二維四次方振子系統之量子渾沌學之研究

Abstract

此論文中我們透過二維之偶合四次方振子作為模型，得以討論古典渾沌所對應其自身量子行為的相關特徵，希望藉由本徵態在位置空間，及相空間的分佈來發掘一些古典不穩定軌道之新關聯，因此我們採用一些定量的方法來量測本徵態在空間中所佔據的有效體積。 首先以second moment of Husimi distribution的方法來計算量子態在相空間中所分佈之有效體積，並輔以Renyi length和位置空間的不準度(position uncertainty)等方式討論其本徵態在位置空間中之有效體積分佈情形。 我們還特別討論了一個著名的系統，其位能形式為x2y2/2。 此系統顯現出在特定能量下具有無限大的相空間體積，同時在座標軸上有著無限延展之通道等性質。 再者其古典行為除了有強渾沌的特性外也兼具著粒子會由這些通道逃逸而出的傾向。 我們藉由adiabatic approximation及對角化之數值計算，進一步發現到波函數在通道內形成局域化之最遠位置和自身能量的關係。 另外我們利用Shannon entropy，position uncertainty和Husimi distribution作為理解波函數分佈的有用資訊。 我們發現到波函數的局域化形態極易受到無窮延伸的通道所影響，這可由和一個同樣具有有限延展之通道之位能形式0.01(x4+y4)/4+x2y2/2比較得之。 另一方面此四次振子具備齊次(homogenous)性質，此這系統之古典分歧(bifurcations)行為可以由Yoshida公式來預測。 我們從自身之量子態的幾何訊息如相空間或位置空間中所佔之有效體積之分佈行為，指出可積系統的幾何相關量之於能譜呈現規律排列之特性。 也可發現到當系統逐漸轉變成渾沌系統時，會逐漸破壞這些規律關係。 同時也可見當此古典系統發生分歧時，所對應之量子態所佔之最小有效體積和古典分歧有著密切關聯。 最後我們也發現到有效體積在位置空間中和其古典遍歷假設的比值，顯示出可積系統下的波函數相較於不可積系統的波函數更顯現出局域化的傾向，同樣的反映出古典可積和不可積系統之軌道分佈行為。

We study the correspondence between quantum and classical systems which have chaotic behaviors classically. The two-dimensional quartic oscillators are chosen as the model in this thesis. It is hoped that we can find some new connections between dispersion of eigenstates and classical unstable orbits. Several methods are adopted to measure the dispersion of eigenstates quantitatively; the volume occupied by a quantum eigenstate in phase space is calculated from the inverse second moment of Husimi distribution. The configuration space volume is with respect to the uncertainty and Renyi length of the eigenstate. We study a famous system with potential x2y2/2 which has the four infinite channels along the axes and an infinite phase space volume. The classical motion of this system not only reveals a chaotic behavior but also expels particle from the central region through the channels. We find some features of its localized eigenstates with the furthest extensions along the channels by the quantum adiabatic approximation and numerical calculation. From the information of eigenstates dispersion, they are Shannon entropy, position uncertainty and Husimi distribution. We show that the localization of eigenstate is strongly affected by the infinite channels. We also illuminate the obvious difference from a similar system with the potential 0.01(x4+y4)/4+x2y2/2 which has channels with finite spatial extension. On the other hand, this quartic oscillator is also a homogenous system, its classical bifurcations can be predicted by the Yashida formula. We also calculate the second moment of Husimi distribution, uncertainty and Renyi length to obtain geometric information from the corresponding quantum spectrum themselves. For the integrable systems, these geometric quantities appear to be in regular relation to the energies. When the system becomes chaotic gradually, parts of these regular relations are destroyed quickly. We indicated that the most localized states have some connections with classical bifurcations. Furthermore, we also find that the ratios of quantal dispersions in configuration space to ergodic limits have similar characteristic to the distributions of classical orbits in integrable and non-integrable systems.