我們首先揭露在量子力學裡N+1維的Hilbert space 的幾何是
S^(2N+1)。Hopf fibration S^(2N+1)/S^1 =CP^N 的存在意味著
S^(2N+1)可以被看成是一個以S^1為它的fiber，CP^N為它的base manifold 的 total bundle space。我們接著建構明確的CP^N 座標以及精確的Hopf projection。藉由這些推導的結果，我們得以以base manifold及fiber的觀點來看待量子力學的各種物理態。這種有別於一般的研究方法揭露了量子力學系統與幾何物件之間的密切關係。在這篇論文中，我們推導出幾何相以及更具一般性的幾何因數的確切形式以及Anandan-Aharonov 定義與non-integrable相位因數的關係。最後，我們提供了幾個用來檢驗與示範這些關係式的例子。

The geometry of the Hilbert space of a generic (N+1)-dimensional quantum mechanical system is revealed to be S^(2N+1). The existence of the Hopf fibration S^(2N+1)/S^1
=CP^N means that one can view the Hilbert space as the total bundle space of the fiber bundle with S^1 as its fiber and CP^N as the base manifold. Explicit coordinates for CP^N together with the precise Hopf projection
maps are then constructed. There is consequently an alternative way of expressing generic quantum states in terms of the coordinates of the base manifold and the fiber. This alternative method of studying quantum states and their evolution reveals the intimate connection between quantum mechanical systems and geometrical objects. The exact expressions of the geometric phases, and more generally the geometrical factors for open paths,
and the precise correspondence between the Anandan-Aharonov
definition and non-integrable phase factor of the geometric Kahler connection are derived. Explicit physical examples are then used to verify and exemplify the formalism.

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