Chapter 29 Quiz: Relativistic Quantum Mechanics

Instructions: This quiz covers the core concepts from Chapter 29. 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. The Klein-Gordon equation is rejected as a single-particle wave equation for the electron primarily because:

(a) It is not Lorentz covariant (b) It does not reproduce the correct energy-momentum relation (c) Its probability density can be negative (d) It predicts the wrong spin for the electron

Q2. The minimum dimension of the Dirac matrices $\alpha_i$ and $\beta$ is $4 \times 4$ because:

(a) There are four spacetime dimensions (b) The electron has four quantum numbers (c) Four anticommuting, traceless, Hermitian matrices with eigenvalues $\pm 1$ require at least a $4 \times 4$ representation (d) The Dirac equation has four independent solutions for each momentum

Q3. The Clifford algebra relation satisfied by the gamma matrices is:

(a) $[\gamma^\mu, \gamma^\nu] = 2g^{\mu\nu}\mathbb{I}_4$ (b) $\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}\mathbb{I}_4$ (c) $\gamma^\mu\gamma^\nu = g^{\mu\nu}\mathbb{I}_4$ (d) $\{\gamma^\mu, \gamma^\nu\} = 2\delta^{\mu\nu}\mathbb{I}_4$

Q4. In the Dirac theory, the spin of the electron is:

(a) Put in by hand as an additional postulate, just as in non-relativistic quantum mechanics (b) A consequence of the first-order time derivative in the Schrodinger equation (c) A necessary consequence of requiring a first-order, Lorentz-covariant wave equation (d) Only present when the electron interacts with an electromagnetic field

Q5. The Dirac equation predicts the electron g-factor to be:

(a) $g_s = 1$ (the classical value) (b) $g_s = 2$ exactly (c) $g_s = 2.0023$ (including the leading QED correction) (d) $g_s = 0$ (the electron has no magnetic moment in the Dirac theory)

Q6. In the exact Dirac spectrum of hydrogen, which of the following pairs of states are degenerate?

(a) $2S_{1/2}$ and $2P_{3/2}$ (b) $2S_{1/2}$ and $2P_{1/2}$ (c) $2P_{1/2}$ and $2P_{3/2}$ (d) $1S_{1/2}$ and $2S_{1/2}$

Q7. The Lamb shift ($2S_{1/2}$ vs. $2P_{1/2}$ splitting in hydrogen) is caused by:

(a) Spin-orbit coupling, which Dirac neglected (b) The interaction of the electron with quantum fluctuations of the electromagnetic vacuum (c) The finite size of the proton (d) The gravitational field of the proton

Q8. In the Dirac sea picture, a positron is interpreted as:

(a) An electron moving backward in time (b) A proton that has lost its quarks (c) A hole (missing electron) in the filled sea of negative-energy states (d) A photon that has acquired mass through the Higgs mechanism

Q9. Pair creation ($\gamma \to e^- + e^+$) requires a minimum photon energy of approximately:

(a) $m_ec^2 \approx 0.511$ MeV (b) $2m_ec^2 \approx 1.022$ MeV (c) $4m_ec^2 \approx 2.044$ MeV (d) $\alpha m_ec^2 \approx 3.7$ keV

Q10. Which of the following is NOT a reason why quantum field theory is necessary?

(a) Particle number is not conserved in relativistic processes (b) The Lamb shift requires quantization of the electromagnetic field (c) The Schrodinger equation is not Lorentz covariant (d) Causality requires contributions from both particles and antiparticles

True/False (4 questions)

For each statement, indicate whether it is True or False and provide a brief justification (1-2 sentences).

Q11. The Klein-Gordon equation is physically useless and plays no role in modern physics.

Q12. The Dirac equation correctly predicts that the electron has spin-1/2, but the value $g_s = 2$ must still be put in by hand.

Q13. In the Dirac theory of hydrogen, states with the same $n$ and $j$ but different $l$ are degenerate (have the same energy).

Q14. The Feynman-Stückelberg interpretation of negative-energy solutions applies only to fermions and not to bosons.

Short Answer (4 questions)

Q15. Explain why the Schrodinger equation being first-order in time is important for the probabilistic interpretation of quantum mechanics. Then explain why Lorentz covariance, combined with this first-order-in-time requirement, forces the spatial part of the Dirac equation to also be first-order. (3-5 sentences)

Q16. The fine structure of hydrogen arises from three distinct physical effects in the non-relativistic perturbation theory of Chapter 18: the relativistic kinetic energy correction, spin-orbit coupling, and the Darwin term. Each depends on $l$. Yet the Dirac equation gives a fine-structure formula that depends only on $j$. Explain how these two results are consistent. (3-5 sentences)

Q17. Describe the Lamb-Retherford experiment of 1947 and explain its significance for the development of QED. What specific prediction of the Dirac equation did it disprove? (3-5 sentences)

Q18. The spin-statistics theorem states that half-integer-spin particles must be fermions and integer-spin particles must be bosons. In non-relativistic quantum mechanics, this is an axiom. In quantum field theory, it can be derived. What assumptions go into the QFT proof, and what would go wrong if you tried to quantize a spin-1/2 field as a boson? (3-5 sentences)

Applied Scenarios (2 questions)

Q19. A cosmic ray photon with energy $E_\gamma = 5.0$ MeV enters the Earth's atmosphere and passes near a nitrogen nucleus.

