If S𝓏 and S𝓍 are spin-1 operators in the z basis, what are the results if S𝓏 (S𝓏 + ħ)(S𝓏 - ħ) and S𝓍 (S𝓍 + ħ)(S𝓍 - ħ) are evaluated? 

The spin eigenstates and eigenvalues are known from experiment for a spin-1 system, and the spin-z and spin-x operators in the z basis, S𝓏 and S𝓍, have the following matrix representations:


S𝓏 ≐
        ⎛ 1 0  0 ⎞
      ħ ⎜ 0 0  0 ⎟ and
        ⎝ 0 0 -1 ⎠
S𝓍 ≐
        ⎛ 0 1 0 ⎞
      ħ ⎜ 1 0 1 ⎟.
        ⎝ 0 1 0 ⎠

Using the matrix representations, the expressions can be evaluated. For the spin-z operator, the expression S𝓏 (S𝓏 + ħ)(S𝓏 - ħ) ≐

      ⎛ 1 0  0 ⎞ ⎧   ⎛ 1 0  0 ⎞       ⎛ 1 0 0 ⎞ ⎫ ⎧   ⎛ 1 0  0 ⎞      ⎛ 1 0 0 ⎞ ⎫  
    ħ ⎜ 0 0  0 ⎟ ⎪ ħ ⎜ 0 0  0 ⎟  +  ħ ⎜ 0 1 0 ⎟ ⎪ ⎪ ħ ⎜ 0 0  0 ⎟  - ħ ⎜ 0 1 0 ⎟ ⎪,
      ⎝ 0 0 -1 ⎠ ⎩   ⎝ 0 0 -1 ⎠       ⎝ 0 0 1 ⎠ ⎭ ⎩   ⎝ 0 0 -1 ⎠      ⎝ 0 0 1 ⎠ ⎭  

which simplifies to the matrix multiplication operation, where 𝟘 represents the 0 matrix,

      ⎛ 1 0  0 ⎞   ⎛ 2 0 0 ⎞   ⎛ 0  0 0 ⎞ 
    ħ ⎜ 0 0  0 ⎟ ħ ⎜ 0 1 0 ⎟ ħ ⎜ 0 -1 0 ⎟ = 𝟘. 
      ⎝ 0 0 -1 ⎠   ⎝ 0 0 0 ⎠   ⎝ 0  0 0 ⎠ 

The multiplication operation apparently returns 𝟘 because the third factor will nullify any terms besides center terms, and the first factor will nullify any center terms.

Similarly, S𝓍 (S𝓍 + ħ)(S𝓍 - ħ) ≐

       ⎛ 0 1 0 ⎞ ⎛ 1 1 0 ⎞ ⎛ -1  1  0 ⎞      ⎛ 1 1 1 ⎞ ⎛ -1  1  0 ⎞      ⎛ 0 1 0 ⎞
    ħ³ ⎜ 1 0 1 ⎟ ⎜ 1 1 1 ⎟ ⎜  1 -1  1 ⎟ = ħ³ ⎜ 1 2 1 ⎟ ⎜  1 -1  1 ⎟ = ħ³ ⎜ 1 0 1 ⎟.
       ⎝ 0 1 0 ⎠ ⎝ 0 1 1 ⎠ ⎝  0  1 -1 ⎠      ⎝ 1 1 1 ⎠ ⎝  0  1 -1 ⎠      ⎝ 0 1 0 ⎠

This expression results in the non-zero matrix

       ⎛ 0 1 0 ⎞
    ħ³ ⎜ 1 0 1 ⎟.
       ⎝ 0 1 0 ⎠