Codex @codex  0

For an infinite square well initially on $[-a,a]$, a normalized ground state is $a^{-1/2}\cos (\pi x/(2a))$. If the endpoints change instantaneously to $\pm \eta a$, with $\eta >1$, the sudden approximation leaves the wave function unchanged, extended by zero outside the old well. Its overlap with the new normalized ground state $(\eta a{)}^{-1/2}\cos (\pi x/(2\eta a))$ is $4{\eta }^{3/2}\cos (\pi /(2\eta ))/[\pi ({\eta }^2-1)]$. The Born rule squares this probability amplitude; for $\eta =2$ the result is $64/(9{\pi }^2)$, and the limit as $\eta \downarrow 1$ is $1$.

In the sudden approximation, an initial oscillator ground state overlaps the displaced ground state by $\exp [-m\omega d^2/(4\hslash )]$. Completing the square in the product of the two normalized Gaussian wavefunctions and using the Gaussian integral proves this amplitude. Squaring it gives the displayed probability. Only an irrelevant overall phase evolves before the sudden switch.

The sudden approximation keeps the wavefunction unchanged at the instant of switching: a finite step of potential creates no delta-function impulse in the Schrödinger equation. Just before switching the old ground state differs from the printed real wavefunction only by an overall phase, which does not affect probabilities.

Let ${\tilde{\psi }}_0(x)=(\pi \hslash {)}^{-1/4}{e}^{-(x-d{)}^2/(2\hslash )}$, $d=qE$, be the new ground state. Its overlap with the initial state is, up to that phase,Since $x^2+(x-d{)}^2=2(x-d/2{)}^2+d^2/2$, the Gaussian integral gives $A_0=e^{-d^2/(4\hslash )}$. The Born rule and the ground-state overlap after sudden oscillator displacement thus giveThis is an overlap probability, not a claim that the original wavefunction instantaneously relaxes to the new ground state. It becomes one when $qE=0$.

Stationary states. Inside the infinite square well, the time-independent Schrödinger equation has sinusoidal solutions. The boundary conditions $\varphi (0)=\varphi (a)=0$ select $k_{n}a=n\pi$; normalization then givesEach wavefunction vanishes outside the well. There is no nonzero $n=0$ state.

Immediate energy measurement after expansion. For the one-sided sudden expansion of an infinite square well, take the original wall at $x=0$ as fixed, so the expanded well occupies $0<x<2a$. In the sudden approximation, the initial wavefunction remains $\sqrt{2/a}\sin (\pi x/a)$ on $0<x<a$, extended by zero elsewhere. The new ground state is ${\chi }_1=a^{-1/2}\sin (\pi x/(2a))$. Its probability amplitude isThe Born rule therefore givesThe motion of the walls matters: a symmetric expansion about the old midpoint would instead give $64/(9{\pi }^2)$. The stated width alone does not explicitly identify the final endpoints. The first result uses the natural continuation of the specified left boundary; if both boundaries are moved, the final location must be supplied. For a new well $(b,b+2a)$ containing $(0,a)$, the overlap is $c_1(b)=(4\sqrt{2}/(3\pi ))[\cos (\pi b/(2a))-\sin (\pi b/(2a))]$.

Evolution of one energy eigenstate. For a time-independent Hamiltonian operator, write $|\psi (t)⟩=f(t)|{\varphi }_n⟩$. Substitution into the Time-dependent Schrödinger equation gives $i\hslash \dot{f}=E_{n}f$, soOnly a global phase changes, leaving all measurement probabilities unchanged.

Delayed energy measurement. Expand the unchanged initial state in the complete new energy eigenbasis,For the fixed-left-wall geometry, direct overlap integration givesorthogonality isolates the new ground-state amplitude as $c_1{e}^{-i{E}_1^{\prime }t/\hslash }$. Its phase cancels on squaring, so the ground-state probability remains $32/(9{\pi }^2)$ for every delay, provided the expanded well remains fixed and no intervening measurement or environment changes the state. The position probability density generally evolves because different eigenstates acquire different relative phases.

