的量子特性，所以它通常可以用来计算维格纳函数的演化。

== Analytical solutions ==

== Analytical solutions ==