Abstract: For any pure state in the state space of spin angular momentum with spin quantum number

0.5，the component operator of spin angular momentum along a certain direction can be

found，and the pure state is the eigenstate of the component operator. However，for the pure

state in the state space of spin angular momentum with quantum number 1 or more，it is found that

only some states have corresponding conclusions. Therefore，pure states can be divided into two

categories. The existence and preparation methods of these two different pure states are dis-

cussed. It is found that there is no good experimental method to prepare the eigenstates which

cannot be the compo- nent operators of angular momentum in a certain direction，which poses a

thorny problem for experimental physi-cists.

Key words: spin angular momentum, state space, the superposition principle of states, preparation of quantum states