= Solution
\b[Noninteracting <spectral line>.] Treat $\mathbf k$ as a <wavevector>, so physical <crystal momentum> is $\hbar\mathbf k$. Write $\Omega_{\mathbf k}=\varepsilon_{\mathbf k}/\hbar$; an <energy> zero at the <chemical potential> can be chosen throughout. The temporal factor of a noninteracting state is $e^{-i\Omega_{\mathbf k}t}$. With the <Fourier transform> convention $\widetilde\psi(\omega)=\int dt\,e^{i\omega t}\psi(t)$,
$$
\widetilde\psi_{\mathbf k}(\omega)=2\pi\psi_{\mathbf k}(0)\delta(\omega-\Omega_{\mathbf k}).
$$
Thus its normalized <frequency> <spectral function> is
$$
\boxed{A_0(\mathbf k,\omega)=\delta(\omega-\Omega_{\mathbf k}).}
$$
One should not square the delta distribution as if it were an ordinary finite-time transition amplitude. More precisely, the fermionic <retarded Green function> is $G_0^R(\mathbf k,t)=-i\Theta(t)e^{-i\Omega_{\mathbf k}t}$. Its <Fourier transform> is $(\omega-\Omega_{\mathbf k}+i0^+)^{-1}$, whose negative imaginary part divided by $\pi$ is the same delta peak.
\b[Interactions and lifetime.] Interactions change the dispersion through the real part of the <self-energy> and permit decay into other excitations through its imaginary part. In the simplest quasiparticle-pole model, <exponential decay> of the retarded amplitude gives
$$
G^R(\mathbf k,t)=-i\Theta(t)e^{-i\Omega_{\mathbf k}^*t-\Gamma_{\mathbf k}t},\qquad \Gamma_{\mathbf k}>0.
$$
Performing the causal integral,
$$
G^R(\mathbf k,\omega)=-i\int_0^\infty e^{[i(\omega-\Omega_{\mathbf k}^*)-\Gamma_{\mathbf k}]t}\,dt=\frac1{\omega-\Omega_{\mathbf k}^*+i\Gamma_{\mathbf k}}.
$$
The corresponding <Lorentzian quasiparticle spectral peak> is
$$
\boxed{A(\mathbf k,\omega)=-\frac1\pi\operatorname{Im}G^R=\frac1\pi\frac{\Gamma_{\mathbf k}}{(\omega-\varepsilon_{\mathbf k}^*/\hbar)^2+\Gamma_{\mathbf k}^2}.}
$$
Its total <frequency> integral is one. The exponential amplitude lifetime is $1/\Gamma_{\mathbf k}$; the probability lifetime is $1/(2\Gamma_{\mathbf k})$. Describing $\Gamma_{\mathbf k}$ as a scattering rate therefore requires the lifetime convention to be specified. The <energy> half-width is $\hbar\Gamma_{\mathbf k}$.
A general interacting <Electron> is more accurately described by $G^R=Z_{\mathbf k}/(\omega-\Omega^*_{\mathbf k}+i\Gamma_{\mathbf k})+G^R_{\mathrm{inc}}$ near the pole. The coherent peak then integrates to the <quasiparticle residue> $Z_{\mathbf k}$ and an incoherent continuum carries the remaining weight. In an energy-unit <self-energy> convention,
$$
G^R=\frac1{\omega-\varepsilon^0_{\mathbf k}/\hbar-\Sigma^R(\mathbf k,\omega)/\hbar},\qquad \varepsilon^*_{\mathbf k}=\varepsilon^0_{\mathbf k}+\operatorname{Re}\Sigma^R(\mathbf k,\varepsilon^*_{\mathbf k}/\hbar).
$$
Linearizing the real denominator gives $Z=[1-\hbar^{-1}\partial_\omega\operatorname{Re}\Sigma^R]^{-1}$ and $\Gamma=-Z\operatorname{Im}\Sigma^R/\hbar$ at the pole. The requested unit-weight Lorentzian is the normalized <quasiparticle> model, not the most general exact interacting spectrum.
\b[Peak position and width.] At fixed $\mathbf k$, put $\gamma=\gamma(\mathbf k)>0$ and $\Omega=\hbar k^2/(2m^*)$. Differentiation gives
$$
\frac{\partial A}{\partial\omega}=-\frac{2\gamma(\omega-\Omega)}{\pi[(\omega-\Omega)^2+\gamma^2]^2}.
$$
It is positive for $\omega<\Omega$ and negative for $\omega>\Omega$, so the only maximum is at $\Omega$, with height $1/(\pi\gamma)$. Setting $A$ to half this height gives $(\omega-\Omega)^2=\gamma^2$. Therefore
$$
\boxed{\omega_{\max}=\frac{\hbar k^2}{2m^*},\qquad \mathrm{HWHM}_\omega=\gamma(\mathbf k),\qquad \mathrm{FWHM}_\omega=2\gamma(\mathbf k).}
$$
In <energy> units the maximum is $\hbar^2k^2/(2m^*)$ and the half-width is $\hbar\gamma(\mathbf k)$. <Frequency> dependence of the <self-energy> would require reconsidering both the shape and peak position.
