Show that the Schwinger representation operators satisfy the spin commutation relations


 * 1) Expand the commutator \([\hat{S}^{+}, \hat{S}^{-}]\): \([\hat{S}^{+}, \hat{S}^{-}] = \hat{a}^{\dagger}\hat{b}\hat{b}^{\dagger}\hat{a} - \hat{b}^{\dagger}\hat{a}\hat{a}^{\dagger}\hat{b}\)
 * 2) Bring like operators together: \([\hat{S}^{+}, \hat{S}^{-}] = \hat{a}^{\dagger}\hat{a}\hat{b}\hat{b}^{\dagger} - \hat{a}\hat{a}^{\dagger}\hat{b}^{\dagger}\hat{b}\)
 * 3) Use the standard commutator to swap terms: \([\hat{S}^{+}, \hat{S}^{-}] = \hat{a}^{\dagger}\hat{a}\left(\hat{b}^{\dagger}\hat{b} + 1\right) - \left(\hat{a}^{\dagger}\hat{a}+1\right)\hat{b}^{\dagger}\hat{b}\)
 * 4) Expand: \([\hat{S}^{+}, \hat{S}^{-}] = \hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b} + \hat{a}^{\dagger}\hat{a} - \hat{a}^{\dagger}\hat{a}\hat{b}^{\dagger}\hat{b} - \hat{b}^{\dagger}\hat{b}\)
 * 5) Cancel like terms: \([\hat{S}^{+}, \hat{S}^{-}] = \hat{a}^{\dagger}\hat{a} - \hat{b}^{\dagger}\hat{b} = 2\hat{S}^{z}\)