利用齊次方法展開求氦原子波函數之精確解

Abstract

目前，氦原子波函數的精確解仍未被完整的解出。如何制定一個有效率的方法來精確地計算出一個雙電子系統的波函數成為了量子化學領域的一個重要議題。如果這個簡單的雙電子系統問題得到解答，那將會有助於了解更進一步的多電子系統。本篇論文的主題是利用齊次方法展開來解氦原子的薛丁格方程式，為了求得精確解，我們將波函數 展開為 ，其中 是在波函數之中對長度單位的次方為 的項。將這個展開代入氦原子的薛丁格方程式之中，便能進一步的求得在本篇論文之中所介紹的主要計算方程式 。我們利用這個主要方程式來計算 在 值為0、1、2以及3之時。在 值為0時，方程式為 ，而我們也能解出 ，在這個部份的論文內容我們有進一步討論有關動能算符的齊次解。而 值為1時，方程式為 ，而解為 。而當我們試圖更進一步的求解 值為2時，方程式為 ，而它的解因為太複雜，我們便不在摘要中提及。利用我們的方法所求出之 只是其中一種解，在某些情況下會有奇異點，在該章節中我們提到利用我們的方法所求出的 與Paul Abbott等人所找到的 做討論與分析，同時在本篇論文之中我們將Paul Abbott等人所找到的 作圖並證明了它的正確性。在後續的章節我們也利用 來進一步的求解 ，我們在此制定了一個專門用來求解單項式時的反函數表，只要能找到此張表內足夠的項，我們將能更進一步的求得精確的氦原子波函數，這項發現說明了求解氦原子的薛丁格方程式是可行並且有價值的。

The exact wave function of helium atom has not been fully determined yet. Formulating an efficient way of describing and computing the wave functions of two-electron systems is important. Solving this fundamental problem can be useful for accurate treatment of many-electron systems. Using homogeneity expansion of the wave function to solve the Schrödinger equation is the main topic in this thesis. In order to solve the Schrödinger equation to helium atom, we assume that the wave function can be expanded as , where is the component of the total wave function homogeneous of order . By substituting the expansion into Schrödinger equation, the working equation of our method, , is produced. We attempt to solve this equation to find with 0, 1, 2 and 3. The first equation is and its solution is . For , we give a short discussion of homogeneous solutions for the kinetic energy operator ( ) in interparticle coordinates (IC). Next, we discuss two methods of finding , separately for terms independent of and terms dependent on . By solving the equation , we can find that . The equation defining is ; its solution is too complicated to quote it here. The solution found by our method is a particular solution. This solution has singularities at some special points. In order to make our particular solution well-behaving, we need to find homogeneous solutions to remove the singularities of when and . A detailed discussion of found by Abbott et al. leads us to find homogeneous solutions we need. We also make a plot of and prove that is a well-behaving wave function successfully. Application of for solving is another important issue studied in this thesis. An inversion table for monomial terms of homogeneity one is created for finding a particular solution of . Enough terms in monomial tables of every homogeneity have the ability to generate particular solutions of with every value of . This discovery shows that using inversion table to solve the Schrödinger equation is feasible and valuable.