• Tags: bohr, radius, energy, derivation
• Difficulty: Moderate

For hydrogen-like atom (nuclear charge Ze, one electron): Centripetal force = Coulomb force: $mv^{2}$/r = k$Ze^{2}$/$r^{2}$. Angular momentum quantization: mvr = nℏ. From these two equations: r_n = $n^{2}$ℏ^{2}/(mk$Ze^{2}$) = $n^{2}$a_{0}/Z, where a_{0} = ℏ^{2}/(m$ke^{2}$) = 0.529 Å. Velocity: v_n = k$Ze^{2}$/(nℏ) = $Zv_{0}$/n, where v_{0} = $ke^{2}$/ℏ = $2.18 \times 10^{6}$m/s = c/137. Energy: E_n = -$mk^{2}$Z^{2}$e^{4}$/(2$n^{2}$ℏ^{2}) = -13.6$Z^{2}$/$n^{2}$eV. The negative energy indicates a bound state. The kinetic energy KE = ½$mv^{2}$= k$Ze^{2}$/(2r) = 13.6$Z^{2}$/$n^{2}$eV (positive). The potential energy PE = -k$Ze^{2}$/r = -27.2$Z^{2}$/$n^{2}$eV. Note: E = KE + PE, and PE = -2KE (virial theorem for 1/r potential).$