Introduction to Quantum Mechanics · Problem 4.41

Quantum Mechanics in Three Dimensions — David J. Griffiths

Physics — Quantum Mechanics · spin_commutator

Problem

Determine the commutator of

S^2

with

S_z^{(1)}

(where

\mathbf{S} \equiv \mathbf{S}^{(1)} + \mathbf{S}^{(2)}

). Generalize your result to show that

\left[S^2, \mathbf{S}^{(1)}\right] = 2i\hbar\left(\mathbf{S}^{(1)} \times \mathbf{S}^{(2)}\right)

. (4.185) Comment: Because

S_z^{(1)}

does not commute with

S^2

, we cannot hope to find states that are simultaneous eigenvectors of both. In order to form eigenstates of

S^2

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

\mathbf{S}^{(1)} + \mathbf{S}^{(2)}

does commute with

S^2

, which only confirms what we already knew (see Equation 4.103).

Related theory

Spin Spin Splitting And The N 1 Rule

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