Effect Upon The Spin Operator and Upon States
Operators can be represented by matrices. From linear algebra one knows that a certain matrix can be represented in another basis through the transformation
where is the basis transformation matrix. If the vectors respectively are the z-axis in one basis respectively another, they are perpendicular to the y-axis with a certain angle between them. The spin operator in the first basis can then be transformed into the spin operator of the other basis through the following transformation:
From standard quantum mechanics we have the known results and where and are the top spins in their corresponding bases. So we have:
Comparison with yields .
This means that if the state is rotated about the y-axis by an angle, it becomes the state, a result that can be generalized to arbitrary axes. It is important, for instance, in Sakurai's Bell inequality.
Read more about this topic: Rotation Operator (quantum Mechanics)
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