Review Questions

1. What is a hydrogenic ion? Provide a few examples of hydrogenic ions.

2. For Bohr's atomic model, what were the two postulates from which he derived his stable electron orbits for the hydrogen atom. In words, describe Bohr's stable orbit/states and how their energies depend on the principal quantum number, n.  Write down the expression for the photon energy emitted or absorbed for a transition between upper state n and lower state n'.

3. What is the mathematical expression for the fine structure constant, ⍺? For Bohr's model, to what power does the electron binding energy depend on ⍺? 

4. What is the definition of the excitation energy? What is the value of the excitation energy of the ground state? Define the ionization energy? Compare the potential energy to the ionization energy; how do they differ in their definitions and meaning?

5. In the Schrödinger atomic model the binding energy for an electron in principal quantum state n of the Schrödinger atom are the same as for the Bohr atom, yet the physics of the model is entirely different.  For a given state n, what do the geometric quantum numbers l and m mean physically, i.e., what are they describing (or what is their physical origin)? If for a given electron, its principal quantum number is n = 3, what are the possible l and m states it can populate?

6. What are the dipole selection rules for transitions in the Schrödinger model? Using spectroscopic notation, write down the Ly⍺ transition. Now write down the three energy-degenerate H⍺ transitions.

7. The Dirac atomic model is also a wave model, but with a Hamiltonian that includes physics not appearing in Schrödinger's Hamiltonian. What physics did Dirac introduce to obtain his model of the hydrogen atom that resulted in energy shifts and fine structure splitting? How did this result in the new quantum number j and fine structure splitting? 

8. Now that we know Ly⍺ is a fine structure doublet, write down the transitions of the two doublet members using spectroscopic notation. Which transition is the higher energy transitions and why? Using spectroscopic notation, write down the transitions in the Hβ multiplet that are energy degenerate with one another. What makes these transitions energy degenerate?

9. What quantized physics did Feynman's QED add to the hydrogen model? What is a vacuum polarization?  What is a "polarization loop" and how does it shift the electron biding energy levels in the hydrogen atom?

10. Using spectrosocopic notation, write down the transition known as the famous Lamb shift. What degeneracy has been broken such that the Lamb shift transition has non-zero energy?

11. What is an "isotope shift"? What is the physical mechanism that induces an isotope shift?  What is a "mass polarization shift"? For hydrogen, which dominates, the isotope shift or mass polarization shifts?

12. What is the physics underlying the 21-cm hyperfine transition of hydrogen (why do the two spin orientations of the ground-state electron have different binding energies)? In what way does the 21-cm hyperfine transition classify as a forbidden transition? 

13. In your own words, summarize Table 20.4 with emphasis on the order of magnitude of the "correction" and the scaling with the fine structure constant.

14. Compare and contrast bound-free and free-free absorption for a hydrogen atom. What physically does the continuous quantum number k (or k') represent for free electron states? 

Problems

1. For a bound free transition, what is k for an electron that is ejected with zero kinetic energy?