• Tags: hydrogen-like, He+, Li2+, scaling
• Difficulty: Moderate

Hydrogen-like ions have one electron: $He^{+}$ (Z=2), $Li^{2+}$ (Z=3), $Be^{3+}$ (Z=4). All Bohr model formulas apply with Z factor: r_n = $n^{2}$a_{0}/Z, v_n = $Zv_{0}$/n, E_n = -13.6$Z^{2}$/$n^{2}$ eV. For $He^{+}$: ground state energy = -13.6 × 4 = -54.4 eV. Ionization energy = 54.4 eV. For $Li^{2+}$: ground state energy = -13.6 × 9 = -122.4 eV.

Wavelengths scale as 1/$Z^{2}$: all spectral lines shift to shorter wavelengths (higher energies) for higher Z. The 2→1 transition in $He^{+}$ has 4× the energy of the same transition in H, giving λ/4 wavelength. JEE frequently asks to identify which hydrogen-like ion has a spectrum that overlaps with hydrogen in certain transitions.