\b[<Fermi gas> and <Fermi liquid>.] A noninteracting <Fermi gas> has a delta-function <spectral line> at its bare dispersion for every <momentum>. A normal three-dimensional <Fermi liquid> has a shifted coherent peak plus incoherent spectral weight; close to the <Fermi surface> its width scales as $(\varepsilon-\mu)^2+(\pi k_BT)^2$. At zero <temperature> the peak becomes arbitrarily sharp at the <Fermi surface>, while it broadens farther away. <Spectral functions> include addition and removal weight; occupied-only photoemission contains an extra Fermi factor.
\Image[/past-exam-of-the-quantum-condensed-matter-course-of-the-university-of-cambridge/2017-quasiparticle-spectra.png]
{title=A free spectral line and broadened quasiparticle peaks, with a separate incoherent background}
\b[Photoemission measurement.] A <photon> of known <energy> $h\nu$ ejects an <Electron>. Measure its <kinetic energy> $E_{\mathrm{kin}}$ and emission direction. For <binding energy> $E_B\geq0$ below the <Fermi level> and the appropriate calibrated <work function> $\Phi$, <conservation of energy> gives
$$
\boxed{E_{\mathrm{kin}}=h\nu-\Phi-E_B.}
$$
<Conservation of momentum> parallel to a periodic surface gives
$$
\mathbf k_{i,\parallel}+\mathbf q_{\gamma,\parallel}=\mathbf k_{\mathrm{out},\parallel}+\mathbf G_\parallel,\qquad k_{\mathrm{out},\parallel}=\frac{\sqrt{2m_0E_{\mathrm{kin}}}}{\hbar}\sin\theta,
$$
where $\mathbf G_\parallel$ is a surface reciprocal-lattice vector. At suitable <photon> energies the <photon> <momentum> can be neglected; at higher energies it must be retained. The surface breaks perpendicular translation symmetry, so the perpendicular <crystal momentum> generally needs a final-state model. A two-dimensional band can be reconstructed directly from the parallel components.
Within the <sudden approximation> and <electric-dipole approximation>, <photoemission intensity and spectral function> are related by
$$
I(\mathbf k,E)\propto|\mathcal M(\mathbf k,h\nu,\text{polarization})|^2 f(E)A_E(\mathbf k,E),\qquad A_E(\mathbf k,E)=\frac1\hbar A(\mathbf k,E/\hbar).
$$
Here $E$ is measured relative to the <chemical potential>. Instrumental resolution convolves the product and inelastic scattering adds background. Thus <ARPES> maps the occupied <electron removal spectral function>, weighted by a <matrix element>; it does not measure the full unoccupied spectrum or an unweighted <spectral function> automatically. Bright ridges give dispersions, and intrinsic widths give decay information after instrumental effects are removed.
\b[Reading the supplied image.] Its bright ridges form a graphene-like cone. The crossing near $k_y=0$ is the <Dirac point>, not in general the <Fermi level>. In the paper's plotted <energy> coordinates its <energy> is roughly $E_D\sim1.4$–$1.5\,\mathrm{eV}$. A velocity follows from a branch slope independently of the <energy> zero:
$$
v=\frac1\hbar\left|\frac{dE}{dk_y}\right|,\qquad 1\,\mathrm{eV\,\mathring A}/\hbar=1.519\times10^5\,\mathrm{m\,s^{-1}}.
$$
For example, the lower branch moves by about $0.3\,\mathrm{eV}$ between $|k_y|\sim0.03$ and $0.11\,\mathrm{\mathring A^{-1}}$. This gives $dE/dk\sim3.8\,\mathrm{eV\,\mathring A}$ and
$$
\boxed{v_F\sim6\times10^5\,\mathrm{m\,s^{-1}}\quad\text{as a rough linear-cone estimate}.}
$$
Reading other points changes the estimate by tens of percent. The original exam image has unequal <energy> increments on nearly equally spaced labelled ticks, so it does not justify a precise fitted velocity. The value is an estimate of the cone slope extrapolated to the Fermi crossing, rather than a measured derivative exactly at a visibly identified Fermi cutoff.
To determine a <Fermi energy>, first calibrate the Fermi edge, then subtract the chosen reference <energy>. Neither a Fermi-level marker nor an explicit energy-zero convention is supplied in this crop. If its upper termination near $1.9\,\mathrm{eV}$ is intended to represent the Fermi cutoff, the conditional graph answer is \b[$E_F\simeq1.9\,\mathrm{eV}$ on that axis], or \b[$E_F-E_D\sim0.4\,\mathrm{eV}$] relative to the Dirac crossing. Identifying the crossing itself as the <Fermi energy> would incorrectly assume undoped <graphene>.
The research article cited by the exam identifies this as linearly polarized <graphene> <ARPES> and independently states $E_F-E_D\simeq0.4\,\mathrm{eV}$ for the substrate-doped sample. That resolves the relative <energy> when this external calibration is admitted.https://hrg.pusan.ac.kr/articles/JPCM_0953-8984_26_33_335501.pdf[Section 3, printed p.2 and figure 1(b): <Dirac point> below the <Fermi energy>]{ref} \b[The crop alone does not uniquely fix an absolute <Fermi energy>]; the source-supported relative value and the conditional $1.9\,\mathrm{eV}$ axis reading should be distinguished.
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