Advanced Quantum Mechanics

For the ground multiplets of Ce3+ (4f¹, J = 5/2) and Pr3+ (4f², J = 4), does every crystal-field Hamiltonian that commutes with Θ leave exact doublets for J = 5/2 and, at a site of no symmetry, generically only singlets for J = 4, and do those singlets carry zero magnetic moment, so that a weak magnetic field moves them quadratically while it splits each doublet linearly?

Theta built as exp(−i pi J_y/ℏ) K in each multiplet and checked to square to −1 and +1; a few hundred crystal fields per multiplet drawn at random by symmetrising Hermitian matrices under Theta; the splitting within each J = 5/2 doublet at the level of rounding error, and no residual degeneracy for J = 4; ⟨J⟩ in every J = 4 singlet zero to rounding; Zeeman shifts plotted against B on log-log axes with fitted slopes near 1 for the doublets and 2 for the singlets; and a random T-odd term shown to split every doublet while added T-even terms of any size never do, so that the pairing is traced to Θ² = −1 and not to the form of the field.

Does the rotation of linear polarisation cancel on a double pass through a sucrose solution and double on a double pass through a glass rod inside a solenoid, as reciprocity, the classical-field counterpart of topic 1.09, predicts for a medium even under T and fails to predict once a magnetic field, odd under T, acts?

Rotation angles read with a polariser and analyser, the double pass made with a mirror and a non-polarising beam splitter and the empty path measured as a control for the splitter's own polarisation effects; single-pass rotation against concentration times path length for the solution and against coil current, with the field calibrated by a Hall probe, for the rod; the fitted specific rotation and Verdet constant with uncertainties; double-to-single-pass ratios compared with the predicted 0 and 2; the sign change of the Faraday rotation when the current reverses; and a statement of why the chiral solution responds as a pseudoscalar, odd under P but even under T, without any law of physics violating parity.

For neutrons on a spin-zero nucleus described by a Woods-Saxon central potential with a Thomas-form spin-orbit term of realistic strength, at what energy and angle is the scattered beam most strongly polarised, and does P vanish, as this unit predicts, when only s waves contribute, when the spin-orbit term is switched off, and in the first Born approximation?

Radial equations integrated by Numerov's method for each l and j = l plus or minus 1/2 up to an lₘₐₓ of about kR + 5, with phase shifts converged in step size, matching radius and lₘₐₓ; g, h, d σ/d Omega and P(θ) at several energies; a map of P against energy and angle with its maximum located; the three null checks carried out; the elastic optical theorem σ = (4 π/k) Im g(0) verified to better than 0.1 per cent; and, with an imaginary central term added, the reaction cross-section computed and the optical theorem with absorption checked.

Near 90 degrees, how fast does the visibility of Mott's interference fall as eta grows for a detector of fixed angular resolution, does it follow the exp(−2 η² s²) suppression that a Gaussian acceptance of width s radians predicts, and does the averaged cross-section then approach the classical sum of the two Rutherford terms for both spinless bosons and unpolarised spin-half fermions?

Mott cross-sections divided by the classical sum, computed for spin-zero and unpolarised spin-half particles from 30 to 150 degrees for eta from 0.1 to 100; the oscillation period near 90 degrees fitted and compared with π/η; the exact 90-degree ratios of 2 and 1/2, independent of η, checked; the cross-sections convolved with Gaussian acceptances of 0.2 to 2 degrees and the fringe visibility plotted on a log scale against eta s with the predicted exponent tested; and eta computed for α-α, carbon-12 on carbon-12 and proton-proton at stated energies to say which experiments could resolve the fringes.

Does a numerical solution of the Klein-Gordon radial equation for a spinless particle of the electron's mass reproduce the exact point-Coulomb 1s, 2s and 2p energies at small Z, and as Z alpha passes 1/2, where the point-nucleus 1s formula turns complex, does a uniformly charged nucleus of radius 1.2 A¹⁄³ fm give a real 1s energy, and how strongly does that energy then depend on the radius?

