Addition of Spin- and Addition of Angular Momenta What's the deal?

Addition of Spin- Operators

Let and be two spin- operators whose total spin AM is given by

There are four possible product states:

in which the first (second) symbol refers to the first (second) spin. These product states are eigenstates of the operators . truein

truein Problem :21

• Evaluate the action of the operator on each of the product states.
• Show that



Hence show that



The states and therefore have total spin S=1 and whose z-component of total spin takes the value of and respectively.

• Given that show that



(The resulting state has been normalised to unity by inserting the factor . Show that the z-component of this state is zero. Using the notation where S designates the total spin and m its z-component, we have



• There is an additional state which is orthogonal to those given above:



By acting on this state with and show that this state has spin zero. So we have found all the eigenstates of and . The states above corresponding to S=1 are referred to as triplet states and that corresponding to S=0 as a singlet state.

• Consider the following projection operators:



Show that projects onto triplet states while projects onto singlet states.

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