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Let $x$ be signed displacement along the rails, measured from the centre. The diagram in the original paper shows a guide extending across the centre, so the signed coordinate can pass through zero. Hooke's law gives the radial force $-kx$. In the rotating reference frame, the radial centrifugal force is $+m{\Omega }^{2}x$. The Coriolis force is perpendicular to the rails and is balanced by their reaction; it has no radial component. Therefore Newton's second law givesEquivalently, the radial component of acceleration in polar coordinates in the inertial frame is $\ddot{x}-{\Omega }^{2}x$. The radial spring oscillator on a rotating guide has simple harmonic motion only when $k/m-{\Omega }^2>0$, withThe angular frequency is in $\mathrm{rad}{\mathrm{s}}^{-1}$ and the frequency in $\mathrm{Hz}$. The critical angular speed isStrictly this is an upper bound, not an attained maximum for oscillations: at equality $\ddot{x}=0$, so a displaced object released from rest stays at its displaced position relative to the rails; above it, the central equilibrium is unstable.

A normal mode is a motion in which all generalized coordinates oscillate at one frequency with fixed amplitude ratios. Small oscillations about stable equilibrium in a time-independent conservative system give $\mathsf{M}\ddot{\mathbf{\theta }}+\mathsf{K}\mathbf{\theta }=0$, with positive mass matrix and symmetric stiffness matrix, allowing a generalized normal-mode eigenproblem.

For the three-to-one mass double-pendulum modes, the angles are absolute displacements from vertical, as printed. To quadratic order,For $\mathbf{\theta }=\mathbf{A}\cos \omega t$ and $q={\omega }^2\ell /g$, the determinant isThusThe low mode has the strings leaning together, with the lower angle twice as large; the high mode has opposite angular signs. Both reverse together after half their respective periods.

For the specified release, decompose $(\alpha ,\alpha )=(3\alpha /4)(1,2)+(\alpha /4)(1,-2)$. Thereforeand the lower mass's horizontal displacement isThis is the supremum of two incommensurate harmonic modes, conventionally described as the maximum-amplitude estimate. Strictly, ${\omega }_{+}/{\omega }_{-}=\sqrt{3}$ is irrational, so the opposite cosine extrema are approached arbitrarily closely but never coincide at a finite time. The quoted value is a supremum of the ideal linearized motion.

Different measurements reveal different aspects of electronic band structure; a convincing determination combines their constraints.

Angle-resolved photoemission. In angle-resolved photoemission spectroscopy, photons of known energy eject occupied electrons. Measuring their kinetic energy and emission angle reconstructs binding energy and the component of crystal momentum parallel to the surface:Here $\Phi$ is the sample work function when kinetic energy is referenced to the nearby vacuum; an analyzer energy scale requires its own work-function calibration. Momentum is conserved modulo a surface reciprocal lattice vector, neglecting photon momentum. Tracking the peaks maps occupied dispersion relations and the Fermi surface. The measured intensity also contains the matrix element, the Fermi-Dirac distribution and the spectral function, so an absent bright feature does not establish an absent band. For a three-dimensional solid, the perpendicular momentum requires a final-state model. Inverse photoemission spectroscopy probes unoccupied states by injecting electrons and detecting emitted photons.

Optical spectroscopy. Photon absorption and ellipsometry measure interband transitions, optical selection rules and the complex dielectric response. Because optical photon momentum is small on the scale of the Brillouin zone, direct transitions approximately preserve crystal momentum. Their onset constrains a direct band gap; a threshold involving a phonon can reveal an indirect band gap. Peak positions constrain energy differences between bands, but the response also depends on the joint density of states and optical matrix elements, so it does not uniquely determine every $E_n(\mathbf{k})$.

