交叉型WKB(AWKB)近似對薛丁格方程式之理論研究

Abstract

WKB近似是一個在求解一維與時間無關之薛丁格方程式時很好用的方法，以 ℏ 為基底做指數型微擾級數展開。但WKB近似的困難點在我們想進一步得到高階近似時出現。不只因為WKB近似隨著階數增加其連接公式變得冗長且極難推導，也因為高階近似在反曲點比起低階近似有更劇烈的發散行為。因此本論文中我們嘗試重新審視第一階WKB近似的函數結構。藉由對波函數給出一個猜想且搭配針對波函數一階與二階微分的條件得到一組交叉耦合微分方程組。我們使用微擾法求解此耦合方程組並且發展出一個具有第一階WKB近似基本型式另加上額外修正項的新的近似方法作為第一階微擾解，我們稱其為交叉型WKB。首先，我們將AWKB應用於簡諧振子及具有 Morse 位能勢的非簡諧振子系統。接著說明AWKB與WKB近似在穿隧問題上的不同處。比起第一階WKB近似解，AWKB不僅在前兩個束縛態模型的反曲點附近較緩和地發散，提供更好的近似，此新的技巧在求解高階近似時也不需要再處理複雜冗長的連接公式。

WKB approximation is a useful technique to solve the one dimensional time independent Schrödinger equation in the form of exponential perturbative series, order by order in powers of ℏ. But the difficulty of WKB theory emerges as one proceed to higher order WKB approximation. Not only because its connection formulas becomes cumbersome to derive as the order of WKB approximation increases but also because higher order WKB approximation usually has stronger divergent behavior than lower order one in the neighborhood of turning points. In this thesis we attempt to re-examine the functional form of first order WKB approximation. A set of coupled differential equations is obtained by considering an ansatz of wave function with two suitable conditions on both first and second order derivative of wave function. And we use the perturbation method to tackle the coupled differential equations and develop a new approximation, consisting of basic formulation of first order WKB approximation with extra correction terms as the first order perturbation solution, called Alternating WKB (AWKB) approximation. First, the AWKB method is applied to the harmonic oscillator and anharmonic oscillator with Morse potential. Next, we show the difference between WKB and AWKB approximation in the tunneling problem. Compared with the first order WKB approximate solution, not only does the AWKB method provide the better approximation, which diverges slower than first order WKB approximation near the turning points in the first two bound state models, but this new scheme does also has no necessity to deal with the complicate connection formulas as one resort to the higher order approximation.