Chapter 29 Exercises: Relativistic Quantum Mechanics

Contributors to Quantum Mechanics: From Wavefunctions to Qubits

Chapter 29 Exercises: Relativistic Quantum Mechanics

Part A: Conceptual Questions (7 problems)

These questions test your understanding of the core ideas. Minimal calculations required.

A.1 Explain in your own words why the Schrodinger equation is not Lorentz covariant. What specific feature of its structure reveals the asymmetry between space and time? Why is this acceptable for atomic physics but not for high-energy physics?

A.2 The Klein-Gordon equation is Lorentz covariant and reproduces the correct relativistic energy-momentum relation. Why, then, was it rejected as the fundamental equation for the electron? Identify the specific physical requirement it violates and explain why this requirement is non-negotiable in quantum mechanics.

A.3 Dirac demanded that his equation be first-order in both time and space derivatives. Explain the physical motivation for the first-order-in-time requirement. Then explain why Lorentz covariance forces the equation to also be first-order in spatial derivatives once the time requirement is imposed.

A.4 In non-relativistic quantum mechanics (Chapter 13), the electron's spin and its g-factor $g_s = 2$ are experimental inputs — they must be put in by hand. How does the Dirac equation change this situation? What does it mean to say that "spin is a relativistic effect"? Is it literally true that a non-relativistic electron has no spin?

A.5 A friend claims: "The Dirac sea picture is absurd — an infinite number of invisible particles filling all of space — so the Dirac equation must be wrong about antimatter." Respond to this argument. Is the Dirac sea necessary for the prediction of antimatter? What replaces it in modern physics?

A.6 The Klein paradox shows that a Dirac electron encountering a sufficiently strong potential step has a transmission coefficient greater than 1. Your friend says this violates conservation of probability. Is your friend correct? If not, what physical process resolves the paradox?

A.7 List at least four distinct reasons why single-particle relativistic quantum mechanics is inadequate and quantum field theory is necessary. For each reason, briefly explain what specific physical phenomenon or theoretical requirement forces the upgrade.

A.8 The CPT theorem guarantees that every particle has an antiparticle with the same mass and lifetime but opposite charge and magnetic moment. Some particles (like the photon) are their own antiparticle. What property must a particle have to be its own antiparticle? Can a charged particle be its own antiparticle? Can a fermion be its own antiparticle? (The latter question leads to the concept of Majorana fermions — research the term briefly if needed.)

Part B: Applied Problems (13 problems)

These problems require direct application of the chapter's key equations and results.

B.1: Klein-Gordon Plane Waves

A free spin-0 particle of mass $m$ is described by the Klein-Gordon equation.

(a) Verify by direct substitution that $\phi(\mathbf{r}, t) = A\exp[i(\mathbf{k}\cdot\mathbf{r} - \omega t)]$ is a solution, and find the dispersion relation $\omega(\mathbf{k})$.

(b) Show that the dispersion relation admits both positive and negative frequency solutions. What is the minimum $|\omega|$ for any value of $\mathbf{k}$?

(c) For the positive-frequency solution, compute the Klein-Gordon probability density $\rho_{\text{KG}}$ (Eq. 29.7) and verify that it is positive. Then do the same for the negative-frequency solution and show that $\rho_{\text{KG}} < 0$.

(d) The neutral pion ($\pi^0$) has mass $m_{\pi^0} = 135.0$ MeV/$c^2$. What is the Compton wavelength $\lambda_C = h/(m_{\pi^0}c)$ of the pion? Compare this to the range of the nuclear force ($\sim 1.4$ fm). Comment on the connection.

B.2: Gamma Matrix Algebra

Using only the Clifford algebra relation $\{\gamma^\mu, \gamma^\nu\} = 2g^{\mu\nu}\mathbb{I}_4$ (not any specific representation), prove the following:

(a) $(\gamma^0)^2 = \mathbb{I}_4$ and $(\gamma^i)^2 = -\mathbb{I}_4$ for $i = 1, 2, 3$.

(b) $\gamma^\mu\gamma_\mu = 4\mathbb{I}_4$ (sum over $\mu = 0, 1, 2, 3$).

(c) $\gamma^\mu\gamma^\nu\gamma_\mu = -2\gamma^\nu$.

(d) $\text{Tr}(\gamma^\mu) = 0$ for all $\mu$.

(Hint for (d): Use the anticommutation relation with $\gamma^5$ and the cyclic property of the trace.)

