The difference J z 1 − − J z 0 comes from successive application of J x − − i J y or J x + i J y which lower or raise the eigenvalue of J z by ℏ ℏ so that, J z 1 − − J z 0 = 0 , ℏ ℏ , 2 ℏ ℏ , … … Let J z 1 − − J z 0 = 2 j ℏ ℏ , where j = 0 , 1 2 , 1 , 3 2 , … … . Then using J z 0 = − − J z 1 and the above, J z 0 = − − j ℏ ℏ and J z 1 = j ℏ ℏ , and the allowable eigenvalues of J z are J z ′ = − − j ℏ ℏ , − − j ℏ ℏ + ℏ ℏ , − − j ℏ ℏ + 2 ℏ ℏ , … … , j ℏ ℏ . Expressing J z ′ in terms of a quantum number m j , and substituting J z 0 = − − j ℏ ℏ into J 2 ′ − − ( J z 0 ) 2 + ℏ ℏ J z 0 = 0 from above,