5. Fine structure and hyperfine structure of hydrogen atoms (given by Dirac equation - relativity
quantum mechanics)
(1) Calculate the energy levels of n = 1, 2, and 3 for hydrogen atom when only considering the
electrostatic interaction (Coulomb force) between the nucleus and the electron. The potential
energy at infinite is set to zero.
(2) Calculate the fine structure for each energy level of n = 1, 2, and 3. The relativity mass correction
($\Delta E_m$), the Darwin term ($\Delta E_a$), and the spin-orbit coupling ($\Delta E_{ls}$) should be calculated, and then
the sum $\Delta E$ of these three terms should be derived. Please show the procedures how you do the
calculation, and then put the results into a table.
(3) Draw a diagram to show the coarse and fine structures of the energy levels (only the final results
for the fine structure). Please mark in the energy level diagram the state symbols and the shift
relative to the coarse energy levels (n = 1, 2, 3). Use Joule and MHz as the energy unit.
(4) Derive the hyperfine structure for the hydrogen ground state, and calculate the hyperfine splitting
between the hyperfine states. Convert the energy split to frequency and wavelength.
