竞争势量子系统的分裂动能方法
Split Kinetic Energy Method for Quantum Systems with Competing Potentials
摘要：
对于具有竞争势的量子系统，传统的微扰理论往往得到一个渐近级数，随后的数值结果变得不确定。针对这类问题，本文提出了一种基于能量剖分思想的一般求解方案。代替将势能分成“未扰动”和“扰动”项，执行动能的划分。通过将动能项部分地分配到每个单独的势中，哈密顿量可以表示为具有相应竞争势的子系统哈密顿量的和。通过使用各个子系统哈密尔顿算子的基组的线性组合来扩展总波函数。我们首先说明的解决方案的过程中使用一个简单的系统，由一个粒子的作用下的双δ函数电位。然后，将该方法应用于强磁场中的带电谐振子和氢分子离子的原型系统。与通常的摄动法相比，新格式收敛到特征值和特征函数的精确解的速度都快得多。当适当扩展时，这个新的解方案对于处理强耦合量子系统是非常有用的。
英文摘要：
For quantum systems with competing potentials, the conventional perturbation theory often yields an asymptotic series and the subsequent numerical outcome becomes uncertain. To tackle such kind of problems, we develop a general solution scheme based on a new energy dissection idea. Instead of dividing the potential energy into "unperturbed" and "perturbed" terms, a partition of the kinetic energy is performed. By distributing the kinetic energy term in part into each individual potential, the Hamiltonian can be expressed as the sum of the subsystem Hamiltonians with respective competing potentials. The total wavefunction is expanded by using a linear combination of the basis sets of respective subsystem Hamiltonians. We first illustrate the solution procedure using a simple system consisting of a particle under the action of double delta-function potentials. Next, this method is applied to the prototype systems of a charged harmonic oscillator in strong magnetic field and the hydrogen molecule ion. Compared with the usual perturbation approach, this new scheme converges much faster to the exact solutions for both eigenvalues and eigenfunctions. When properly extended, this new solution scheme can be very useful for dealing with strongly coupling quantum systems.
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