Determine the commutator of S 2 with S z (1) (where S S (1) + S

Determine the commutator of S² with S_z⁽¹⁾ (where S Ξ S⁽¹⁾ + S⁽²⁾). Generalize your result to show that

Because S_z⁽¹⁾ does not commute with S², we cannot hope to find states that are simultaneous eigenvectors of both. In order to form eigenstates of S² we need linear combinations of eigenstates of S_z⁽¹⁾. This is precisely what the Clebsch– Gordan coefficients (in Equation 4.183) do for us. On the other hand, it follows by obvious inference from Equation 4.185 that the sum S⁽¹⁾ + S⁽²⁾ does commute with S², which only confirms what we already knew (see Equation 4.103).]

Equation 4.183

Equation 4.103

Transcribed Image Text:

[185] = = 2ih (s(1)