Chapter 18 Quiz: Degenerate Perturbation Theory and Fine Structure

Instructions: This quiz covers the core concepts from Chapter 18. For multiple choice, select the single best answer. For true/false, provide a brief justification (1-2 sentences). For short answer, aim for 3-5 sentences. For applied scenarios, show your work.

Multiple Choice (10 questions)

Q1. Non-degenerate perturbation theory fails for degenerate states because:

(a) The first-order energy correction $E_n^{(1)} = \langle n | \hat{H}' | n \rangle$ is undefined for degenerate states (b) The second-order formula has terms with vanishing denominators when states share the same energy (c) Degenerate states cannot be eigenstates of any Hamiltonian (d) The perturbation expansion parameter $\lambda$ is no longer small for degenerate systems

Q2. In degenerate perturbation theory, the first-order energy corrections are obtained by:

(a) Taking the diagonal matrix elements of $\hat{H}'$ in any convenient basis of the degenerate subspace (b) Diagonalizing the full Hamiltonian $\hat{H}_0 + \hat{H}'$ exactly (c) Diagonalizing the matrix of $\hat{H}'$ restricted to the degenerate subspace (d) Using the second-order formula with a regularized denominator

Q3. The fine structure of hydrogen is of order:

(a) $\alpha E_n$ where $\alpha \approx 1/137$ (b) $\alpha^2 E_n$ (c) $\alpha^3 E_n$ (d) $(m_e/m_p)\alpha^2 E_n$

Q4. The relativistic kinetic energy correction to hydrogen energy levels:

(a) Raises all energy levels (b) Lowers all energy levels (c) Raises levels with high $l$ and lowers levels with low $l$ (d) Has no net effect when averaged over all $l$ values

Q5. Spin-orbit coupling vanishes for $s$-states ($l = 0$) because:

(a) $s$-states have zero probability density at the nucleus (b) An electron with no orbital angular momentum produces no magnetic field at its own position (c) The Thomas precession exactly cancels the spin-orbit interaction for $l = 0$ (d) The Darwin term absorbs the $l = 0$ contribution

Q6. The Darwin term in hydrogen:

(a) Affects all states equally (b) Affects only states with $l = 0$ (c) Affects only states with $l \geq 1$ (d) Vanishes for all states (it is a gauge artifact)

Q7. The total fine structure correction for hydrogen depends on:

(a) $n$ only (b) $n$ and $l$ only (c) $n$ and $j$ only (d) $n$, $l$, and $j$

Q8. The Lande $g$-factor for a $^2P_{3/2}$ state (with $l = 1$, $s = 1/2$, $j = 3/2$) is:

(a) $1$ (b) $2/3$ (c) $4/3$ (d) $2$

Q9. The hydrogen 21 cm line arises from:

(a) The fine structure splitting of the $n = 1$ level (b) The transition between the $n = 1$ and $n = 2$ levels (c) The hyperfine splitting of the hydrogen ground state (spin-flip transition) (d) The Lamb shift between $2s_{1/2}$ and $2p_{1/2}$

Q10. In the Paschen-Back (strong-field) Zeeman effect, the good quantum numbers are:

(a) $j$ and $m_j$ (b) $l$, $s$, $j$, and $m_j$ (c) $m_l$ and $m_s$ (d) $F$ and $m_F$

True/False (4 questions)

Q11. TRUE or FALSE: The Thomas precession increases the spin-orbit coupling energy by a factor of 2.

Q12. TRUE or FALSE: In the weak-field Zeeman effect, all sublevels of a given $j$ level are equally spaced in energy.

Q13. TRUE or FALSE: The fine structure of hydrogen lifts all degeneracies of a given $n$ level (except for the $m_j$ degeneracy).

Q14. TRUE or FALSE: The hyperfine splitting of hydrogen's ground state is approximately 1000 times smaller than the fine structure splitting of the $n = 2$ level.

Short Answer (4 questions)

Q15. Explain what "good quantum numbers" means in the context of degenerate perturbation theory. Give an example of quantum numbers that are "good" for the weak-field Zeeman effect but "bad" for the strong-field Zeeman effect.

Q16. The fine structure formula $E_{\text{FS}}^{(1)} = -(E_n^{(0)})^2/(2m_ec^2) \times (4n/(j+1/2) - 3)$ shows that for a given $n$, the state with the smallest $j$ has the lowest (most negative) energy. Explain qualitatively why this is the case by considering the sign of the spin-orbit interaction.

Q17. The spontaneous transition rate for magnetic dipole (M1) transitions scales as $\omega^3$, where $\omega$ is the transition frequency. Use this to explain why the 21 cm hyperfine transition is so extraordinarily slow (lifetime $\sim 10^7$ years), whereas optical transitions typically have lifetimes of $\sim 10^{-8}$ seconds.

Q18. Describe the physical origin of the Darwin term using the concept of Zitterbewegung. Why does this "smearing" of the electron position lead to an energy correction that depends on $\nabla^2 V$ rather than $V$ itself?

Applied Scenarios (2 questions)

Q19. A precision spectroscopy experiment measures the hydrogen Balmer-$\alpha$ line ($n = 3 \to n = 2$) with a resolution of 0.001 cm$^{-1}$.

(a) The gross (Bohr) energy difference for $3 \to 2$ is $E_3 - E_2 = -13.6(1/9 - 1/4)$ eV. Convert to cm$^{-1}$.

(b) Including fine structure, how many distinct spectral components does this line have? List all allowed transitions using the selection rules $\Delta l = \pm 1$, $\Delta j = 0, \pm 1$.

(c) Calculate the fine structure splitting of the $n = 3$ level (difference between $j = 5/2$ and $j = 1/2$) in cm$^{-1}$. Would this splitting be resolved by the experiment?

(d) Calculate the fine structure splitting of the $n = 2$ level ($j = 3/2$ vs. $j = 1/2$) in cm$^{-1}$.

(e) What is the total spread (in cm$^{-1}$) of the Balmer-$\alpha$ fine structure components?

Q20. A hydrogen atom in the $n = 2$ level is placed in a magnetic field.

(a) In the weak-field regime ($B = 0.01$ T), calculate the Zeeman splitting of the $2p_{3/2}$ level. How many sublevels are there? What is the energy separation between adjacent sublevels?

(b) In the strong-field regime ($B = 10$ T), list all eight $n = 2$ Paschen-Back levels with their energies (measured relative to the unperturbed $E_2^{(0)}$, and expressing energies in terms of $\mu_B B$).

(c) At approximately what field strength does the transition from weak-field to strong-field behavior occur? Justify your answer by comparing $\mu_B B$ to the fine structure splitting.

(d) A radio astronomer observes the 21 cm line from a hydrogen cloud in a magnetic field. The line splits into three components. What field regime is this (weak or strong relative to the hyperfine splitting)? Explain.