Quantum oscillations and cyclotron resonance. In clean samples at low temperature, the de Haas-van Alphen effect measures oscillations in magnetization and the Shubnikov-de Haas effect measures oscillations in electrical resistivity as the inverse magnetic field is varied. Their period in inverse magnetic field is $1/F$, where Onsager quantization relates $F$ to an extremal Fermi-surface area perpendicular to the field:Rotating the magnetic field maps the geometry of extremal orbits. The thermal damping determines the cyclotron effective mass,with the appropriate magnitude for a hole orbit. Cyclotron resonance supplies the corresponding dynamical mass through ${\omega }_c=eB/m_c$. These are detailed probes near the Fermi level, rather than measurements of all occupied bands. Diffraction supplies the reciprocal lattice needed to label the measured momenta, but diffraction by itself is not a measurement of electronic dispersion.

Use a representative conduction-electron density $n\sim 10^{29}{\mathrm{m}}^{-3}$, roughly one mobile electron per atom in a dense metal, and the bare electron mass for this estimate. The plasma frequency is thenThis is an order-of-magnitude answer because no particular metal or carrier density is specified. It corresponds to an ordinary frequency $f_p\simeq 2.8\times 10^{15}\mathrm{Hz}$, excitation energy $\hslash {\omega }_p\simeq 12\mathrm{eV}$, and free-space wavelength near $1.1\times 10^{-7}\mathrm{m}$.

The physical oscillation is collective motion of the electron gas against the positive ionic background. Displacing the electrons relative to the ions produces a restoring electric field. In the simplest uniform-displacement picture, $\ddot{x}+{\omega }_p^{2}x=0$: the frequency increases with the density of mobile electric charge and decreases with carrier inertia. Quantizing this longitudinal collective oscillation gives a plasmon.

The same scale controls the ideal collisionless Drude model optical response. Its cold collisionless plasma dielectric response is ${ϵ}_r(\omega )=1-{\omega }_p^2/{\omega }^2$. Below ${\omega }_p$, the negative relative permittivity makes a transverse electromagnetic wave evanescent, producing strong reflection; above it, the ideal model admits propagation. Real-metal collisions, interband photon absorption and background polarization modify this simple optical edge. Thus ${\omega }_p$ is a collective charge-response scale, not a single-electron scattering rate or a sharp transparency threshold for every real metal.

A normal Fermi liquid has interacting fermions whose low-energy excitations can be followed continuously from those of a Fermi gas, provided no phase transition destroys that correspondence. A Landau quasiparticle is a particle together with the surrounding many-body response. It has the appropriate charge and spin, a renormalized dispersion and a long lifetime close to the Fermi surface. Interactions remain in both the quasiparticle parameters and the interactions between quasiparticles; they have not been discarded.

Near an isotropic Fermi surface,The last relation defines the local isotropic Fermi-liquid effective mass. The Landau quasiparticle energy functional expresses how changing occupations shifts other quasiparticle energies:with normalization of the sum absorbed into $f$. Its angular components give Landau parameters describing collective density and spin response.

The Pauli exclusion principle severely restricts the available final states for low-energy collisions. In an ordinary three-dimensional Fermi liquid with regular interactions,At zero temperature the decay width decreases faster than $|\xi |$, making quasiparticles increasingly sharp near the Fermi surface. Their coherent spectral weight is the quasiparticle residue $0<Z\le 1$, and the true electron momentum distribution has a discontinuity of height $Z$ at zero temperature. The rest of the spectral weight is incoherent. The quasiparticle occupation itself still follows a Fermi-Dirac distribution.

The low-temperature electronic heat capacity is linear, $C_V/\mathcal{V}=({\pi }^2/3)k_B^2{N}^{\ast }(0)T$, where $N^{\ast }(0)$ is the spin-summed quasiparticle density of states. In an isotropic spin-one-half model, the uniform spin susceptibility and compressibility take the formsThus enhanced effective mass and residual interactions can separately change observable responses. The familiar $T^2$ electrical resistivity requires a mechanism relaxing electrical current, such as allowed Umklapp processes; collisions in a perfectly clean Galilean-invariant continuum conserve total momentum and do not alone produce a finite dc resistivity.

