利用分離動能法求解競爭型位能量子力學系統

Abstract

對於競爭型位能的量子力學系統，傳統上都是用微擾理論的級數近似解，但有可能此級數無法收斂，諸如此類問題，本研究室發展基於分離能量的全新方法，而非像是微擾理論的分解位能項。此方法是分離動能項裡的質量，我們稱此方法為分離動能法(Kinetic Energy Partition (KEP))，此方法不像是按照傳統方式把位能分成'微擾項'與'非微擾項'，因為在競爭型位能下，無法區分哪個位能為'微擾項'與'非微擾項'。分離後的動能項分別對應一個位能使形成子漢米爾頓系統，所以此系統的總漢米爾頓可拆解成個別的子系統漢米爾頓相加，總波函數可以由個別子漢米爾頓系統所對應的波函數線性疊加而成。 直覺上可以先把質量都分解成兩個正質量的動能項，此結構比較符合物理，在此我們有利用五個例子來驗證，把動能項裡的質量拆解成兩個都是正質量的粒子是否可行，利用分離動能法所得到的能量與波函數，會與正合解的能量與波函數作為比較。五個例子分別為:雙負delta函數位能、三負delta函數位能、簡諧振子在無限深位能井、氫原子模型、一維史塔克效應，利用分離動能法可以精準解出，簡諧振子在無限深位能井的基態與第一激發態能量，比某些微擾展開方法解來得更精準，氫原子模型與一維史塔克效應，利用分離動能法則須搭配某些邊界條件，則可解出與正合解基態能量誤差不到5%的精準度，對於兩個或兩個以上的競爭型負delta函數來說，分離動能法在delta函數脈衝較小情況下也是精準的。 按照數學上分析，也可以把質量拆解成一個正質量與一個負質量，只要質量總合不變即可，負質量項的動能項搭配一項位能使形成子漢米爾頓系統，在某些位能下，漢米爾頓系統沒有束縛解，但是因為負質量關係，使得位能函數顛倒(對x軸鏡射)，而得到束縛解，我們利用四個例子來驗證:反對稱delta函數位能、正delta函數在無限深位能井、氦原子模型、一維氦原子，delta函數位能因為波函數簡單，且只有一個能量即為基態能量，對於KEP方法來說相當簡單，對於正delta函數位能是沒有束縛解，但利用分離動能法使得其中一項動能為負，得到E>0之束縛態。對於氦原子來說，是個最簡單的多體物理模型，也是許多多體理論裡的基礎例子，氦原子中有一項庫倫排斥能ke^{2}/r_{12}，因庫倫排斥能沒有束縛解，則可以利用負質量只得位能變為負值，是得能量有束縛解，利用分離動能法各個子漢米爾頓系統的波函數各取一個(基態波函數)，再利用橢圓座標轉換，可計算出氦原子基態能量為-70.2541eV，與實驗值-79.0143eV相差不遠，利用最佳化方法可得到基態能量為-78.9985eV則與實驗值非常接近。一維氦原子為簡化雙座標系統積分的近似，也可提供一維化學實驗的理論背景，而我們也利用KEP方法分析一維氦原子。

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 and triple delta -function potentials . Next, this method is applied to harmonic oscillator in the infinite potential box , the hydrogen atom and 1D Stark effect. Compared with the exact solution, 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. In the mathematical allowed, we can shall dividing mass with 'negative' term and 'positive' term. Then the total Hamiltonian is conserved. Some kind of Hamiltonian system didn't have discrete energy solution. But introduce with negative mass, we can regard as the potential energy inverse of x-axes. So the Hamiltonian system can be solved. We have 4 case to illustrate this method. The first case is antisymmetry delta -function potentials, this is a simply case to explain how to apply 'negative' mass in the quantum system. The second case is 'positive' $delta -function potential in the infinite potential box. The third case is Helium atom, we used negative mass to solved the electron repulsive potential energy. Optimize Helium system position we get -78.965eV Helium ground state energy, just 0.02% error with experimental value. The last case is 1D helium atom, it is a mathematical model. This model can reduce the double system integration, so there is a very persuasive example.