A shooting or finite-difference radial solver that handles the quadratic dependence on E by iterating on the energy, checked against the exact formula to at least six figures for Z from 1 to 60 with the nuclear radius taken to zero; the finite-nucleus 1s energy for Z up to 92 beside the point formula; the fractional change of the 1s binding when the radius is doubled, plotted against Z and compared below and above Z α = 1/2; and the computed wavefunction near the nucleus on either side of that value.

If a one-dimensional Gaussian packet of rms width w is started with d φ/dt = −i(mc²/ℏ) φ, as the non-relativistic picture suggests, what fraction of its conserved charge sits in negative frequencies, does that fraction fall as (ℏ/mcw)⁴ for wide packets with the coefficient an expansion by hand predicts, and below what width does it exceed 1 per cent?

Each Fourier mode split into positive- and negative-frequency amplitudes from phi and d φ/dt at t = 0; the negative-frequency fraction of the charge against w/(ℏ/mc) on log-log axes with fitted slope and coefficient set beside the hand expansion; the density evolved exactly mode by mode, showing the interference at about 2mc²/ℏ that Unit 8 meets as Zitterbewegung; and the most negative local density reached by a packet built from positive frequencies alone as w shrinks.

Do eigenvectors of the 4 x 4 free Dirac Hamiltonian, diagonalised numerically for speeds from 0.01c to 0.999c, reproduce the analytic u and v spinors, their normalisation and the spin sums p-slash plus or minus mc, and at what speed does the ratio of small to large components first exceed the low-speed estimate v/2c by 10 per cent, and at what speed does it reach one half?

Eigenvalues against |p| showing two doubly degenerate branches at plus and minus √(p² c² + m² c⁴), a comparison made through the projectors onto each degenerate pair rather than through individual eigenvectors, which a solver may mix arbitrarily within a pair, v(p, s) read from the negative-energy eigenvectors of H at momentum −p, since those of H at p are v(−p, s) and give u-bar v nonzero, residuals of u-bar u = 2mc, u-bar v = 0 and both spin sums at machine precision, and the numerical component ratio plotted against v/c beside v/2c and the closed form γ v/((γ + 1) c), with the two crossing speeds read off and checked against that closed form.

Can a numerical minimiser find four Hermitian N x N matrices that square to the identity and anticommute in pairs for N = 2, 3, 4, 5, 6 and 8, does it succeed only where topic 4.02's trace and Pauli-counting arguments allow, and is N = 6, which those arguments do not exclude, also impossible?

The residual of the algebra minimised from many random starts for each N, the smallest final residual plotted against N with zero reached only at N = 4 and 8, the nonzero floor at N = 6 explained by the quoted result that every representation is a direct sum of four-dimensional ones, a solution found at N = 4 carried into the Dirac representation by a similarity matrix built by summing over the sixteen products of gammas and checked, once normalised, to be unitary, and topic 4.03's trace identities evaluated in both representations to machine precision.

Does an explicitly constructed S(Λ) satisfy the covariance condition to machine precision for random rotations and boosts, does the product of the S matrices for two transformations equal the S of their product only up to a sign, and for two perpendicular boosts does the leftover rotation match the closed-form Wigner angle and approach v₁ v₂/(2c²) at low speed?

Gamma matrices built in the Dirac and Weyl representations and checked against the Clifford algebra, residuals of S⁻¹ γ^μ S - Λ^μ_ν γ^ν for a few thousand random transformations, the sign of S(Λ₁) S(Λ₂) S(Λ₁ Λ₂)⁻¹, each S built from the principal rotation angle and rapidity of its Λ, recorded for random pairs and for rotations whose angles sum past 2 π, the rotation extracted from a composed boost by polar decomposition with its angle plotted against v₁ v₂/c² and the small-velocity slope fitted, and boosted rest spinors compared component by component with u(p, s) and v(p, s).