Zero sound is a collisionless collective density oscillation sustained by the quasiparticle interaction; ordinary first sound occurs when collisions establish local equilibrium during a cycle. Metals and normal liquid helium-3 supply important examples. The description fails in phases with no sufficiently sharp low-energy quasiparticles, such as generic interacting one-dimensional systems, and must be modified after pairing or another ordering transition.

A numerical evolution has numerical stability if small perturbations in initial data, forcing or arithmetic are not amplified without control over the time interval of interest. For this constant-coefficient decaying linear system, an absolute-stability test requires the homogeneous numerical amplification factors to have magnitude at most one, with no growing Jordan-block contribution on the unit circle. Stability is distinct from accuracy: a strongly overdamped numerical solution can be stable while representing the physical oscillation poorly.

For $|x|\ll s_0$ the quadratic Hamiltonian describes a simple harmonic oscillator withAn exactly circular orbit closes for every nonzero $l$. A nonzero infinitesimal radial oscillation closes at leading order precisely when $\sqrt{3}\sin \theta$ is rational. This condition concerns the cone angle: changing $s_0$ or $l$ cannot change the leading-order ratio. Thus there are no special nonzero $l$ values selecting closure for a fixed cone at this order.

At finite radial amplitude the system is equivalent, after unrolling the cone with planar angle $\chi =\varphi \sin \theta$, to a planar central force with potential proportional to $s$. Its angle advance per radial period changes continuously from $2\pi /\sqrt{3}$ near a circle towards $\pi$ at very large radial energy. Hence the physical advance changes from $2\pi /(\sqrt{3}\sin \theta )$ towards $\pi /\sin \theta$. This nonzero interval contains rational multiples of $2\pi$, so tuning a larger initial $|p_s|$ can produce closed orbits even when the small-amplitude ratio is irrational. Generic amplitudes still produce a precessing orbit; neither energy conservation nor increasing $|p_s|$ guarantees closure.

Define the plasma parameter as the number in a Debye sphere, $N_D=(4\pi /3)n_e{\lambda }_D^3$; some conventions omit $4\pi /3$. A good classical plasma has many particles in a screening volume, $N_D\gg 1$, and a system size much larger than ${\lambda }_D$, so collective fields dominate individual encounters.

A collision here means cumulative Coulomb deflection that significantly changes a particle's velocity, not necessarily a literal hard-sphere impact. A collisionless description requires the phenomenon's timescale to be short compared with the appropriate collision time, and its length scale short compared with the mean free path. Parametrically $\boxed{{\omega }_p{t}_c\sim N_D/\log \Lambda }$ up to convention-dependent constants, with Coulomb logarithm $\log \Lambda$ representing the range of impact parameters. Many oscillations can then occur before collisional relaxation. The formula itself presupposes the weak-coupling regime.

Rapid longitudinal oscillations in a weakly collisional plasma compress the velocity distribution primarily along one direction. There is insufficient time for collisions to redistribute that energy among three directions; the effective one-dimensional adiabatic exponent is $1+2/1=3$. Ordinary monatomic-gas sound with local collisional equilibrium has three translational degrees of freedom, giving $1+2/3=5/3$. Thus $\Gamma =3$ belongs to the stated kinetic/time-scale limit, not to every thermodynamic process involving ionized gas.

Rewrite the real-energy probability as $T=[1+U_0^2{\sin }^2(\zeta a)/(4E(E+U_0)){]}^{-1}$. Thus $\boxed{T_{\max }=1}$ at $\zeta a=j\pi$, namely $E_j={\hslash }^2{j}^2{\pi }^2/(2m{a}^2)-U_0\ge 0$, with a strictly positive resonance energy in the ordinary scattering problem. Between resonances it is less than one and its oscillations approach one as energy increases. Generically $T(0^{+})=0$; a well tuned to a zero-energy resonance instead has $T(0^{+})=1$. The original transmission plot in S1 illustrates these maxima for a chosen well, not a universal specified depth.