(a) Is the photon's energy sufficient for electron-positron pair production? Show your calculation.

(b) If pair production occurs, what is the maximum kinetic energy available to be shared between the electron and positron? (Neglect the recoil of the nucleus.)

(c) The Dirac equation predicts $g_s = 2$ for the electron. The leading QED correction modifies this to $g_s \approx 2(1 + \alpha/(2\pi))$. Compute the difference $\Delta g = g_s^{\text{QED}} - g_s^{\text{Dirac}}$ and express it as a fraction of $g_s^{\text{Dirac}}$.

Q20. You are asked to compute the fine-structure splitting of the $n = 3$ level of hydrogen.

(a) List all the possible states at $n = 3$ using spectroscopic notation ($nL_J$). For each state, give the values of $l$, $j$, and the degeneracy $2j + 1$.

(b) Using the approximate fine-structure formula from Eq. (29.28), compute the energy of each distinct $j$ level relative to the unperturbed Bohr energy $E_3 = -13.6/9$ eV. Express your answers in eV and in cm$^{-1}$.

(c) Which states are degenerate according to the Dirac equation? Which degeneracy is broken by the Lamb shift?

(d) Sketch an energy-level diagram showing the Bohr level, the Dirac fine-structure levels, and the Lamb-shift corrections (qualitative splitting only for the Lamb shift).

Answer Key

Q1: (c). The Klein-Gordon probability density $\rho_{\text{KG}}$ involves $\partial\phi/\partial t$ and can take negative values for negative-frequency solutions, making it unacceptable as a probability density.

Q2: (c). We need four objects ($\alpha_1, \alpha_2, \alpha_3, \beta$) that are Hermitian, traceless, square to the identity, and mutually anticommute. In $2\times 2$, the Pauli matrices provide only three such objects. The minimum dimension allowing four is $4\times 4$.

Q3: (b). The Clifford algebra is defined by the anticommutation relation $\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}\mathbb{I}_4$, where $g^{\mu\nu}$ is the Minkowski metric.

Q4: (c). Spin emerges automatically when one constructs a wave equation that is simultaneously first-order in time (for positive-definite probability density) and Lorentz covariant. The four-component spinor structure, which encodes spin-1/2, is forced by these two requirements.

Q5: (b). The Dirac equation predicts $g_s = 2$ exactly. The deviation from 2 (the anomalous magnetic moment) is a QED radiative correction, first computed by Schwinger.

Q6: (b). The exact Dirac spectrum $E_{n,j}$ depends only on $n$ and $j$, not on $l$. The $2S_{1/2}$ ($l=0, j=1/2$) and $2P_{1/2}$ ($l=1, j=1/2$) states share the same $n$ and $j$ and are therefore degenerate. This degeneracy is broken by the Lamb shift.

Q7: (b). The Lamb shift arises from the electron's interaction with the quantized electromagnetic vacuum — specifically, the electron self-energy (emission/reabsorption of virtual photons) and vacuum polarization (virtual electron-positron pairs). The Dirac equation, which treats the EM field classically, predicts zero splitting.

Q8: (c). In the Dirac sea picture, the vacuum is a filled sea of negative-energy electron states. Removing one electron creates a hole that behaves as a particle with positive energy, positive charge, and the electron's mass — a positron.

Q9: (b). The minimum energy for $e^-e^+$ pair creation is $2m_ec^2 \approx 1.022$ MeV (in the presence of a nucleus to conserve momentum). This is the combined rest mass energy of the electron and positron.

Q10: (c). While true, the non-covariance of the Schrodinger equation is already addressed by the Klein-Gordon and Dirac equations within single-particle relativistic QM. It is not, by itself, a reason to go all the way to QFT. The other three options — variable particle number, the Lamb shift, and causality — specifically require the field-theoretic framework.

Q11: False. The Klein-Gordon equation fails as a single-particle wave equation but succeeds brilliantly as a quantum field equation for spin-0 particles. It correctly describes pions, kaons, and the Higgs boson within QFT.

Q12: False. The Dirac equation predicts both spin-1/2 and $g_s = 2$ automatically. The g-factor emerges from the non-relativistic limit of the Dirac equation without any additional input.

Q13: True. The exact Dirac spectrum $E_{n,j}$ depends on $n$ and $j$ only, not on $l$. For example, $2S_{1/2}$ ($l=0$) and $2P_{1/2}$ ($l=1$) are degenerate because both have $j = 1/2$. This accidental degeneracy is lifted by the Lamb shift.

Q14: False. The Feynman-Stückelberg interpretation — reinterpreting negative-energy solutions as positive-energy antiparticles — is completely general and applies to both fermions and bosons. It is the Dirac sea picture that is specific to fermions (since it relies on the Pauli exclusion principle).

