Numerical Methods for the Wigner Equation with Unbounded Potential
Numerical Methods for the Wigner Equation with Unbounded Potential
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DOI：
10.1007/s10915-018-0853-0
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发表时间：
2018-10
影响因子：
2.5
通讯作者：
Zhenzhu Chen;Yunfeng Xiong;Sihong Shao
中科院分区：
文献类型：
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作者：
Zhenzhu Chen;Yunfeng Xiong;Sihong Shao
由于量子隧穿的存在，无界势常被用来严格限制量子动力学，产生束缚态或定态。然而，现有的精确维格纳求解器通常是针对局域势或多项式型势设计的。本文利用伪微分算子的两种等价形式：积分形式和级数形式（即，Moyal扩展）。无穷远处的无界部分由多项式近似或建模，然后剩余的局部势支配中心区域。充分利用了Moyal展开式对于多项式势可化为有限级数的事实。为了精确求解伪微分算子和线性微分算子，采用了相空间的谱配置格式和显式四阶Runge-Kutta时间离散。我们能够证明，由此产生的全离散光谱方案保持质量和能量。对几个典型的量子系统进行了高精度的模拟，得到了宏观可测量的可靠估计。
Unbounded potentials are always utilized to strictly confine quantum dynamics and generate bound or stationary states due to the existence of quantum tunneling. However, the existed accurate Wigner solvers are often designed for either localized potentials or those of the polynomial type. This paper attempts to solve the time-dependent Wigner equation in the presence of a general class of unbounded potentials by exploiting two equivalent forms of the pseudo-differential operator: integral form and series form (i.e., the Moyal expansion). The unbounded parts at infinities are approximated or modeled by polynomials and then a remaining localized potential dominates the central area. The fact that the Moyal expansion reduces to a finite series for polynomial potentials is fully utilized. In order to accurately resolve both the pseudo-differential operator and the linear differential operator, a spectral collocation scheme for the phase space and an explicit fourth-order Runge–Kutta time discretization are adopted. We are able to prove that the resulting full discrete spectral scheme conserves both mass and energy. Several typical quantum systems are simulated with a high accuracy and reliable estimation of macroscopically measurable quantities is thus obtained.