Given the Pauli equations IV: 1 − − A ⋅ ⋅ A = − − ( 2 E μ μ Z 2 e 4 ) ( L 2 + ℏ ℏ 2 ) and III: ( A × × A ) j = − − ( 2 i ℏ ℏ E μ μ Z 2 e 4 ) L j , and starting with the equation A − − A + | ℓ ℓ ∗ ∗ , ℓ ℓ ∗ ∗ ⟩ ⟩ = 0 and expanding, one obtains (assuming ℓ ℓ ∗ ∗ is the maximum value of the angular momentum quantum number consonant with all other conditions) ( 1 + 2 E μ μ Z 2 e 4 ( L 2 + ℏ ℏ 2 ) − − i 2 i ℏ ℏ E μ μ Z 2 e 4 L z ) | ? , ℓ ℓ ∗ ∗ , ℓ ℓ ∗ ∗ ⟩ ⟩ = 0 , which leads to the Rydberg formula E n = − − μ μ Z 2 e 4 2 ℏ ℏ 2 ( ℓ ℓ ∗ ∗ + 1 ) 2 , implying that ℓ ℓ ∗ ∗ + 1 = n = ? , where n is the traditional quantum number.