Q15: A first-order-in-time equation requires only $\psi(\mathbf{r}, 0)$ as an initial condition, which means the probability density $\rho = |\psi|^2$ is determined by the state alone and is automatically non-negative. A second-order equation (like Klein-Gordon) also requires $\dot{\psi}(\mathbf{r}, 0)$, introducing a probability density that depends on $\dot{\psi}$ and can be negative. Lorentz covariance treats time and space on equal footing, so if the equation is first-order in $\partial/\partial t$, it must also be first-order in $\partial/\partial x^i$. This is what forces the introduction of matrix coefficients ($\alpha_i$, $\beta$) and the four-component spinor structure.

Q16: In the perturbative approach of Chapter 18, the three fine-structure corrections (relativistic kinetic energy, spin-orbit, and Darwin) are computed separately, and each depends on both $l$ and $j$. However, when all three are summed, the $l$-dependence cancels, leaving a result that depends only on $j$. This cancellation appears "miraculous" in perturbation theory but is guaranteed by the Dirac equation, which treats all three effects simultaneously and exactly. The Dirac fine-structure formula depends only on $n$ and $j$ from the outset because the Dirac equation respects a higher symmetry (related to the Dirac quantum number $\kappa$) that makes the $l$-independence manifest.

Q17: In 1947, Willis Lamb and Robert Retherford used microwave spectroscopy to measure the energy difference between the $2S_{1/2}$ and $2P_{1/2}$ states of hydrogen, finding a splitting of about 1057 MHz. The Dirac equation predicts these two states to be exactly degenerate, so the observed splitting was a direct failure of the Dirac theory. This result catalyzed the development of quantum electrodynamics (QED), since the Lamb shift could be explained by the interaction of the electron with quantum fluctuations of the electromagnetic vacuum. Bethe's estimate of the self-energy contribution, computed shortly after the Shelter Island conference, agreed with experiment to within 2% and helped establish QED as the correct theory of electromagnetic interactions.

Q18: The QFT proof of the spin-statistics theorem requires Lorentz invariance, locality of interactions (microcausality), positive-definite energy (the Hamiltonian is bounded below), and the existence of a unique vacuum state. If one quantizes a spin-1/2 field using bosonic commutation relations (rather than fermionic anticommutation relations), the resulting theory has either negative-energy states (the Hamiltonian is unbounded below) or violations of microcausality (measurements at spacelike-separated points can influence each other). Either outcome is physically unacceptable, forcing spin-1/2 fields to obey Fermi-Dirac statistics.

Q19: (a) The threshold for pair production is $2m_ec^2 = 2 \times 0.511 = 1.022$ MeV. Since $E_\gamma = 5.0$ MeV $> 1.022$ MeV, the photon has sufficient energy. Yes. (b) The kinetic energy available is $K_{\text{total}} = E_\gamma - 2m_ec^2 = 5.0 - 1.022 = 3.978$ MeV, shared between the electron and positron (neglecting nuclear recoil). (c) $\Delta g = 2 \times \frac{\alpha}{2\pi} = \frac{\alpha}{\pi} = \frac{1}{137.036\pi} \approx 2.323 \times 10^{-3}$. As a fraction of $g_s^{\text{Dirac}} = 2$: $\Delta g / g_s \approx 1.161 \times 10^{-3} \approx 0.116\%$.

Q20: (a) The $n = 3$ states are: $3S_{1/2}$ ($l=0, j=1/2$, degeneracy 2), $3P_{1/2}$ ($l=1, j=1/2$, degeneracy 2), $3P_{3/2}$ ($l=1, j=3/2$, degeneracy 4), $3D_{3/2}$ ($l=2, j=3/2$, degeneracy 4), $3D_{5/2}$ ($l=2, j=5/2$, degeneracy 6). Total degeneracy: $2(3)^2 = 18$. (b) Using $\Delta E = -\frac{mc^2\alpha^4}{2n^4}\left(\frac{n}{j+1/2} - \frac{3}{4}\right)$: For $j=1/2$: $\Delta E = -mc^2\alpha^4/(2\cdot 81)(3/1 - 3/4) = -mc^2\alpha^4 \cdot 9/(8\cdot 81) \approx -1.51 \times 10^{-5}$ eV; For $j=3/2$: $\Delta E = -mc^2\alpha^4/(2\cdot 81)(3/2 - 3/4) = -mc^2\alpha^4 \cdot 3/(8\cdot 81) \approx -5.04 \times 10^{-6}$ eV; For $j=5/2$: $\Delta E = -mc^2\alpha^4/(2\cdot 81)(3/3 - 3/4) = -mc^2\alpha^4 \cdot 1/(8\cdot 81) \approx -1.68 \times 10^{-6}$ eV. (c) Dirac degeneracies: $3S_{1/2}$ and $3P_{1/2}$ (same $j=1/2$); $3P_{3/2}$ and $3D_{3/2}$ (same $j=3/2$). The Lamb shift breaks the $3S_{1/2}$/$3P_{1/2}$ degeneracy (and the $3P_{3/2}$/$3D_{3/2}$ degeneracy, though this is much smaller). (d) Diagram: Single Bohr level $\to$ three Dirac levels (grouped by $j$) $\to$ Lamb shift splits each $j$ group by $l$.