The percentage of spins in the same direction

The percentage of quantum spins in the same direction with different angles: An intuitive understanding 

When the angle between the electron spins and detector is , the percentage of spins in the up direction will be  〖cos〗^2 (θ/2). This value can be derived formally. But it is not always easy to understand this number intuitively. In the following, we will attempt to offer an intuitive understanding. 

The correlation between two unit vectors is cos⁡(θ). cos⁡(θ)= 〖cos〗^2 (θ/2) - 〖sin〗^2 (θ/2)   Spin directions take two values, 1 and -1. This means that the percentage of spins which take the value 1 is 〖cos〗^2 (θ/2) and the percentage of spins which take the value -1 is 〖sin〗^2 (θ/2). This provides a simple and intuitive understanding of where 〖cos〗^2 (θ/2) comes from. 

It is known that in quantum mechanics, the correlation between two unit vectors is cos⁡(θ). Yet it is rarely emphasized that this correlation relation is identical in quantum mechanics and classical theory. This suggests that we might not expect some extraordinary results, something like Bell’s inequality, in quantum mechanics. When Bell’s inequality did occur, we might incline to pause to check the details of the derivation before pronouncing great implications. As greater and greater amount of related literature and projects pile up, it becomes increasingly difficult to mention the possibility that Bell’s inequality might have technical issues.