Computed from explicit spinors, do the four helicity amplitudes for Coulomb scattering split into a non-flip part proportional to cos(θ/2) and a flip part proportional to (mc²/E) sin(θ/2), and does their spin-averaged sum reproduce the trace result 1 - β² sin²(θ/2) at every angle and energy?

Helicity spinors built from Unit 4's u(p, s) for the incoming and outgoing directions, the matrix elements u-dagger(p', h') u(p, h) tabulated for theta from 0 to 180 degrees and E/mc² from 1 to 100, agreement with the trace formula to rounding error, the flip contribution relative to Rutherford at 90 and 180 degrees plotted against E/mc² on log-log axes with its slope of −2 fitted, and the Mott factor plotted against theta for several β, showing the backward zero as beta tends to 1.

For a Gaussian well much shallower than mc² in one dimension, where no spin-orbit term exists, does adding the kinetic correction and the Darwin term (ℏ²/(8m² c²)) V'' to the Schrodinger levels shrink the gap to the exact Dirac levels from order (v/c)² to order (v/c)⁴, and does a fitted Darwin coefficient land on 1/8 rather than the 1/4 that unrenormalised elimination gives once its non-Hermitian term is dropped?

Bound-state energies of H = c σₓ p + σ_z mc² + V(x) from diagonalisation in a plane-wave basis, which avoids the spurious doubled states of naive finite differences, with mc² subtracted on the diagonal before diagonalising; Schrodinger levels and first-order kinetic and Darwin shifts computed on the same basis; residuals for every bound state over at least a decade of c on a log-log plot with fitted slopes; the Darwin coefficient fitted from the order-(v/c)² residual with its uncertainty; the first-order shifts of the full unrenormalised operator, non-Hermitian term included, shown to equal the renormalised ones; and convergence checks in basis size and box length.

Does the product of many small boosts, each taken in the particle's current rest frame, carrying it once around a circular orbit converge to a pure boost times a retrograde rotation through 2 π(γ - 1), and above what speed does the low-speed rate (1/(2c²)) a x v fall more than 10 per cent short of the exact one?

4 x 4 Lorentz matrices composed step by step, the residual rotation extracted from the spatial block after removing the final pure boost, its convergence with the number of steps shown, the angle per orbit compared with 2 π(γ - 1) for beta from 0.01 to 0.99 including speeds where it exceeds a full turn, the 10 per cent speed located, and the defining relation Λ^T g Λ = g checked to rounding error at the end as a guard against accumulated error.

Does the K-α1-K-α2 splitting, the 2P_(3/2)-2P_(1/2) energy difference of an L-shell vacancy, follow the exact Dirac n = 2 doublet with Z replaced by Z - s and one screening constant s for every element, while the order-α⁴ formula needs a screening constant that drifts with Z?

Bragg angles of the K-α doublet from the anodes and fluorescence targets available, such as copper, molybdenum and silver, measured in the highest diffraction orders the crystal allows and converted to energies with its stated lattice spacing; each splitting with an uncertainty from the angular resolution; the screening constant solved from each splitting once with the exact Dirac energies and once with the α⁴ formula, both plotted against Z beside tabulated doublets of heavier elements up to uranium, quoted; and a statement of what screening by the other electrons leaves approximate in the comparison.

How large is the finite-nuclear-size shift of 1s, 2s and 2p_(1/2) in hydrogen-like ions from Z = 1 to 92, by what factor does it exceed Quantum Mechanics II's non-relativistic contact estimate scaled to uranium, and at what Z does a uniformly charged nucleus bring the 1s level down to −mc²?