Expand $A(r)=A(r_0)+A^{\prime }(r_0)s+O(s^2)$ about $qA(r_0)=p_z$. Then $p_z-qA\simeq -q{A}^{\prime }(r_0)s$ and conservation of energy givesHere $\Omega =q{A}^{\prime }(r_0)/m$ can be signed, but the physical oscillation frequency and amplitude are nonnegative. The small-energy condition ensures that this local expansion samples only a narrow range about $r_0$, provided the field varies smoothly there.

The plasma parameter is the mean number of electrons in a Debye sphere. With electron Debye length ${\lambda }_D=({\epsilon }_0{k}_{B}T/n_e{e}^2{)}^{1/2}$,Large $N_D$ describes weakly coupled collective screening; a single electron does not dominate its screening cloud. For a small uniform electron displacement $\xi$ relative to fixed ions, the restoring field gives $m_e\ddot{\xi }=-n_e{e}^2\xi /{\epsilon }_0$, henceThis cold-electron oscillation estimate neglects thermal dispersion and ion motion.

Using the stipulated collision-time approximation,With $C\sim 3n_e{k}_B/2$ and $v^2\sim k_{B}T/m_e$, the kinetic estimate $K=C{v}^2\tau /3$ givesThe elementary single-speed estimate corresponds to $B=1$; velocity averaging and the stated collision model are represented by the supplied $B$. A detailed Coulomb transport calculation also contains a Coulomb logarithm and cannot be inferred precisely from this simplified expression.

The electron thermal energy density is $3n_e{k}_{B}T/2$. Transporting all of it at the thermal speed gives the free-streaming scaleFor heat flowing in the positive $x$ direction, $-dT/dx>0$. Inserting $K$ into $K(-dT/dx)>A{\dot{q}}_{\max }$ givesThis flux-limiting criterion expresses the failure of a local diffusive description when an electron samples a substantial change in temperature over a transport path.

For the stipulated carbon plasma and $B\simeq 10$, setting the gradient equal to $10^{11}\mathrm{K}{\mathrm{m}}^{-1}$ gives a critical flux fraction of order $A_{\mathrm{crit}}\simeq 0.05$. Fourier transport is invalid according to this test when $\boxed{A<A_{\mathrm{crit}}}$; a still larger actual gradient raises the critical value. Thus the source's request for a minimum $A$ has the inequality reversed: the calculation gives an upper threshold, not a positive minimum. An arbitrarily smaller positive $A$ makes the criterion easier to satisfy. The exact numerical threshold is evaluated in the independent check.

Write $M=m_1+m_2$. Integrating the zero-momentum condition gives $Mx+m_{2}l\sin \varphi =M{X}_c$, where $X_c$ is the constant horizontal centre-of-mass position. Therefore the bob coordinates areEliminating the angle gives the elliptic bob path at zero horizontal momentum,The semiaxes are $m_{1}l/M$ horizontally and $l$ vertically. An oscillation traverses only the allowed arc, reversing at its turning points. The support counter-moves to keep the horizontal centre of mass fixed. The heavy-support limit tends to the familiar fixed-pivot circle.

Insert the harmonic variables into the forward map and reduce the squared trigonometric functions. This givesIndeed $p=m\dot{x}$ at this order. The second harmonic and mean displacement of a weakly cubic oscillator show asymmetry of the motion: for positive $\lambda$ the mean shifts towards negative $x$. There is no first-order frequency shift, although the waveform is no longer purely sinusoidal.

The small parameter is $|\lambda |A/(m{\omega }^2)\ll 1$. A purely cubic perturbation is unbounded in one direction, so these formulas describe small local oscillations around the stable origin, not globally confined motion at arbitrary energy. Amplitude $A$ labels the transformed orbit and must be inferred using the inverse map if physical initial position and momentum are supplied.

For constant $r=R>0$, the radial equation requires $\dot{\theta }(\dot{\theta }+2e\mu /m)=0$. Select the stipulated nonzero angular velocity and define the positive cyclotron frequency ${\Omega }_c=2e\mu /m$. ThenThis is a circle in the transverse plane combined with an independent axial oscillation, as in uniform magnetic field with axial harmonic confinement. Viewed from positive $z$, positive charge rotates clockwise. Its three-dimensional path lies on the cylinder $r=R$ and reverses its vertical motion; it is generally not a constant-pitch helix. For $Z=0$ it is a horizontal circle. This orbit has $p_{\theta }=-e\mu R^2$.