B.3: Dirac Anticommutation Verification

In the standard (Dirac) representation:

$$\alpha_i = \begin{pmatrix} 0 & \sigma_i \\ \sigma_i & 0 \end{pmatrix}, \qquad \beta = \begin{pmatrix} \mathbb{I}_2 & 0 \\ 0 & -\mathbb{I}_2 \end{pmatrix}$$

(a) Verify explicitly that $\alpha_1^2 = \mathbb{I}_4$, $\beta^2 = \mathbb{I}_4$, and $\alpha_1\beta + \beta\alpha_1 = 0$.

(b) Verify that $\alpha_1\alpha_2 + \alpha_2\alpha_1 = 0$.

(c) Construct the four gamma matrices $\gamma^0 = \beta$ and $\gamma^i = \beta\alpha_i$. Write them explicitly as $4 \times 4$ matrices.

(d) Construct $\gamma^5 = i\gamma^0\gamma^1\gamma^2\gamma^3$ and verify that $(\gamma^5)^2 = \mathbb{I}_4$ and $\{\gamma^5, \gamma^0\} = 0$.

B.4: Free Particle Dirac Solutions

Consider a free Dirac particle with mass $m$ and momentum $\mathbf{p} = p\hat{z}$ (motion along the $z$-axis).

(a) Write the Dirac equation in the standard representation and show that the positive-energy spinor $u(\mathbf{p})$ satisfies:

$$\left(c\alpha_3 p + \beta mc^2\right)u(\mathbf{p}) = Eu(\mathbf{p})$$

with $E = +\sqrt{p^2c^2 + m^2c^4}$.

(b) Show that the two independent positive-energy spinors can be written as:

$$u^{(1)}(\mathbf{p}) = N\begin{pmatrix} 1 \\ 0 \\ \frac{pc}{E + mc^2} \\ 0 \end{pmatrix}, \qquad u^{(2)}(\mathbf{p}) = N\begin{pmatrix} 0 \\ 1 \\ 0 \\ \frac{-pc}{E + mc^2} \end{pmatrix}$$

and find the normalization constant $N$.

(c) In the non-relativistic limit ($p \ll mc$), show that the lower two components of $u(\mathbf{p})$ are smaller than the upper two components by a factor of order $v/c$. This is why they are called the "small components."

(d) In the ultra-relativistic limit ($p \gg mc$, or equivalently $E \gg mc^2$), what happens to the ratio of upper and lower components?

B.5: Spin from the Dirac Equation

(a) Define the orbital angular momentum $\hat{L}_z = \hat{x}\hat{p}_y - \hat{y}\hat{p}_x$ and the Dirac Hamiltonian $\hat{H}_D = c\boldsymbol{\alpha}\cdot\hat{\mathbf{p}} + \beta mc^2$. Compute $[\hat{L}_z, \hat{H}_D]$ and show that it is not zero (i.e., orbital angular momentum is not conserved).

(b) Define $\hat{S}_z = \frac{\hbar}{2}\Sigma_z$ where $\Sigma_z = \begin{pmatrix} \sigma_z & 0 \\ 0 & \sigma_z \end{pmatrix}$. Compute $[\hat{S}_z, \hat{H}_D]$.

(c) Show that $[\hat{L}_z + \hat{S}_z, \hat{H}_D] = 0$, verifying that total angular momentum $\hat{J}_z = \hat{L}_z + \hat{S}_z$ is conserved.

(d) What is the physical interpretation of this result? Why is it remarkable that $\hat{S}$ appears with eigenvalues $\pm\hbar/2$?

B.6: Dirac Hydrogen Fine Structure

The exact Dirac energy levels for hydrogen are given by Eq. (29.27):

$$E_{n,j} = mc^2\left[1 + \left(\frac{\alpha}{n - j - \frac{1}{2} + \sqrt{(j + \frac{1}{2})^2 - \alpha^2}}\right)^2\right]^{-1/2}$$

(a) For the ground state ($n = 1$, $j = 1/2$), compute $E_{1,1/2}$ numerically using $\alpha = 1/137.036$ and $mc^2 = 0.510999$ MeV. Express your answer as a binding energy $B = mc^2 - E_{1,1/2}$.

(b) Compare your exact result with the Bohr energy $E_1^{\text{Bohr}} = -13.6058$ eV. What is the fractional difference?

