The operators S_x and S_y can be determined using the ladder method. In the case of the spin 1/2 case (fermion), the operator S + acting on a state produces S + | + ⟩ ⟩ = 0 and S + | − − ⟩ ⟩ = ℏ ℏ | + ⟩ ⟩ . Likewise, the operator S − − acting on a state produces S − − | − − ⟩ ⟩ = 0 and S − − | + ⟩ ⟩ = ℏ ℏ | − − ⟩ ⟩ . The matrix representations of these operators are constructed as follows: