Gabriel Barton

Quantum Mechanics of the Inverted Oscillator Potential

Abstract The Hamiltonian ( 1 2m )p 2 − 1 2 mω 2 x 2 yields equations solvable in closed form; one is led to them by questions about the longest mean sojourn time T allowed by quantum mechanics to a system near unstable equilibrium. These equations are then studied further in their own right. After criticism of earlier arguments, one finds, by aid of the Greens function, that T ∼ ω −1 log { l ( h mω ) 1 2 } for sojourn in the region | x | l , where l is the resolving power of the detector. Without appeal to some parameter like l one would get nonsense estimates T ∼ ω −1 (e.g., from the nondecay probability familiar in the decay of metastable states). in this potential wavepackets Gaussian in position do not split on impact: their peaks are either transmitted or reflected, depending on the sign of the energy E ≷ 0; however, they spread so fast that not all the probability ends up on the same side of the origin as the peak. The energy eigenfunctions (parabolic cylinder functions) identify the transmission and reflection amplitudes as T = (1 + e −2π E) −1 2 e iφ , R = −i(1 + e −2πE ) −1 2 e −πE e iφ , where φ = arg Γ( 1 2 − i E ) (in units where 2m = 1 = ω = h ). The density of states for the interval | x | ≤ L is 2π −1 log L + π −1 ϕ′( E ). Wavepackets that are peaked sharply enough in energy travel without dispersion in the asymptotic region | x | > | E |, and do split on impact in the usual way. The travel times and time delays of these packets are determined. For both reflection and transmission, and for both E ≷ 0, the time delays are given by φ′( E ), which is a symmetric function of E , with a positive maximum at E = 0. In particular, packets tunneling under the barrier reemerge sooner if their energy is more negative. This paradox (which occurs also in other tunneling problems) is elucidated as far as possible. Coherent states are constructed by analogy to those of the ordinary oscillator. Though not integrable, their probability distributions do have a recognizable pattern which moves classically. Such states form a complete set only if generated from energy eigenstates with definite parity. If generated from scattering eigenstates, only certain special coherent states are physically admissible, and these do not form a complete set. The effects of resistive (energy dissipating) forces and of thermal agitation are considered briefly. At zero temperature ordinary resistive mechanisms enhance the sojourn time.

Plasma spectroscopy proposed for C60 and C70

The fullerenes C60 (quasispherical though actually icosahedral) and C70 (quasispheroidal) are very like hollow shells (graphiteroles) made from a single hexagonal layer from a graphite crystal. They should support multipolar (l≥1) plasma oscillations, closely related to the plasmons seen in graphite. We estimate their frequencies using the so‐called hydrodynamic model, with (provisionally) just one parameter calibrated on graphite. We predict π plasmons in the range between 6 and 8 eV, and σ plasmons near and above 25 eV. The best, if not the only way to observe them is by electron energy‐loss spectroscopy. Only the dipole (l=1) excitation is allowed optically; in the solid, its frequency should be shifted according to the Lorentz–Lorenz formula. If, improbably, the l=1 π plasmon is observable in solution, its frequency should be unprecedentedly sensitive to the refractive index of the solvent. On C70, the multipoles are split and shifted relative to C60, but only by surprisingly little.

Levinson's theorem in one dimension: heuristics

For partial waves in three dimensions, Levinsons theorem asserts that delta (0)= pi nb, where delta (p) is the phase shift at wavenumber p, and nb the number of bound states ( delta ( infinity )=0 by convention). The corresponding theorem in one dimension calls first for a systematic parametrisation of the transmission amplitude T(p)=cos theta ei tau , and of the left (right)-incidence reflection amplitudes RL,R(p)=i sin theta exp(i tau +or-i rho ), where the phase angles tau , theta , rho are functions of p. If the potential is not everywhere zero, and excluding throughout the exceptional case where it has a zero-energy bound state, heuristic arguments show that RL,R(0)=-1, that theta (0)=1/2 pi (-1/2 pi ) when nb is odd (even), and that tau (0)= pi (nb-1/2); (by convention, tau ( infinity )=0= theta ( infinity )). Thus mod tau (0) mod cannot be less than 1/2 pi , no matter how weak the potential; the transition to the limit of zero potential is non-uniform. In the special case of reflection-symmetric potentials, rho =0, one can subdivide nb=nb(e)+nb(o), and define even- and odd-parity phase shifts E=1/2( tau + theta ) and Delta =1/2( tau - theta ); then E(0)= pi (nb(e)-1/2), Delta (0)= pi nb(o). The appendix shows how E(0) is obtainable by suitably adapting the familiar s-wave argument which exploits the analyticity properties of the Jost solutions.

QED between parallel mirrors : light signals faster than c, or amplified by the vacuum

Because it is scattered by the zero-point oscillations of the quantized fields, light of frequency omega travelling normally to two parallel mirrors experiences the vacuum between them as a dispersive medium with refractive index n( omega ). An earlier low-frequency result that n(0)<1 is combined with the Kramers-Kronig dispersion relation for n and with the classic Sommerfeld-Brillouin argument to show (under certain physically reasonable assumptions) that either n( infinity )<1, in which case the signal velocity c/n( infinity ) exceeds c; or that the imaginary part of n is negative at least for some ranges of frequency, in which case the vacuum between the mirrors fails to respond to a light probe like a normal passive medium. Further, the optical theorem suggests that n exhibits no dispersion to order e4, i.e. that n( infinity )=n(0) up to corrections of order e6 at most.

Perturbative Casimir energies of dispersive spheres, cubes and cylinders

The quantum-electrodynamic binding energies B are determined perturbatively to order (nα)2 for single macroscopic bodies (quasi-continua mimicking atomic solids) having the dispersive dielectric function e(ω){1 + 4πnαΩ2/(Ω2-(ω2-i0)2}, as if each atom were an oscillator of frequency Ω, and n the number density of atoms (pairwise separations ρ). The familiar divergences all persist although they are modified by dispersion (finite rather than infinite Ω); they must be controlled instead by imposing the condition ρ>λ~(minimum lattice spacing) >c/Ω) is dominated by components proportional, respectively, to (nα)2Ω×{-V/λ3, + S/λ2, -a/λ (if there are edges) and ±log (c/2Ωλ)}. These always tend to induce collapse rather than expansion. The pure Casimir components are of order (nα)2c/a, and (like the logarithmic terms) sometimes positive, which makes them impossible to understand if the dominant terms are disregarded. The B are found in closed form for spheres, spherical shells and cubes, up to corrections vanishing with λ. For unit length of an infinitely long right circular cylinder of radius a, the standard V- and S-proportional terms are corrected only by -(nα)2(π2Ω/128a)log (c/2Ωλ); the pure Casimir component, which would be proportional to (nα)2c/a2, vanishes through apparently accidental cancellations peculiar to order (nα)2.

On van der Waals friction. II: Between atom and half-space

We calculate the frictional resistance experienced by an atom, modelled as a harmonic oscillator, moving with constant velocity u at a fixed distance ζ outside and parallel to the surface of a Drude-modelled half-space. Our method applies in the nonrelativistic/nonretarded/electrostatic regime, where u is far below c, and u/ζ is far below any important natural frequency of the atom or of the material. For a dissipative (e.g. for an ohmic) half-space and for the low values of u least unlikely to be of practical interest, this force is dominated by a term proportional to u/ζ 8, found perturbatively to fourth order in the interaction between atom and half-space. It appears to depend rather sensitively on how line shapes are handled.

Casimir energies of spherical plasma shells

We study a model loosely inspired by the giant carbon molecule C60, namely a negligibly thin spherical shell of radius R, carrying a continuous fluid with mass and charge surface-densities m/a2 and e/a2 (plus an inert overall-neutralizing charge distribution), so that a mimics some mean inter-electron spacing, comparable say to the Bohr radius. The Casimir energy B is the total zero-point energy of the exact multipolar normal modes, minus that of empty space, minus the self-energy of the given amount of material at infinite dilution. Subject to a Debye-type cutoff l ? L on angular momenta l, but needing no frequency cutoff, B is a well-defined function of R, x ? e2/mc2a, and X ? R/a, expressible in terms of the multipolar phase shifts ?TE,TMl. We consider it only for X 1 ? L ~ X. Realistically one has x 1, but ? ? 4?xX can be large or small. Then B is always dominated by terms of order stemming from TM modes; but the pattern of corrections as functions of x and X is intricate, and accessible only through the Debye (uniform) expansions of the Bessel functions figuring in the ?l. Historical interest attaches to Boyer components, far-subdominant parts of B having the form (c/R)CB, where CB is a pure number. When ? 1 (as in C60) there are none, because all corrections are at least of order x1/2, and none are proportional to 1/R. But a Boyer component does exist when ? 1 (as in macroscopic shells), with CB = [3/64] ? [(9/4096)(?2/8 ? 1)] + 0.0464. The two terms come from orders 1 and 2 of the Debye expansion; the contribution from order 0 vanishes because of an apparently fortuituous cancellation between TE and TM. The most precise value proposed so far is CB = 0.046?1765; but the significance of comparisons is unclear, because previous calculations mistakenly treat the Boyer component as if it included all of B in a hypothetical perfect-reflector limit x ? ?.

On the fluctuations of the Casimir force

Standard statistical and quantum physics are used to analyse the fluctuations of the Casimir stress S (normal force per unit area) exerted on a flat perfect mirror (conductor) by the zero-point electromagnetic fields in adjacent space, partly in order to help dispel the apologetics that often befog zero-point effects generally. S is measurable only when averaged (S) over finite times T, and also (S) over finite areas of typical linear dimensions a. With only one mirror, the mean-square deviations are ( Delta S2)=constant*h(cross)2/c6T8, the value of such constants depending on how the apparatus averages over time; if a >cT, then (S2)=constant*h(cross)2/c4a2T6. On one of a pair of parallel mirrors a distance L apart, if cT>>a, cT>>L, then ( Delta S2)=( Delta S2)=constant*h(cross)2/c4L2T6. Mirror transparency at frequencies well above 1/T has negligible effect. An appendix outlines the mathematical problems, mostly unsolved, met in attempts to evaluate the full probability distributions underlying such mean-square deviations.

Casimir effects for a flat plasma sheet: I. Energies

We study a fluid model of an infinitesimally thin plasma sheet occupying the xy plane, loosely imitating a single base plane from graphite. In terms of the fluid charge e/a2 and mass m/a2 per unit area, the crucial parameters are q ? 2?e2/mc2a2, a Debye-type cutoff on surface-parallel normal-mode wavenumbers k, and X ? K/q. The cohesive energy ? per unit area is determined from the zero-point energies of the exact normal modes of the plasma coupled to the Maxwell field, namely TE and TM photon modes, plus bound modes decaying exponentially with |z|. Odd-parity modes (with Ex,y(z = 0) = 0) are unaffected by the sheet except for their overall phases, and are irrelevant to ?, although the following paper shows that they are essential to the fields (e.g. to their vacuum expectation values), and to the stresses on the sheet. Realistically one has X 1, the result ? ~ cq1/2K5/2 is nonrelativistic, and it comes from the surface modes. By contrast, X 1 (nearing the limit of perfect reflection) would entail ? ~ ?cqK2log(1/X): contrary to folklore, the surface energy of perfect reflectors is divergent rather than zero. An appendix spells out the relation, for given k, between bound modes and photon phase-shifts. It is very different from Levinsons theorem for 1D potential theory: cursory analogies between TM and potential scattering are apt to mislead.

On the 1D Coulomb Klein–Gordon equation

For a single particle of mass m experiencing the potential −α/|x|, the 1D Klein–Gordon equation is mathematically underdefined even when α 1: unique solutions require some physically motivated prescription for handling the singularity at the origin. The procedure appropriate in most cases is to soften the singularity by means of a cutoff. Here we study the bound states of spin-zero particles in the potential −α/(|x| + R), extending the nonrelativistic results of Loudon (1959 Am. J. Phys. 27 649) to allow for relativistic effects, which become appreciable and eventually dominant for small enough mR: they are totally different from conclusions based hitherto on mathematically simple-seeming matching conditions on the wavefunction at x = 0. For realizable R, all relativistic effects remain very small; but with mR decreasing to order α2 the ground-state energy E decreases through zero, and soon after that mR reaches a finite critical value below which E becomes complex, signalling a breakdown of the single-particle theory. At this critical point of the curve E(mR) the Klein–Gordon norm changes sign: the curve has a lower branch describing a bound antiparticle state, with positive energy −E, which exists for mR between the critical and some higher value where E reaches −m. Though apparently unanticipated in this context, similar scenarios are in fact familiar for strong short-range potentials (1D or 3D), and also for strong 3D Coulomb potentials with α of order unity.