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)
Famous quotes containing the words effect upon, effect, spin and/or states:
“I have no faith in human perfectability. I think that human exertion will have no appreciable effect upon humanity. Man is now only more active—not more happy—nor more wise, than he was 6000 years ago.”
—Edgar Allan Poe (1809–1845)
“At first I intended to become a student of the Senate rules and I did learn much about them, but I soon found that the Senate had but one fixed rule, subject to exceptions of course, which was to the effect that the Senate would do anything it wanted to do whenever it wanted to do it.”
—Calvin Coolidge (1872–1933)
“In tragic life, God wot,
No villain need be! Passions spin the plot:
We are betrayed by what is false within.”
—George Meredith (1828–1909)
“I do seriously believe that if we can measure among the States the benefits resulting from the preservation of the Union, the rebellious States have the larger share. It destroyed an institution that was their destruction. It opened the way for a commercial life that, if they will only embrace it and face the light, means to them a development that shall rival the best attainments of the greatest of our States.”
—Benjamin Harrison (1833–1901)