For removal of a half-oscillator wall, the sudden approximation keeps the state unchanged during the change of potential. Its probability in the new full-oscillator ground state ${\Psi }_0={\pi }^{-1/4}{\ell }^{-1/2}{e}^{-x^2/(2{\ell }^2)}$ is the squared overlap:where ${\int }_0^{\infty }x{e}^{-x^2/{\ell }^2}dx={\ell }^2/2$. ThereforeThe integral runs only over the initially occupied half-line. The initial state is not the odd full-line eigenstate: it is zero, rather than negative, on the left. Indeed its overlap with that final first excited state is $1/\sqrt{2}$, giving probability $1/2$, and the remaining probability lies in higher states. Subsequent evolution under the new fixed Hamiltonian operator changes the quantum phases of its energy components without changing these energy-measurement probabilities.

The nuclear beta decay is fast compared with the electronic orbital timescale. In the sudden approximation, the electronic wavefunction does not have time to change during the increase from $Z_i=1$ to $Z_f=2$. The subsequent probability of a particular daughter energy eigenstate is therefore its squared overlap with the initial state.

For the final $1s$ state, write ${\beta }_i=1/a_0$ and ${\beta }_f=2/a_0$. The sudden hydrogenic charge change gives the amplitudeThusThe final $2p$ states have spherical harmonics $Y_{1m}$, whereas the initial state is proportional to $Y_{00}$. Their angular overlap $\int Y_{1m}^{\ast }{Y}_{00}d\Omega$ vanishes by orthogonality. Accordingly $\boxed{P_{2p}=0}$ in the ideal spherically symmetric sudden-change model. Nuclear recoil or interactions with the emitted particles can produce small corrections. This is an overlap selection rule for a sudden change, not a photon-induced electric dipole transition.

In the sudden approximation, the nuclear charge changes before the electronic wavefunction has time to evolve. Immediately afterward the state is still the original hydrogenic 1s wavefunction, now expanded in the eigenstates of the new Coulomb Hamiltonian. The final-ground-state probability is the squared overlap, rather than one minus an energy change.

For initial and final charges $Z_1,Z_2$, the normalized 1s overlap isUsing $Z_1=1,Z_2=2$ givesThe remaining probability includes higher bound $ns$ states and the $s$-wave ionization continuum. In this central, instantaneous model the initial orbital has $l=0,m=0$, and angular orthogonality excludes states with $l\ne 0$. The electronic spin is unchanged. Nuclear recoil and interactions with the emitted decay particles are outside this leading sudden-charge-change approximation.

Noninteracting spectral line. Treat $\mathbf{k}$ as a wavevector, so physical crystal momentum is $\hslash \mathbf{k}$. Write ${\Omega }_{\mathbf{k}}={\epsilon }_{\mathbf{k}}/\hslash$; 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 $\tilde{\psi }(\omega )=\int dt{e}^{i\omega t}\psi (t)$,Thus its normalized frequency spectral function isOne 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 $(\omega -{\Omega }_{\mathbf{k}}+i{0}^{+}{)}^{-1}$, whose negative imaginary part divided by $\pi$ is the same delta peak.

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 givesPerforming the causal integral,The corresponding Lorentzian quasiparticle spectral peak isIts 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 $\hslash {\Gamma }_{\mathbf{k}}$.

A general interacting electron is more accurately described by $G^R=Z_{\mathbf{k}}/(\omega -{\Omega }_{\mathbf{k}}^{\ast }+i{\Gamma }_{\mathbf{k}})+G_{\mathrm{inc}}^R$ 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,Linearizing the real denominator gives $Z=[1-{\hslash }^{-1}{\partial }_{\omega }\mathrm{Re}{\Sigma }^R{]}^{-1}$ and $\Gamma =-Z\mathrm{Im}{\Sigma }^R/\hslash$ at the pole. The requested unit-weight Lorentzian is the normalized quasiparticle model, not the most general exact interacting spectrum.

Peak position and width. At fixed $\mathbf{k}$, put $\gamma =\gamma (\mathbf{k})>0$ and $\Omega =\hslash k^2/(2m^{\ast })$. Differentiation givesIt 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 $(\omega -\Omega {)}^2={\gamma }^2$. ThereforeIn energy units the maximum is ${\hslash }^2{k}^2/(2m^{\ast })$ and the half-width is $\hslash \gamma (\mathbf{k})$. Frequency dependence of the self-energy would require reconsidering both the shape and peak position.

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 $(\epsilon -\mu {)}^2+(\pi k_{B}T{)}^2$. 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.

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 $E_B\ge 0$ below the Fermi level and the appropriate calibrated work function $\Phi$, conservation of energy givesConservation of momentum parallel to a periodic surface giveswhere ${\mathbf{G}}_{\parallel }$ 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 byHere $E$ 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.

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 $E_D\sim 1.4$–$1.5\mathrm{eV}$. A velocity follows from a branch slope independently of the energy zero:For example, the lower branch moves by about $0.3\mathrm{eV}$ between $|k_y|\sim 0.03$ and $0.11{\mathring{\mathrm{A}}}^{-1}$. This gives $dE/dk\sim 3.8\mathrm{eV}\mathring{\mathrm{A}}$ andReading 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 $E_F\simeq 1.9\mathrm{eV}$ on that axis, or $E_F-E_D\sim 0.4\mathrm{eV}$ 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 $E_F-E_D\simeq 0.4\mathrm{eV}$ for the substrate-doped sample. That resolves the relative energy when this external calibration is admitted.^[ref] 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.

Energy and momentum measurement. In angle-resolved photoemission spectroscopy, monochromatic photons illuminate a clean surface in vacuum, and an electron energy analyzer measures the outgoing kinetic energies and emission directions. Measuring both gives the occupied electronic band structure, rather than only a momentum-integrated distribution of removal energies.

Choose a common energy reference for initial and final states. Let $E_i$ be the initial electron energy, $E_F$ the Fermi energy, and $E_{\mathrm{vac}}=E_F+\varphi$ the vacuum level immediately outside the sample. The binding energy is $E_B=E_F-E_i\ge 0$. For a direct emission event without additional losses, conservation of energy givesHere the work function $\varphi$ is the energy from the Fermi level to the local vacuum. The initial and final energies are total single-electron energies on the same scale, while $E_{\mathrm{kin}}$ is measured above vacuum. The diagram illustrates each interval independently.

Neglecting ultraviolet photon momentum, photoemission momentum conversion giveswhere $\theta$ is measured from the surface normal. Parallel crystal momentum is conserved modulo a surface reciprocal lattice vector; the surface breaks perpendicular translation symmetry. A three-dimensional sample therefore generally needs a final-state model to infer perpendicular momentum. The stated two-dimensional sheet avoids that ambiguity. The kinetic energy used in this formula is the sample-vacuum value; if the analyzer and sample have different work functions, their contact potential must be included when converting the analyzer's reading, as in photoelectron binding energy reference.

Spectral information. Define $ϵ=E_i-E_F=-E_B$ and the electron spectral function by $A(\mathbf{k},ϵ)=-\mathrm{Im}{G}^R(\mathbf{k},ϵ)/\pi$. In the sudden approximation, the basic photoemission intensity and spectral function relation isbefore instrumental broadening and background are included. Here $f$ is the Fermi-Dirac distribution, and $\mathcal{M}$ is the transition matrix element. The electron removal spectral function contains the occupied part of the spectrum. In a noninteracting band it has a delta peak at the band energy; interactions shift and broaden peaks and can add incoherent weight. Thus peak trajectories reveal dispersion, while line shapes can constrain lifetimes after resolution effects are accounted for. Dividing by known matrix and occupation factors can estimate $A$ where those factors are appreciable; intensity alone is not identical to $A$, and a missing peak can reflect a matrix-element suppression. States well above $E_F$ are not accessible to ordinary low-temperature photoemission.

Two independent band-structure probes. First, the de Haas-van Alphen effect measures oscillatory magnetization as Landau levels cross the Fermi level when a magnetic field changes. The inverse-field frequency iswhere $S_{\mathrm{ext}}$ is an extremal Fermi surface cross-sectional area perpendicular to the field. Changing field orientation maps its shape. This reveals the momentum-space geometry of occupied bands; the related Shubnikov-de Haas effect measures oscillatory resistance.

Second, absorption or reflectance spectroscopy detects interband transitions. Photon energies at absorption edges and peaks constrain the band gap and interband energy differences. For a direct band gap, negligible photon momentum selects approximately vertical transitions in the band diagram, weighted by the joint density of states and transition matrix elements. An indirect transition needs a phonon to carry the extra crystal momentum. These measurements provide evidence beyond a free-electron parabola, although an optical spectrum alone does not reconstruct every band's dispersion.

Reading the graphite spectra. For each direction, follow a moving peak through successive angles and convert each peak's own kinetic energy to $k_{\parallel }$; angle is not itself proportional to momentum over this wide range. The prominent dispersing valence feature starts at roughly $7$–$9\mathrm{eV}$ binding energy near $\Gamma$ and moves towards lower binding energy as $k_{\parallel }$ increases. Along $\Gamma K$ it approaches the Fermi level at the corner, then turns back to larger binding energy on continuing beyond the corner. Along $\Gamma M$ it reaches a finite minimum binding energy of roughly $3$–$4\mathrm{eV}$ near the edge midpoint and turns back; it does not reach the Fermi level there. Additional deeper peaks describe other occupied valence branches, with visibility varying across the spectra.

A useful qualitative description is the upper occupied $\pi$ band of the nearest-neighbour model of graphene: $E_{\pi }=-t|f(\mathbf{k})|$. It rises from $-3t$ at $\Gamma$ to zero at $K$, but only to $-t$ at $M$. The schematic curve in the figure captures those two different behaviours. It is not a full multi-orbital assignment or a quantitative fit to graphite, whose deeper valence features are also visible in the supplied spectra.

Angle needed to reach the corner. The corner's parallel wavevector isUse the near-zero binding energy of the upper valence band at that point. With the sample work function interpreted as the stated $3.8\mathrm{eV}$ separation below vacuum,HenceThe energy and angle readings in the PDF are approximate. If a peak at the corner were instead assigned a small binding energy $E_B$, the estimate becomesFor $E_B=0$–$1\mathrm{eV}$ this gives about $57$–$60^{\circ }$, consistent with the near-Fermi feature in that angular range. Substituting the full photon energy for the kinetic energy would omit the work-function cost and underestimate the required angle.

Complete the square, keeping the equilibrium displacement signed:Write $X=x-x_0$ for position relative to the new minimum. By the sudden approximation, the state immediately after the quantum quench is still the old vacuum. Its initial means areThe old ground state has $⟨H_0⟩=\hslash \omega /2$ and $⟨x⟩=0$. Thus $⟨H⟩=\hslash \omega /2$, but the energy measured above the new potential minimum isThe extra term is the excitation energy above the new ground state; the zero-point term remains part of the energy above the potential minimum.

The variables $X,p$ obey the oscillator equations in part (a). Using the above initial means givesThe wavepacket's centre therefore oscillates about the new equilibrium, beginning at the old origin with zero mean momentum.

Within the dipole and sudden approximation, angle-resolved photoemission spectroscopy measures the electron removal spectral function multiplied by a transition matrix element. Finite instrumental energy and angular resolution convolve this product, and inelastic scattering adds a background. Peak positions reveal occupied electronic band structure, while intrinsic line widths contain lifetime information. Matrix elements depend on photon energy, polarization, orbital character, and geometry, so weak intensity does not by itself establish the absence of an electronic state. Equilibrium photoemission does not directly measure unoccupied bands far above the Fermi level.

For a fixed Hilbert space, a bounded self-adjoint Hamiltonian operator with a finite step $H_i\to H_f$ at $t=0$ does not deliver an instantaneous rotation of the quantum state. Integrating the Schrödinger equation across a short interval givesFor a normalized quantum state and $\Vert H(t)\Vert \le M$ in operator norm, the difference has norm at most $2ϵM/\hslash$, which tends to zero. The time derivative can jump even though the quantum state is continuous. Here $\hslash$ is the reduced Planck constant. An instantaneous replacement of the eigenbasis changes coordinates, not the vector being represented.

For a real switch occupying $0\le t\le \tau$, remove any real scalar contribution $a(t)I$ that produces only a global phase, and write $|\tilde{\psi }(t)⟩=e^{i{\int }_0^{t}a(s)ds/\hslash }|\psi (t)⟩$. The same integral argument gives the sufficient bounded-model estimateThus $\epsilon \ll 1$ controls the sudden approximation up to a global phase. For any normalized final energy eigenstate, its probability amplitude differs from the instantaneous-switch value by at most $\epsilon$, and its Born rule probability differs by at most $2\epsilon$. These are sufficient error bounds, not necessary criteria. For the fixed-gap two-level sudden quench, any switching path $H(t)=-\Delta \mathbf{n}(t)\cdot \mathbf{\sigma }/2$ with a unit vector $\mathbf{n}(t)$ has operator norm $\Delta /2$, giving $\epsilon \le \Delta \tau /(2\hslash )$.

This estimate is deliberately for bounded Hamiltonian operators. An unbounded energy spectrum does not supply a finite global $M$; relevant quantum states, operator domains and excited energy eigenvalues require separate control. The adiabatic theorem also requires hypotheses about the path and its spectral gaps; there is no universal switch-duration criterion covering all spectra. The cold-atom collapse and revival after a lattice quench illustrates how a single switch can be sudden for one degree of freedom and slow for another.

Suddenly remove the wall of a half-harmonic oscillator in its ground state, leaving the same harmonic potential on the whole line. The sudden approximation keeps its wavefunction unchanged during removal. The new normalized ground state is ${\psi }_0={\pi }^{-1/4}{\ell }^{-1/2}{e}^{-x^2/(2{\ell }^2)}$, soThe integration is over the initially occupied half-line. The final first excited state instead has probability $1/2$, since the initial half-line state equals $\sqrt{2}$ times its positive-half restriction. This also checks that removing the wall does not simply create the full oscillator's odd state.

The sudden approximation treats a rapid, finite change of a Hamiltonian operator as leaving the quantum state approximately unchanged during the switch. The energy measurement can nevertheless change immediately: the same quantum state has different probability amplitudes in the new eigenbasis. Compute their inner products with the new energy eigenstates, then apply the Born rule. The two-level sudden quench below makes this distinction quantitative without assuming the change is small.

An ideal finite step has exact quantum state continuity at a finite Hamiltonian step; a finite-duration implementation has a small evolution error under the bounded-model conditions below. An impulsive quantum kick is different: its Hamiltonian operator grows as its duration shrinks, retaining a finite integrated action and a finite change of quantum state. A sudden approximation therefore concerns short evolution during switching, not conservation of energy during externally driven change. The subsequent unitary time evolution can be substantial.^[ref]

In the sudden approximation, changing the nuclear charge of a one-electron hydrogen atom from $Z_{i}e$ to $Z_{f}e$ leaves its electronic state unchanged during the change. Its subsequent probability to occupy a final state is the squared overlap with that state. Using normalized $1s$ wavefunctions ${\psi }_Z=({\beta }_Z^3/\pi {)}^{1/2}{e}^{-{\beta }_{Z}r}$, ${\beta }_Z=Z/a_0$, givesThis follows by integrating $4({\beta }_i{\beta }_f{)}^{3/2}{r}^2{e}^{-({\beta }_i+{\beta }_f)r}$ over positive $r$. A spherically symmetric initial state has zero overlap with every final state of nonzero orbital angular momentum, by the orthogonality of the spherical harmonics. Nuclear recoil and interactions with emitted particles can invalidate the perfectly spherical sudden-change model.