A shooting or finite-difference solver for topic 7.04's equations, written for the binding energy E - mc² rather than E, validated against the exact point-Coulomb energies to one part in 10⁹ at Z = 1, 50 and 92 and then run with a uniformly charged sphere of radius 1.2 A¹⁄³ fm; at low Z, where the shift falls below that precision (hydrogen's 1s shift is about 3 parts in 10¹⁰ of its binding), the shift computed instead as the first-order expectation value of the sphere's potential minus the point potential in the exact Dirac states, the two methods checked against each other over the range of Z where both resolve it; the three shifts plotted against Z beside the non-relativistic estimate, which grows as Z⁴ times the square of the nuclear radius and misses the r^{(2 γ - 2)} enhancement; the uranium 1s shift set against the quoted 199 eV; the 2p_(1/2) shift, nonzero through its s-wave small component, shown growing relative to the 2s shift; and the Z at which 1s reaches −mc², with the extrapolation A = 2.5Z stated as an assumption, set against the quoted value near 172.

Do the photons from positron annihilation in matter each carry mc² and leave back to back, and does Compton polarimetry find their polarisations perpendicular, as the odd intrinsic parity of the electron-positron singlet derived in topic 8.07 requires?

Two scintillation detectors energy-calibrated on cesium-137's 662 keV line and sodium-22's own 1275 keV line, giving the annihilation-line centroid with its uncertainty against 511.0 keV; the coincidence rate against the angle between the detectors, peaked at 180 degrees with a width compared with the angle the detector faces subtend; and coincidences of photons Compton-scattered by two small scatterers with scattering planes perpendicular and parallel, their ratio compared with the prediction of the quoted Klein-Nishina cross-section once finite solid angles are folded in, a ratio above 1 indicating odd parity and below 1 even. The source is used under the institution's radiation-safety rules.

In a one-dimensional simulation, does a Dirac wave packet striking a step higher than E + mc² split with R below 1, as the group-velocity choice predicts, rather than above it, and when the step is spread over a linear ramp of width w does the transmitted fraction fall as exp(−π (mc²/V₀)(mc w/ℏ)), the uniform-field exponent behind the critical field?

A split-operator integration of the two-component Dirac equation with α = σₓ and β = σ_z, in units ℏ = m = c = 1, checked by norm conservation and by a free packet moving at c² p/E; reflected and transmitted norms for step heights spanning the transmitting, evanescent and Klein regimes, plotted against both stationary formulas; a check that in the Klein regime the transmitted part has E - V₀ below −mc² yet moves away from the step; the logarithm of the transmitted fraction against w, in units of ℏ/mc, for ramps several Compton wavelengths wide, its fitted slope compared with pi mc²/V₀; and a statement that the pair-creation rate itself lies outside any one-particle simulation.

For a 1+1-dimensional Klein-Gordon field put on a lattice, which breaks Lorentz invariance, how far does the field commutator leak outside the continuum light cone, does the leak vanish as the lattice spacing shrinks, and does the anticommutator that fermionic quantisation would give converge instead to a nonzero spacelike value?

Exact mode sums for N sites with periodic boundaries, in units ℏ = c = 1 at fixed mass m, giving heat maps of the commutator and the anticommutator over x and t with the continuum light cone drawn; the commutator at three fixed spacelike points as the spacing is halved repeatedly, with the fitted rate at which the leak falls; the spacelike anticommutator compared with its closed form (1/π) K₀(m s), s the invariant interval; a check that both continuum limits depend on x and t only through s, the commutator inside the cone also on the sign of t; and a check that the box is long enough for wrap-around not to reach the points tested.

Are the oxygen A-band absorption lines that Bose symmetry of spin-zero oxygen-16 nuclei forbids, those from even-N rotational levels of the ground state, absent from a laser absorption spectrum of air, and what upper limit does the measurement set on the fraction of molecules in the forbidden states?

