論波耳-索末菲半整數與整數量子化條件之過渡

Abstract

在WKB近似下的一維束縛態問題裡所得到的波耳-索末菲半整數與整數量子化條件，分別對應於兩種截然不同的邊界條件，但當實際的邊界條件界於兩者之間時，兩個量子化條件的適用性將成為問題。我們知道的是，在系統兩側的位能牆逐漸傾斜至無窮斜的過程裡，半整數量子化條件的適用性終將被整數量子化條件所取代，但在絕大多數的情況裡，我們對於其中過渡地帶的細節仍未知曉。 本論文旨在探討這種過渡發生的過程與機轉，在探索的過程裡，我們得到了WKB近似的另一種推導，並分別使用了兩種不同的近似方法來計算在過渡發生的過程中，相位在位能牆上的變化。

The Bohr-Sommerfeld half-integer and integer quantization conditions, obtained when the WKB approximation is applied to two types of systems with different boundary conditions, become ambiguous and subtle if neither case is an accurate description of the actual system. As a not-so-steep potential wall gradually becomes infinitely steep, it is a known fact that the half-integer rule should be eventually replaced by the integer rule, but the transition in between, in most cases, is left unknown. This thesis aims to explore the question of how and why such transition occurs. In the process of doing so, we present an alternative derivation of the WKB approximation and apply two different methods to calculate the change in the 'phase loss' during such transition.