變分原理求橢圓體及圓柱體束縛量子系統的本徵能量值

Abstract

本論文研究位於橢圓體及圓柱體束縛量子系統中的氫雜質態 之能階。位於球形的量子點中心的氫雜質原子能態己被研究。現在我們考慮橫截面為圓的橢圓體及有限長度的圓柱體。假設此系統有無窮大的束縛位。我們以一種新的變分法來求此系統的本徵能量，使用的嘗試函數不需滿足任何特定的邊界條件。為了解此方法所得解的正確與否，我們令量子點的半徑變大，發現系統的氫雜質的本徵能量可以趨近三維氫原子的正確極限。對某些系統而言，雖然這個方法並非一般正確的，但在常見的一些實例中結果則很恰當。

This paper research the energies of hydrogenic impurit states in ellipsoid and cylinder confined quantum systems. The energies of hydrogenic impurity states with an impurity atom located at the center of a quantum dot have been studied. Now, we consider the case of an ellipsoid, but which cross section is a circular. This system is assumed to have an infinite confining potential. We use a new variational principle to find the energy eigenvalues of this system. Whereas the trial function in this system need not obey any specific boundary condition. When the radius of the quantum dot becomes large, the ground-state energy of the impurity approaches the correct limit of three-dimensional hydrogen atom. Although this variational method is not a definite (minimum) one, good results are obtained in several examples.