應用超對稱動能分離法求解量子特徵值問題

Abstract

本文將本實驗室先前發展的一個系統性求解量子特徵值的方法—動能分離法(Kinetic Energy Partition)發展到算符基底的框架中。將KEP方法結合算符法產生一個新穎的方法用於處理普遍化位能的算符解法，稱之為超對稱動能分離法(SUSY KEP)，本方法將動能中的質量藉由可調整之質量參數分離成子動能項並與位能重組，構成多個等效子系統，對個別子系統設定其對應的階梯算符，總漢米爾頓重寫成子系統超對稱形式的總和，求解特徵值問題本質上即為找尋一組對應特徵值的基底線性組合，在此方法中考慮之基底為子系統階梯算符，透過計算，最終得到對應總系統之特徵值、座標空間、降階算符以及基態全解。 開頭先討論簡單模型，逐漸增加粒子數與維度討論其特性。其中多電子系統中引用「負質量」的概念來處理排斥交互作用力的部分。本研究使用模型為：單體雙簡諧位能、單體多簡諧位能系統以及不同電子數的Moshinsky原子模型來檢視SUSY KEP在處理低維度多作用力與高維度的多體問題是否可行。 本研究使用SUSY KEP 方法在兩種不同類型的模型上，第一種類為單體雙簡諧位能與之延伸的單體多簡諧位能，此類模型驗證在單體系統中，即使位能個數上升，藉由SUSY KEP方法所解出的基態解仍然準確，並且從久期矩陣所得到基態能量對於大尺度個數N的冪次關係也與計算結果所預測的趨勢以及精確解的冪次相同。第二類為Moshinsky原子模型，此模型特點為利用簡諧作形式位能來模擬電子與原子核之間所有的作用力，對於包含2顆電子的Moshinsky雙體問題，SUSY KEP方法能夠快速並且準確的計算其基態解。將電子數目提升，嘗試將SUSY KEP運用在多體問題，依然能夠準確的計算出系統基態，以數值分析法擬合出基態能量與大尺度個數N的關係，整體趨勢也與精確解相同。

This thesis extends a systematic method for solving quantum eigenvalue problems, Kinetic Energy Partition (KEP) method, developed by our lab, into operator-based framework. By combining KEP method with the operator method, a novel approach for treating general potential within the operator formalism is developed and referred to as Supersymmetric Kinetic Energy Partition (SUSY KEP) method. In this approach, the kinetic energy is partitioned into sub-kinetic-energy terms through adjustable mass factor and recombined with the potential energy to construct equivalent subsystems. Ladder operators are introduced for each subsystem, allowing the total Hamiltonian to be rewritten as a sum of subsystem Hamiltonians in supersymmetric form. Solving the eigenvalue problem is essentially reduced to finding a linear combination of basis states associated with the eigenvalues. In this method, the chosen basis consists of the ladder operators of the subsystems. Through calculations, the eigenvalues of the total system, the corresponding coordinate space, the annihilation operators, and the complete ground-state are obtained. This study begins with simple models and progressively increases the number of particles and dimensions to investigate their physical properties. In multi-electron systems, the concept of negative mass is introduced to treat repulsive interaction forces. The models considered in this work include a single-particle double harmonic oscillator potential, single-particle multi-harmonic oscillator systems, and the Moshinsky atom model with different numbers of electrons. These models are used to examine the feasibility of the SUSY KEP method in low-dimensional systems with multiple interactions as well as high-dimensional many-body problems. In this study, we applied SUSY KEP method to two different types of models. The first type includes single-particle double-harmonic potentials and its extension to single-particle multi-harmonic potentials. For these models, we verified that the ground-state solutions obtained by SUSY KEP method remain accurate even as the number of potential terms increases. Moreover, the ground-state energies derived from the secular matrix exhibit a power-law dependence on the large-scale potential number N, which is consistent with both the predicted trend and the exact solutions. The second type is the Moshinsky atom model, in which all interactions between electrons and the atomic nucleus are modeled by harmonic potentials. For the two-electron Moshinsky problem, the SUSY KEP method can compute the ground-state solution efficiently and accurately. Extending the approach to systems with more electrons, the SUSY KEP method still accurately determines the system's ground state. Numerical analysis shows that the relationship between the ground-state energy and the large-scale particle number N follows the same overall trend as the exact solution.