(c) Expand $E_{1,1/2}$ to order $\alpha^4$ and verify that you recover:

$$E_{1,1/2} \approx mc^2\left(1 - \frac{\alpha^2}{2} - \frac{\alpha^4}{8}\right)$$

What is the numerical value of the $\alpha^4$ correction in eV?

(d) For $n = 2$, compute the fine-structure splitting $\Delta E = E_{2,3/2} - E_{2,1/2}$ in eV and in MHz. Compare with the accepted experimental value of approximately $10969$ MHz (or $\approx 4.5 \times 10^{-5}$ eV).

B.7: Pair Creation Threshold

(a) Explain why a single photon in vacuum cannot create an electron-positron pair, even if $E_\gamma > 2m_ec^2$. (Hint: Consider conservation of both energy and momentum in the center-of-momentum frame.)

(b) In the presence of a heavy nucleus of mass $M \gg m_e$, the minimum photon energy for pair creation is essentially $2m_ec^2$. Calculate this threshold energy in eV and the corresponding photon wavelength. In what part of the electromagnetic spectrum does this lie?

(c) In the collision of two photons ($\gamma + \gamma \to e^- + e^+$), what is the minimum total energy needed if the photons have equal energy and collide head-on? What about if one photon is at rest in the lab frame (impossible, but this sets a scale) and the other is highly energetic?

(d) The Schwinger critical field $E_S$ is the electric field strength at which the vacuum becomes unstable against spontaneous pair creation. It is given by $E_S = m_e^2c^3/(e\hbar)$. Compute $E_S$ in V/m. How does this compare to the electric field at the surface of a uranium nucleus ($\sim 10^{21}$ V/m)?

B.8: Anomalous Magnetic Moment

The Dirac equation predicts $g_s = 2$ exactly. The leading QED correction (Schwinger, 1948) gives:

$$g_s = 2\left(1 + \frac{\alpha}{2\pi}\right)$$

(a) Compute the numerical value of $g_s$ to seven decimal places using $\alpha = 1/137.036$.

(b) The current experimental value is $g_s = 2.00231930436256(35)$. What is the fractional agreement between the Schwinger one-loop result and experiment?

(c) The theoretical value of $g_s - 2$, computed to tenth order in QED perturbation theory (five-loop diagrams!), agrees with experiment to 10 significant figures. Express the precision of this agreement as a ratio. If this same precision applied to measuring the distance from Earth to the Moon ($384,400$ km), what would the uncertainty be?

(d) Why is the agreement between QED predictions and $g - 2$ measurements considered the most precise test of any physical theory? What assumptions of QED would be falsified if a discrepancy were found?

B.9: Klein Paradox

Consider a Dirac electron with energy $E > 0$ incident on a step potential:

$$V(x) = \begin{cases} 0 & x < 0 \\ V_0 & x > 0 \end{cases}$$

(a) For $V_0 < E - mc^2$ (weak potential), explain qualitatively what happens. Is this regime paradoxical?

(b) For $E - mc^2 < V_0 < E + mc^2$ (intermediate potential), the wave function decays exponentially in the barrier region, similar to non-relativistic tunneling. Explain why.

(c) For $V_0 > E + mc^2$ (strong potential, the "Klein paradox" regime), the wave function becomes oscillatory again inside the barrier, and the reflection coefficient exceeds 1. Explain the physical interpretation: what particles are being created, and where does the energy come from?

(d) Estimate the potential strength $V_0$ needed for the Klein paradox in units of $m_ec^2$. Is this achievable with ordinary electromagnetic fields? Where in nature might such strong fields occur?

B.10: Zitterbewegung

The velocity operator in the Dirac theory is $\hat{v}_i = c\alpha_i$ (from $\hat{v} = i[\hat{H}, \hat{x}]/\hbar$).

(a) Show that the eigenvalues of $\alpha_i$ are $\pm 1$, which implies that a measurement of any component of the Dirac electron's velocity always yields $\pm c$. Why is this surprising for a massive particle?

(b) Compute $\langle\hat{v}_x\rangle$ for a positive-energy Dirac electron at rest. Show that $\langle\hat{v}_x\rangle = 0$, even though a measurement always gives $\pm c$. Explain how this is consistent.

(c) The phenomenon of Zitterbewegung (German for "trembling motion") refers to the rapid oscillation of $\hat{v}_i(t)$ at frequency $\omega_Z = 2mc^2/\hbar$. Compute $\omega_Z$ and the corresponding period $T_Z$ for an electron. What length scale does $cT_Z$ correspond to?