For the specified parameter relation,Thus there are $|n|$ axial oscillations per transverse revolution, and the nondegenerate three-dimensional trajectory closes after $2\pi /{\Omega }_c$. Since $\lambda >0$, the integer $n=0$ is excluded; changing its sign changes nothing. More generally commensurate cyclotron and axial oscillations close for any rational frequency ratio, whereas irrational ratios give quasiperiodic motion when both components are present. An integer ratio is sufficient, not the only closure condition. The same frequency argument also applies to transverse cyclotron circles centred away from the chosen cylindrical origin.

Use the forward spatial transform $\hat{f}(k)=\int e^{-ikx}f(x)dx$ and inverse $f(x)=\int e^{ikx}\hat{f}(k)dk/(2\pi )$. The retarded transformed Green function obeysIt vanishes for negative $\tau$. Continuity at zero excludes an unwanted ${\delta }^{\prime }$ term; integrating across zero then fixes $G_k^{\prime }(0^{+})=c^2$. Define $\gamma =\alpha c^2$ and $d_k=\sqrt{{\gamma }^2-c^2{k}^2}$. The two homogeneous exponents are $-\gamma \pm d_k$. Imposing the two matching conditions gives the Fourier-mode Green function of the telegraph equation,For an explicit time-Fourier route, use $G_k(\tau )=\int e^{-i\omega \tau }{\mathcal{G}}_k(\omega )d\omega /(2\pi )$. ThenFor $k\ne 0$, both poles lie below the real axis. Closing above for $\tau <0$ gives zero; closing below for $\tau >0$ is clockwise, giving $-i$ times the residue sum. Since ${\mathcal{G}}_k=-c^2/[(\omega -{\omega }_{+})(\omega -{\omega }_{-})]$, this iswhich reduces to the same damped sine or hyperbolic sine above. The $k=0$ pole on the real axis is interpreted by the retarded limiting prescription, or directly by the initial-value matching above.

For $|k|<\alpha c$ the response is overdamped and nonoscillatory. For $|k|>\alpha c$, put ${\Omega }_k=\sqrt{c^2{k}^2-{\gamma }^2}$; analytic continuation gives $G_k=\Theta (\tau )c^2{e}^{-\gamma \tau }\sin ({\Omega }_k\tau )/{\Omega }_k$, a real damped oscillation. At equality the finite limit is $\Theta (\tau )c^2\tau e^{-\gamma \tau }$.

All nonzero Fourier modes decay. The uniform mode instead has $G_0=(1-e^{-2\gamma \tau })/(2\alpha )$ for positive time, leaving a constant temperature increment. At small wavenumber the slow decay exponent is $-k^2/(2\alpha )+O(k^4)$, so the diffusion limit of the damped heat-transmission equation has diffusivity $1/(2\alpha )$. The full hyperbolic equation has finite propagation speed $c$, unlike the limiting parabolic heat equation.

The spatial transform of $\cos (px)$ is $\pi [\delta (k-p)+\delta (k+p)]$. Convolution with the impulsive source and the inverse transform therefore give simply $T(x,t)=\cos (px)G_p(t-t_0)$. Take $p\ge 0$ without loss of generality and set $\tau =t-t_0$.

For $\alpha >p/c$, let $d_p=\sqrt{{\alpha }^2{c}^4-p^2{c}^2}$. ThenFor $\alpha <p/c$, let ${\Omega }_p=\sqrt{p^2{c}^2-{\alpha }^2{c}^4}$. ThenBoth satisfy $T(t_0^{+})=0$ and ${\partial }_{t}T(t_0^{+})=c^2\cos (px)$, so the impulse changes the time derivative rather than discontinuously changing temperature. The first case rises and then relaxes without sign oscillations for a nonzero mode; the second oscillates as its amplitude decays. The limiting equality gives $T=\Theta (\tau )c^2\tau e^{-\alpha c^2\tau }\cos (px)$, and $p=0$ gives the nondecaying uniform increment discussed above.

