reestablish fundamental brakets

    spin Z operators

    raising/lowering operators (lowering erased before pic)

    commutations relations (erased before pic)

Now setup a two-spin system

    (pic) Look at general operator expressions ?? in different spaces ??

    Take direct product of A and B

(pic) Developed 𝐒 using raising lowering operators

    (pic: last y should be an x) Proved this.


(pic x2) Still missing a 1/2 somewhere! But moving on to see proper solutions of direct product.

    Eigenvalues are
        
        𝐒² = ħ² S(S+1) = ⎧ 2ħ²
                         ⎨ 
                         ⎩ 0ħ²
    
        𝐒 = ⎧ 1  (3 eigenstates)
            ⎨
            ⎩ 0  (1 eigenstate)
    
    (pic) Plugging back to direct product matrix

(pic) FOUND PROBLEM of -1/2 from the up/down raising/lowering operator interactions

(pic) finished diagonalizing operator product