A 763 nm diode laser tuned across allowed lines and the predicted position of a forbidden one, positions computed from quoted rotational constants and the frequency axis calibrated on neighbouring allowed lines or an etalon; the weak lines of the 16O18O isotopologue, whose nuclei are not identical and which has both N parities, identified so they are not mistaken for violations; the absorbance at the forbidden position compared with the noise floor and with the strength the line would have if even-N levels were populated, interpolated from neighbouring allowed lines with Boltzmann and line-strength factors; the resulting upper limit on the forbidden fraction; and the published laser-spectroscopy limits, from 1996 onwards, quoted for comparison.

Does a self-consistent radial Hartree-Fock code reach helium's −77.87 eV and −ε₁s = 24.98 eV, is the correlation energy, measured against quoted exact non-relativistic energies, nearly independent of Z from Li+ to Ne8+ while the total energy grows as Z², and does Hartree-Fock bind H-minus, which Quantum Mechanics II proved is bound?

Total and orbital energies converged in grid spacing, outer radius and iteration count until the fourth decimal in hartree stops changing; the virial ratio -⟨V⟩/⟨T⟩ = 2, which a converged Hartree-Fock solution satisfies, as an internal check; helium compared with the published Hartree-Fock limit, quoted; correlation energies for Z = 1 to 10 against quoted exact non-relativistic energies, plotted against Z; and the H-minus energy set against the −13.6 eV threshold, with a statement of why an energy above it proves nothing about binding.

Built from creation and annihilation matrices with Jordan-Wigner signs, does exact diagonalisation reproduce the dimer's (U - √(U² + 16t²))/2, and as U/t runs from 0 to 20, how do restricted and unrestricted Hartree-Fock energies and the exact singlet-triplet gap compare with 4t²/U on the dimer and on a four-site ring at half filling, taken with antiperiodic boundary conditions so that its non-interacting ground state is a closed shell?

c and c-dagger for every spin-orbital built as matrices on the full Fock space with all anticommutators checked to machine precision; the Hamiltonian assembled from them, N and S_z conservation verified, and each sector diagonalised; the dimer spectrum against the analytic formulas; restricted and unrestricted self-consistent energies, with the U at which the unrestricted solution first breaks spin symmetry, 2t for the dimer; the exact gap divided by 4t²/U plotted against U/t; and the correlation energy, exact minus restricted, plotted on the same axis.

Does the normalised coincidence rate g⁽²⁾(τ) of a helium-neon laser stay at 1 for every delay, while laser light scattered from a rotating ground-glass disc reaches 2 at zero delay and falls over a correlation time set by the disc's speed, and does either result require the quantised field of this unit?

Coincidence histograms from two single-photon detectors behind a polariser and a 50:50 beam splitter, normalised by the product of the singles rates and the bin width, for the laser and for the scattered light at three disc speeds; the fitted zero-delay value and correlation time with uncertainties, the correlation time set against the illuminated spot size on the disc divided by its surface speed, the loss of contrast when the collecting aperture spans several speckles or the polariser is removed, depolarisation by the ground glass lowering g⁽²⁾(0) towards 1.5, dead-time and afterpulsing checked on the flat laser trace, and a statement that a classical field of fluctuating intensity predicts g⁽²⁾(0) of at least 1, so only a value below 1, Unit 14's antibunching, would demand quantisation.

For a resonant two-level atom in a cavity mode prepared in a coherent state of mean photon number n-bar from 4 to 100, does the atomic inversion collapse after about √(2)/g and revive near 2 pi √(n-bar)/g, and does the revival vanish when the Poisson sum over discrete n is replaced by a classical Rabi drive averaged over a continuous spread of intensities with the same mean and variance?

The inversion computed as the Poisson-weighted sum of cos(2g √(n + 1) t), checked against direct integration of the Schrodinger equation in a truncated Fock space with the truncation shown not to matter; collapse and revival times fitted for each n-bar and plotted against it on log-log axes with their slopes; the same inversion for the classical drive with a continuous intensity distribution, showing whether it collapses and whether it ever revives; and the effect of a detuning of a few g on the revival amplitude.

