Chin Siu A
Department of Physics, Texas A&M University, College Station, Texas 77843, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2008 Jun;77(6 Pt 2):066401. doi: 10.1103/PhysRevE.77.066401. Epub 2008 Jun 3.
The exponential splitting of the classical evolution operator yields symplectic integrators if the canonical Hamiltonian is separable. Similar splitting of the noncanonical evolution operator for a charged particle in a magnetic field produces exact energy-conserving algorithms. The latter algorithms evaluate the magnetic field directly with no need of a vector potential and are more stable with far less phase errors than symplectic integrators. For a combined electric and magnetic field, these algorithms from splitting the noncanonical evolution operator are neither fully symplectic nor exactly energy conserving, yet they behave exactly like symplectic algorithms in having qualitatively correct trajectories and bounded periodic energy errors. This work shows that exponential-splitting algorithms of any order for solving particle trajectories in a general electric and magnetic field can be systematically derived by use of the angular momentum operator of quantum mechanics. The use of operator analysis in this work fully comprehends the intertwining interaction between electric and magnetic forces and makes possible the derivation of highly nontrivial integrators.
如果正则哈密顿量是可分离的，经典演化算符的指数分裂会产生辛积分器。对于磁场中带电粒子的非正则演化算符进行类似的分裂会产生精确的能量守恒算法。后一种算法直接计算磁场，无需矢量势，并且比辛积分器更稳定，相位误差要小得多。对于组合电场和磁场，这些由非正则演化算符分裂得到的算法既不是完全辛的，也不是精确能量守恒的，但它们在具有定性正确的轨迹和有界的周期能量误差方面，表现得与辛算法完全一样。这项工作表明，通过使用量子力学的角动量算符，可以系统地推导出用于求解一般电场和磁场中粒子轨迹的任意阶指数分裂算法。在这项工作中使用算符分析充分理解了电力和磁力之间的交织相互作用，并使得推导高度不平凡的积分器成为可能。
