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 ≐
      ħ ⎛ 1 0  0 ⎞
        ⎜ 0 0  0 ⎟ and
        ⎝ 0 0 -1 ⎠
S𝓍 ≐
      ͟ħ͟  ⎛ 0 1 0 ⎞
      √2 ⎜ 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) ≐

    ħ ⎛ 1 0  0 ⎞ ⎧ ħ ⎛ 1 0  0 ⎞     ħ ⎛ 1 0 0 ⎞ ⎫ ⎧ ħ ⎛ 1 0  0 ⎞    ħ ⎛ 1 0 0 ⎞ ⎫  
      ⎜ 0 0  0 ⎟ ⎪   ⎜ 0 0  0 ⎟  +    ⎜ 0 1 0 ⎟ ⎪ ⎪   ⎜ 0 0  0 ⎟  -   ⎜ 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,

    ħ ⎛ 1 0  0 ⎞ ħ ⎛ 2 0 0 ⎞ ħ ⎛ 0  0 0 ⎞ 
      ⎜ 0 0  0 ⎟   ⎜ 0 1 0 ⎟   ⎜ 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) ≐

   ͟ħ͟   ⎛ 0 1 0 ⎞ ͟ħ͟  ⎛ √2  1  0  ⎞ ͟ħ͟  ⎛ -√2   1   0 ⎞ 
   √2  ⎜ 1 0 1 ⎟ √2 ⎜  1 √2  1  ⎟ √2 ⎜   1 -√2   1 ⎟.
       ⎝ 0 1 0 ⎠    ⎝  0  1 √2  ⎠    ⎝   0   1 -√2 ⎠ 

Performing the multiplication operation on the last two matrices returns the expression

  ͟ħ͟³͟   ⎛ 0 1 0 ⎞ ⎛ -1  0  1 ⎞
  2√2  ⎜ 1 0 1 ⎟ ⎜  0  0  0 ⎟ = 𝟘.
       ⎝ 0 1 0 ⎠ ⎝  1  0 -1 ⎠

Because each pair of row and column in this matrix has alternating 0s and ±1s, every multiplication operation will return 0. The second expression is therefore equivalent to the zero matrix 𝟘.