(d) Zitterbewegung arises from interference between positive-energy and negative-energy components of a wave packet. In QFT, it is reinterpreted as virtual pair creation/annihilation. Explain why Zitterbewegung is unobservable for a single free electron, and what conditions would be needed to observe analogous effects.

B.11: Relativistic Corrections to Hydrogen Energy Levels

The $1S_{1/2}$ ground state energy of hydrogen from the Dirac equation differs from the Bohr result. Using the exact Dirac formula (Eq. 29.27):

(a) Compute the exact ground state energy $E_{1,1/2}$ as a binding energy $B = mc^2 - E$.

(b) Expand to order $\alpha^4$: show that $B \approx \frac{1}{2}mc^2\alpha^2(1 + \frac{1}{4}\alpha^2)$.

(c) The $\alpha^4$ correction to the binding energy of the $1S$ state is often quoted as $-\frac{1}{8}mc^2\alpha^4$. Compute this in eV and in cm$^{-1}$. Compare its magnitude to the fine structure splitting of $n = 2$.

(d) For a hydrogen-like ion with nuclear charge $Z$ (replace $\alpha \to Z\alpha$ in the formula), compute the ground state binding energy for He$^+$ ($Z = 2$), Li$^{2+}$ ($Z = 3$), and U$^{91+}$ ($Z = 92$). For which of these is the Dirac correction a significant fraction of the total binding energy?

B.12: Four-Current Conservation

The Dirac equation has an associated conserved four-current $j^\mu = c\bar{\psi}\gamma^\mu\psi$, where $\bar{\psi} = \psi^\dagger\gamma^0$ is the Dirac adjoint.

(a) Show that the zeroth component $j^0 = c\psi^\dagger\psi$ is positive-definite. This is the probability density. Why is this an improvement over the Klein-Gordon equation?

(b) Starting from the Dirac equation and its adjoint, prove the continuity equation $\partial_\mu j^\mu = 0$.

(c) Show that $\rho = \psi^\dagger\psi = |\psi_1|^2 + |\psi_2|^2 + |\psi_3|^2 + |\psi_4|^2$. Interpret each term physically for a positive-energy electron at rest.

B.13: The Dirac Equation in Natural Units

In particle physics, it is standard to use natural units where $\hbar = c = 1$.

(a) Rewrite the Dirac equation in natural units. Show that it becomes $(i\gamma^\mu\partial_\mu - m)\psi = 0$.

(b) In natural units, what are the dimensions of $m$, $\psi$, and $\gamma^\mu$?

(c) The Compton wavelength of the electron is $\lambda_C = \hbar/(m_ec) = 1/(m_e)$ in natural units. Express $\lambda_C$ in fm. How does it compare to the Bohr radius $a_0 = 1/(\alpha m_e)$ in natural units?

(d) Restore the factors of $\hbar$ and $c$ in the expression $E = \sqrt{p^2 + m^2}$ to obtain $E = \sqrt{p^2c^2 + m^2c^4}$. Verify the dimensions.

Part C: Advanced Problems (7 problems)

These problems require deeper analysis, proof, or synthesis across multiple topics.

C.1: Covariance of the Dirac Equation

Under a Lorentz transformation $x^\mu \to x'^\mu = \Lambda^\mu_{\ \nu}x^\nu$, the Dirac spinor transforms as $\psi(x) \to \psi'(x') = S(\Lambda)\psi(x)$, where $S(\Lambda)$ is a $4\times 4$ matrix.

(a) Show that the Dirac equation $(i\hbar\gamma^\mu\partial_\mu - mc)\psi = 0$ is covariant (has the same form in the primed frame) if and only if:

$$S^{-1}(\Lambda)\gamma^\mu S(\Lambda) = \Lambda^\mu_{\ \nu}\gamma^\nu$$

(b) For an infinitesimal Lorentz transformation $\Lambda^\mu_{\ \nu} = \delta^\mu_\nu + \omega^\mu_{\ \nu}$ (with $\omega_{\mu\nu} = -\omega_{\nu\mu}$), show that $S(\Lambda) = \mathbb{I}_4 + \frac{i}{4}\omega_{\mu\nu}\sigma^{\mu\nu}$ satisfies the condition in (a), where $\sigma^{\mu\nu} = \frac{i}{2}[\gamma^\mu, \gamma^\nu]$.

(c) For a rotation by angle $\theta$ about the $z$-axis, find $\omega_{\mu\nu}$ and show that $S = \exp\left(\frac{i\theta}{2}\Sigma_z\right)$. What happens when $\theta = 2\pi$? What does this tell you about the spinor nature of $\psi$?