For a two-level emitter coupled to N discrete modes whose coupling-weighted density rises as ω³ up to a cutoff, over what window of times does the exact one-excitation solution follow exp(−Γ t) within 1 per cent, how does that window shrink as Γ/ω₀ is raised towards 0.1, and does the fitted frequency shift match the principal-value sum and grow with the cutoff as topic 12.06 predicts?

Topic 12.05's equations integrated, or the (N + 1) x (N + 1) matrix diagonalised, with the mode spacing fine enough that the recurrence time lies beyond the run, checked by halving it; ln |cₑ|² against t with residuals from exp(−Γ t), the initial curvature compared with 1 - (Δ H)² t²/ℏ², and any late-time departure plotted on log-log axes against the power the spectrum's lower edge predicts; the fitted shift against the principal-value integral for three cutoffs; and the final photon spectrum fitted to a Lorentzian whose width and centre are compared with Gamma and ω₀', its total probability checked against 1.

Does the radiative rate that the Einstein relation of topic 12.03, in Strickler and Berg's form for a broad molecular band, predicts from rhodamine 6G's measured absorption spectrum in ethanol agree within combined uncertainty with the rate obtained from its lifetime measured by time-correlated single-photon counting and its fluorescence quantum yield?

Absorbance spectra at three or more concentrations inside the linear Beer-Lambert range, converted to molar extinction with path-length and concentration uncertainties propagated; the emission spectrum supplying the average of ν⁻³ over the fluorescence band and the solvent's refractive index entering as n²; decay curves from a pulsed diode laser with the instrument response recorded on a scatterer and deconvolved, fitted to one exponential with residuals shown; the quantum yield quoted or measured against a standard; and predicted and measured radiative rates compared, with the band formula's assumptions, mirror-image absorption and emission from one electronic transition with no change of geometry, stated and checked against the spectra.

Does a sum over hydrogen's bound and continuum dipole matrix elements reproduce ln(K₀/Ry) = 2.81 for 2s and 2.98 for 1s, what fraction of each weight lies above the ionisation threshold, and how large must the basis be before the logarithm settles to 0.01?

p-wave pseudostates from diagonalising the radial Hamiltonian in an even-tempered Slater or Laguerre basis whose exponents span several decades, the Thomas-Reiche-Kuhn sum and the sum-rule denominator 2/n³ in atomic units reproduced to a stated precision, length and velocity forms of pₘₙ compared as a test of the basis, ln(K₀/Ry) for 1s and 2s tabulated against basis size with an extrapolation, the continuum share of the weights, and the resulting 1s and 2s shifts with the cutoff at mc² set beside 8.17 GHz and 1057.8 MHz together with the 9 per cent spread a factor of two in the cutoff produces.

Can saturated-absorption or polarisation spectroscopy of Balmer-α in a deuterium discharge separate the 2S_(1/2)-3P_(3/2) and 2P_(1/2)-3D_(3/2) components, which the Dirac theory places at one frequency, and does their separation agree with a 2s shift near 1.05 GHz and exclude zero?

A 656 nm external-cavity diode laser scanned across the line, its frequency axis calibrated by a confocal Fabry-Perot of known free spectral range and checked for scan linearity, the Doppler-free spectrum with crossover resonances identified from shared levels, fitted centres and widths of the two components as discharge pressure and pump power are varied, deuterium's 41 MHz 2S hyperfine splitting included in the line shape, and the fitted separation with its uncertainty compared with zero, with Bethe's 1040 MHz and with the quoted QED Lamb shift less the difference of the 3P_(3/2) and 3D_(3/2) shifts, a few MHz, also quoted.

For protons in glycerine, mineral oil and water doped with copper sulphate, do inversion recovery and the Hahn echo show exponential relaxation with T₂ at most 2T₁, and how much of the free-induction decay time T₂* is reversible dephasing in the magnet's inhomogeneous field, which an echo undoes, rather than decoherence, which it cannot undo?

