延伸分離動能法求解多體系統

Abstract

本文延伸本實驗室發展的一個創新且有系統的方法－動能分離法(Kinetic Energy Partition)，用於求解量子力學中特徵值問題。其中利用動能項中質量的拆解，將拆解後的動能項分別與對應的位能項形成子系統，將總漢米爾頓系統拆解為多個子系統之和。首先從最簡單的單體問題，逐漸增加複雜度包含粒子數、維度以及交互作用模式來討論。其中在多電子系統中，使用“負質量”的觀念，來處理斥力位能的部分，使其能量有束縛解，以降低具有連續能增的基底函數的數目。以及在多體系統中，為了化簡動能分離法的過程，使用絕熱近似法(Adiabatic Approximation)將動能分離法作近似。本研究使用摩辛斯基原子及虎克原子和狄拉克原子等模型來檢視動能分離法的可行性，其誤差大致在5%以內的精準度。並且首次使用動能分離法挑戰三體問題，使用一維摩辛斯基原子以證明此方法是有機會推廣至多體問題。此外另一個研究部分是變分的動能分離法(Variational KEP)，將探討動能分離法的理論基礎，研究其和變分法的關係。因為動能分離法的一個特點是能夠引入新的變分參數，我們相信變分的動能分離法能夠改進原來的動能分離法，使它達到更高準確度，並使用最簡單的例子是雙負狄拉克函數位能來驗證其可行性。

This study extends a novel and systematic scheme, kinetic energy partition (KEP) method, developed by our group for solving the general quantum eigenvalue problems. The key point of the KEP method is to split the mass factor into effective ones, each to be associated with partial kinetic energy terms, and the full Hamiltonian of the system can be wrote as the sum of subsystem Hamiltonians. Starting from the simple one-particle problems, we gradually increase the complexity in particle number, dimension and interaction patterns. For the many-body system, we propose to use the idea of “negative mass” to deal with the repulsive interaction potential, and in order to reduce the number of basis sets with continuous-energy. In addition, to simplify the procedure of KEP method, we employ an adiabatic approximation. We will test the utility of the KEP method with the models such as Moshinsky atoms, Hookium atom and Dirackium atom which error within 5%. Furthermore, to challenge the three-body problem first, use the one-dimensional Moshinsky atoms to prove it has the opportunity to confront the quantum many-body problems. Moreover, the other part is variational kinetic energy partition method (VKEP). We study the theoretical background of the KEP method by studying its relation with the variational principles. Owing to the new variational parameters provided by the KEP method, we believe VKEP method can improve KEP method to make it more precise. We use the simplest case of double delta potential to verify its feasibility.