C.2: Non-Relativistic Limit: Recovering the Pauli Equation

Write the four-component Dirac spinor as $\psi = \begin{pmatrix} \phi \\ \chi \end{pmatrix}$ where $\phi$ and $\chi$ are two-component spinors. In the presence of an electromagnetic field:

(a) Show that the Dirac equation becomes the coupled system:

$$(E - mc^2 - e\Phi)\phi = c\boldsymbol{\sigma}\cdot(\hat{\mathbf{p}} - \frac{e}{c}\mathbf{A})\chi$$ $$(E + mc^2 - e\Phi)\chi = c\boldsymbol{\sigma}\cdot(\hat{\mathbf{p}} - \frac{e}{c}\mathbf{A})\phi$$

(b) In the non-relativistic limit ($E \approx mc^2 + \varepsilon$ with $\varepsilon, e\Phi \ll mc^2$), show that the small component $\chi$ can be eliminated to give:

$$\chi \approx \frac{\boldsymbol{\sigma}\cdot(\hat{\mathbf{p}} - \frac{e}{c}\mathbf{A})}{2mc}\phi$$

(c) Substitute back into the equation for $\phi$ and use the identity $(\boldsymbol{\sigma}\cdot\mathbf{A})(\boldsymbol{\sigma}\cdot\mathbf{B}) = \mathbf{A}\cdot\mathbf{B} + i\boldsymbol{\sigma}\cdot(\mathbf{A}\times\mathbf{B})$ to derive the Pauli equation:

$$\left[\frac{(\hat{\mathbf{p}} - \frac{e}{c}\mathbf{A})^2}{2m} + e\Phi - \frac{e\hbar}{2mc}\boldsymbol{\sigma}\cdot\mathbf{B}\right]\phi = \varepsilon\phi$$

Identify the magnetic moment and confirm $g_s = 2$.

C.3: Dirac Hydrogen — Exact Solution Structure

The exact solution of the Dirac equation for hydrogen uses the quantum numbers $n$, $l$, $j$, $m_j$, and the Dirac quantum number $\kappa$.

(a) The operator $\hat{K} = \beta(\hat{\boldsymbol{\Sigma}}\cdot\hat{\mathbf{L}} + \hbar)$ commutes with $\hat{H}_D$, $\hat{J}^2$, and $\hat{J}_z$. Show that $\hat{K}^2 = \hat{J}^2 + \frac{\hbar^2}{4}$, which means the eigenvalues of $\hat{K}$ are $\kappa = \pm(j + \frac{1}{2})$, with $\kappa > 0$ for $j = l - 1/2$ and $\kappa < 0$ for $j = l + 1/2$.

(b) Using the quantum number $\kappa$, the principal quantum number can be written as $n = n_r + |\kappa|$ where $n_r = 0, 1, 2, \ldots$ is the radial quantum number. Verify this relationship for the states $1S_{1/2}$, $2S_{1/2}$, $2P_{1/2}$, and $2P_{3/2}$ by identifying $n_r$, $\kappa$, $l$, and $j$ for each.

(c) Explain why the Dirac spectrum depends on $|\kappa|$ (or equivalently $j$) but not on the sign of $\kappa$ (or equivalently not on $l$ for fixed $j$). This is the origin of the accidental degeneracy between $2S_{1/2}$ and $2P_{1/2}$.

C.4: The Dirac Equation and CPT

(a) Define the charge conjugation transformation $\psi \to \psi_C = C\bar{\psi}^T$ where $C = i\gamma^2\gamma^0$. Show that if $\psi$ satisfies the Dirac equation with charge $+e$ in an external field, then $\psi_C$ satisfies the Dirac equation with charge $-e$.

(b) The parity transformation sends $\psi(\mathbf{r}, t) \to P\psi(-\mathbf{r}, t)$ with $P = \gamma^0$. Show that the Dirac equation is invariant under this transformation for a spherically symmetric potential.

(c) The time-reversal transformation sends $\psi(\mathbf{r}, t) \to T\psi^*(\mathbf{r}, -t)$ with $T = i\gamma^1\gamma^3$. Show that $T^2 = -\mathbb{I}_4$, which is the hallmark of half-integer spin (Kramers' degeneracy).

(d) Verify that the combined CPT transformation sends a positive-energy electron solution into a positive-energy positron solution, consistent with the CPT theorem.

