First, we will review spin for non-relativistic quantum mechanics. Spin, an intrinsic property similar to angular momentum, is defined by a spin operator S that plays a role on a system similar to the operator L for orbital angular momentum. The operators S 2 and S z whose eigenvalues are S 2 | s , m ⟩ ⟩ = s ( s + 1 ) ℏ ℏ 2 | s , m ⟩ ⟩ and S z | s , m ⟩ ⟩ = m ℏ ℏ | s , m ⟩ ⟩ respectively. These formalisms also obey the usual commutation relations for angular momentum [ S x , S y ] = i ℏ ℏ S z , [ S y , S z ] = i ℏ ℏ S x , and [ S z , S x ] = i ℏ ℏ S y . The raising and lowering operators, S + and S − − , are defined as S + = S x + i ⋅ ⋅ S y and S − − = S x − − i ⋅ ⋅ S y respectively. These ladder operators act on the state in the following S + | s , m ⟩ ⟩ = ℏ ℏ s ( s + 1 ) − − m ( m + 1 ) | s , m + 1 ⟩ ⟩ and S − − | s , m ⟩ ⟩ = ℏ ℏ s ( s + 1 ) − − m ( m − − 1 ) | s , m − − 1 ⟩ ⟩ respectively.