: Final remarks : lecture_8 : The general problem

The simple example revisited

Consider, again, the example of adding to spin-1/2 angular momenta. This time, we will use the Clebsch-Gordan coefficients directly to determine the unitary transformation. Start with the state . Expanding gives

Only one term gives , which is clearly and . Thus,

The Clebsch-Gordan coefficients, by special case is just 1, so

as expected.

The state is expanded as

This time, since , two terms contribute, and . Hence,

However, since , we can use special case , and we find

so that

It is straightforward to show that . The state is expanded to give

Again there are two terms that contribute. However, to determine the two Clebsch-Gordan coefficients:

we can use special case , since . In this case,

Hence, the state is given by

: Final remarks : lecture_8 : The general problem
Mark Tuckerman Ê¿À®17Ç¯2·î18Æü