A cosine source has both positive and negative regions, so this linear solution can be understood as a temperature perturbation about a background; its negative values are not assertions of negative absolute temperature. The temporal oscillations arise from the second-time-derivative term. Dropping that term changes the initial-jump structure as well as removing the underdamped regime.

Let $\beta =1/(k_{B}T)$. Each independent fermion orbital is either empty or occupied once, so its grand partition factor is $1+e^{-\beta ({ϵ}_j-\mu )}$. Multiplying factors and taking the logarithm givesSince $1-n_F=[1+e^{-\beta (ϵ-\mu )}{]}^{-1}$, this is the requested form. Spin or other orbital degeneracy is already included in $g$.

For the Fermi-window convolution of the grand potential, defineDirect differentiation gives $F^{\prime }=+n_F$. The supplied PDF hint has the opposite sign; using that sign would contradict its grand-potential formula. Integrating twice with $A(0)={\Phi }_0(0)=0$ and vanishing upper boundary terms,At zero temperature $-{\partial }_{ϵ}{n}_F\to \delta (ϵ-\mu )$, confirming ${\Phi }_0(\mu )=\Phi (0,\mu )$.

Use the source's implicit unit normalization of the constant density of states; restoring a constant baseline $g_0$ multiplies the following potentials by $g_0$. For the sinusoidal density of states, direct integration givesPhysical nonnegative density of states requires $|\alpha |\le 1$. Write $t=k_{B}T$ and $y=ϵ-\mu$. The Fermi windowis even, with full-line moments $\int wdy=1$, $\int ywdy=0$ and $\int y^{2}wdy={\pi }^2{t}^2/3$. The last integral follows, for example, by expanding $e^x/(1+e^x{)}^2$ on $x>0$ into its alternating exponential series. When $t\ll \mu$, extending the lower integration limit to $-\infty$ introduces only exponentially small band-edge corrections. The supplied Fourier transform givesThe odd sine term vanishes. Substitution provesThe thermal damping of a sinusoidal density of states is therefore $X/\sinh X$ in the full oscillatory potential. The requested ratio instead concerns the thermal change $\Phi (T,\mu )-{\Phi }_0(\mu )$. Its oscillatory thermal-change amplitude ratio isFor positive $X$ it decreases smoothly from the limit $|\alpha |$ to zero:At exactly $T=0$ both thermal changes vanish; the quoted value is their limiting ratio. The amplitude is the envelope in chemical potential. At a particular fixed $\mu$, the signed contribution ratio has $\alpha \cos (2\pi \mu /\Delta )$ in place of $|\alpha |$; at an oscillation node the actual oscillatory contribution is zero. The large-$X$ behaviour still requires $k_{B}T\ll \mu$, so the useful range is $1\ll X\ll 2{\pi }^2\mu /\Delta$.

Each supplied eigenvector gives relative displacements at $x=1,2,3$ in a standing wave. Insert zero at $x=0,4$ to describe the whole sampled string. A real normal mode is $u_j(t)=C{v}_j\cos (\omega t+\varphi )$: its shape stays fixed while its amplitude oscillates.

These semidiscrete fixed-string normal modes are samples of $\sin (k\pi x/4)$, with $k=3,2,1$ in the order listed. More precisely, their interior vectors are $v_j^{(k)}=2^{-1/2}\sin (k\pi j/4)$, which have unit Euclidean norm and form an orthonormal basis of ${\mathbb{R}}^3$. Their eigenvalues and angular frequencies obeyThus larger absolute value of a negative eigenvalue means faster oscillation, rather than faster decay. The numerical angular frequencies approximate the continuous fixed-string values $k\pi c/L$; the finite difference spatial approximation changes them. Only three independent interior normal modes exist on this grid, and any interior displacement or initial velocity vector can be expanded in this orthonormal basis.