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:
“This is the great truth life has to teach us ... that gratification of our individual desires and expression of our personal preferences without consideration for their effect upon others brings in the end nothing but ruin and devastation.”
—Hortense Odlum (1892–?)
“Movies are one of the bad habits that corrupted our century. Of their many sins, I offer as the worst their effect on the intellectual side of the nation. It is chiefly from that viewpoint I write of them—as an eruption of trash that has lamed the American mind and retarded Americans from becoming a cultured people.”
—Ben Hecht (1893–1964)
“Words can have no single fixed meaning. Like wayward electrons, they can spin away from their initial orbit and enter a wider magnetic field. No one owns them or has a proprietary right to dictate how they will be used.”
—David Lehman (b. 1948)
“With steady eye on the real issue, let us reinaugurate the good old “central ideas” of the Republic. We can do it. The human heart is with us—God is with us. We shall again be able not to declare, that “all States as States, are equal,” nor yet that “all citizens as citizens are equal,” but to renew the broader, better declaration, including both these and much more, that “all men are created equal.””
—Abraham Lincoln (1809–1865)