The relativistic fine-structure correction to the hydrogen atom energy levels lifts the degeneracy between states having the same principal quantum number $n$ but different
(A)Total angular momentum quantum number $j$
(B)Orbital angular momentum quantum number $l$ only (leaving $j$ degenerate)
(C)Magnetic quantum number $m_j$ in the absence of an external field
### Step-by-Step Derivation & Verification: 1. **Fine Structure Components**: The fine structure correction Hamiltonian consists of three terms: $H_{\text{FS}} = H_{\text{rel}} + H_{\text{SO}} + H_{\text{Darwin}}$ 2. **First-Order Energy Shift**: $E_{\text{FS}}^{(1)} = \frac{(E_n^{(0)})^2}{2 m c^2} \left[\frac{4n}{j + 1/2} - 3\right] = E_n^{(0)} \frac{\alpha^2}{n^2} \left[\frac{n}{j + 1/2} - \frac{3}{4}\right]$ 3. **Degeneracy Structure**: - The energy depends only on $n$ and $j$. - It does NOT depend on orbital angular momentum $l$ individually. For example, $2s_{1/2}$ ($l=0, j=1/2$) and $2p_{1/2}$ ($l=1, j=1/2$) possess identical Dirac fine structure energies. - States with different $j$ (e.g., $2p_{1/2}$ vs $2p_{3/2}$) have their degeneracy lifted by fine structure. 4. **Conclusion**: The fine structure lifts degeneracy between states of different total angular momentum $j$. Option (A) is correct.