Quantum States and Complex Projective Space

Abstract

In this paper we propose the idea that there is a corresponding relation between quantum states and points of the complex projective space, given that the number of dimensions of the Hilbert space is finite. We check this idea through analyzing some of the basic principles and concepts of quantum mechanics, including the principle of superposition, representations and inner product of quantum states, and give some interesting examples. Based on our point of views we are able to generate the evolution equation of quantum states -- the Heisenberg equation. We also discuss the act of dynamical operators on quantum states.