I don't have any problems dealing with mechanical calculations(I think), but yet I have some conceptual problems with mixed state(= statistical mixture?) and pure state in QM. - pure state : |Φ> = 1/sqrt2 ( |↑> + |↓> ) - mixed state : ρ = 1/2 ( |↑><↑| + |↑><↑| ) What is the difference besides the coefficients? And, some denote that |↑><↑| is an operator but others a state. I am really confused. Somebody give me clarification.