C.5: Lamb Shift Estimate

The dominant contribution to the Lamb shift comes from the electron self-energy, which can be estimated semi-classically as follows.

(a) Quantum fluctuations of the electromagnetic vacuum cause the electron's position to "jitter" by an amount $\delta r$. Show that the mean-square displacement is roughly:

$$\langle(\delta r)^2\rangle \sim \frac{\alpha}{\pi}\left(\frac{\hbar}{mc}\right)^2\ln\left(\frac{mc^2}{E_{\text{binding}}}\right)$$

where $E_{\text{binding}} \sim \alpha^2mc^2$ is the typical atomic binding energy. Evaluate this numerically.

(b) The effect of this jitter on the energy levels is:

$$\Delta E \approx \frac{1}{6}\langle(\delta r)^2\rangle\langle\nabla^2V\rangle$$

For the Coulomb potential $V(r) = -e^2/r$, show that $\nabla^2V = 4\pi e^2\delta^3(\mathbf{r})$, so only $S$-states ($l = 0$) are affected at this level.

(c) Using $|\psi_{ns}(0)|^2 = 1/(\pi n^3 a_0^3)$ for $S$-states, show that:

$$\Delta E_{nS} \approx \frac{4\alpha^5mc^2}{3\pi n^3}\ln\left(\frac{1}{\alpha^2}\right)$$

Evaluate this for $n = 2$ and compare with the experimental Lamb shift of 1057 MHz.

C.6: Completeness of the Dirac Algebra

(a) Show that the sixteen matrices $\Gamma_A = \{\mathbb{I}_4, \gamma^\mu, \sigma^{\mu\nu}, \gamma^5\gamma^\mu, \gamma^5\}$ satisfy $\text{Tr}(\Gamma_A\Gamma_B) = 4\delta_{AB}$ (with appropriate normalization). This proves they form an orthogonal basis for the space of $4\times 4$ matrices.

(b) Use this completeness to prove the Fierz identity: any product of two Dirac bilinears $(\bar{\psi}_1\Gamma_A\psi_2)(\bar{\psi}_3\Gamma_B\psi_4)$ can be rewritten as a sum over bilinears of the form $(\bar{\psi}_1\Gamma_C\psi_4)(\bar{\psi}_3\Gamma_D\psi_2)$. Write the explicit Fierz rearrangement for $A = B = \mathbb{I}_4$ (scalar-scalar).

(c) Prove that $\text{Tr}(\gamma^\mu\gamma^\nu) = 4g^{\mu\nu}$, $\text{Tr}(\gamma^\mu\gamma^\nu\gamma^\rho\gamma^\sigma) = 4(g^{\mu\nu}g^{\rho\sigma} - g^{\mu\rho}g^{\nu\sigma} + g^{\mu\sigma}g^{\nu\rho})$, and $\text{Tr}(\text{odd number of gammas}) = 0$. These trace identities are the workhorses of QED calculations.

C.7: Synthesis Problem — From Schrodinger to Dirac to QFT

This problem asks you to trace the logical thread of the entire chapter.

(a) Start with the non-relativistic Schrodinger equation for a free particle. Show that it predicts dispersion ($\partial\omega/\partial k$ depends on $k$) and that the group velocity equals $p/m$ (the classical velocity). Then show that it violates Lorentz covariance by demonstrating that $\partial\psi/\partial t$ and $\nabla^2\psi$ transform differently under Lorentz boosts.

(b) Write the Klein-Gordon equation and verify Lorentz covariance. Then show that it admits negative-energy solutions and that the associated probability density is not positive-definite. Explain concretely why you cannot simply "throw away" the negative-energy solutions.

(c) Write the Dirac equation and show that squaring it recovers the Klein-Gordon equation for each component. Explain how the Dirac equation resolves the probability density problem but does not resolve the negative-energy problem.

(d) Explain — in one clear paragraph per point — the three most important reasons why even the Dirac equation is insufficient and quantum field theory is necessary. Make sure your reasons are logically independent and cover different aspects of the physics.

Solutions Notes

Selected solutions are provided in Appendix G. Full solutions to all problems are available in the Instructor's Manual.

Part A problems: Answers should be conceptual paragraphs, not calculations.
Part B problems: Show all mathematical steps clearly. Numerical answers should be given with appropriate significant figures.
Part C problems: These are multi-step proofs or derivations. Partial credit is available for correct intermediate steps even if the final result is not reached.