Inversion-recovery magnetisations fitted to M₀(1 - 2 exp(−t/T₁)), with the null near T₁ ln 2 as a check; Hahn-echo heights against 2 tau fitted to exp(−2 τ/T₂), repeated with a Carr-Purcell-Meiboom-Gill train to remove the extra decay that diffusion through the field gradient adds at long tau; T₂* from the free-induction decay set beside T₂; both relaxation rates for four or more copper sulphate concentrations plotted against concentration with their intercepts; and every pair of times checked against T₂ at most 2T₁ within its uncertainty.

Does the average of N quantum-jump trajectories of a resonantly driven atom converge to the optical Bloch solution with an error falling as 1/√(N), and do the simulated detection times alone reproduce g⁽²⁾(τ), including g⁽²⁾(0) = 0 and its change from monotonic rise to damped oscillation as Ω passes Γ/4?

Trajectories generated by non-Hermitian evolution with random jumps for Ω/Γ from 0.1 to 10; the averaged excited population set against direct integration of the master equation, with the root-mean-square error plotted against N on log-log axes and its slope fitted; the histogram of all pairs of detection-time separations, normalised by the squared mean rate, set beside g⁽²⁾(τ) from the quantum regression theorem; the waiting-time distribution of consecutive detections shown to agree with it only for tau short against the mean waiting time; and a check that halving the time step changes nothing beyond the statistical error.

Computed from discretised momentum sums, does the exchange energy of jellium converge to −0.916/rₛ Ry, does the Hartree-Fock slope of εₖ at k_F grow without limit as the grid is refined, and below what screening wavenumber kₛ, as a fraction of k_F, does the effective mass at k_F at sodium's rₛ of about 4 depart from the free mass by more than 10 per cent?

The exchange energy from grid or Monte Carlo sums over pairs in the Fermi sphere at several resolutions, extrapolated and compared with −0.916/rₛ Ry; εₖ on a fine grid checked against the closed form in F(k/k_F); the finite-difference slope at k_F against grid spacing on a logarithmic axis, growing linearly in ln(1/δ k); the same calculation with a Yukawa-screened v_q for kₛ/k_F from 0.01 to 3, the Thomas-Fermi value of about 1.6 included, giving m*/m and the implied specific-heat coefficient, with the kₛ at which the 10 per cent threshold is crossed; and a statement that a fixed kₛ is a model, the random-phase approximation that would justify it, with dynamic screening, being named only.

For N bosons on a ring of L sites with hopping t and on-site repulsion U, does the single-quasiparticle state that exact diagonalisation finds in each total-momentum sector follow the Bogoliubov dispersion Eₖ = √(eₖ(eₖ + 2Un)) with eₖ = 2t(1 - cos k), is the depletion of k = 0 the predicted one, and at what Un/t does the lowest excitation at k = 2 π/L miss the Bogoliubov value by 10 per cent?

The Hamiltonian built on occupation-number states of the momentum modes and blocked by total momentum, for N = L up to 10 (92 378 states at N = L = 10, about a tenth of that per sector); in each sector the eigenstate carrying the largest weight of the density fluctuation ρ-daggerₖ = sum over p of a-dagger_(p+k) aₚ acting on the ground state, its energy above the ground state plotted against k beside the Bogoliubov curve, with the lowest state of each sector beyond k = 2 π/L, in which several quasiparticles of smaller momentum share k at lower energy, shown separately and not mistaken for the Bogoliubov branch; the condensate fraction ⟨n₀⟩/N from the ground state against the lattice sum of vₖ²; the fractional error of the k = 2 π/L excitation against Un/t on a logarithmic axis with its 10 per cent crossing; a check that at small U that excitation tends to eₖ + U(N - 1)/L, first-order perturbation theory, which is Bogoliubov's eₖ + Un up to a 1/N correction; and a note that a finite ring has a condensate although an infinite one-dimensional gas has none.