mkal@pas.rochester.edu

Photons seem to be a spin-1 particles with the Hamiltonian being the projection of their spin and their momentum H = p S. Therefore while hitting the objects inelastically they should transfer their internal angular momentum and cause the rotation in analogy to Einstein - de Haas effect . Rigid black body disk will start to rotate under the influence of the Circularly Polarized light (S_z= +-1). The smaller and the lighter the object the better and the certain preexcited single electrons in atoms appear to be the best candidates. My research interests emerge from my PhD discovery at University of Rochester of

strongly localized Starkstates for Zeeman Hamiltonian

so called Trojan states, Landau states for electrodynamics with complex magnetic fields and mass tensor particles.

My PhD: The Hamiltonian of the hydrogen atom in the CP field is H=p^2/2-1/r- e x - w Lz. When w=E2-E1 the matrix of it in the basis of two first Hydrogen eigenstates of the type n=1, l=0, n=0 and n=2, l=1, m=1 (only those two states are degenerate for the exact resonance in the Bloch-Rabi rotating frame for the perturbation theory because of the Lz term) is H= {{E1, e d},{e d, E1}} with the eigenstate |X1>=(|100> + |211>)/2^0.5 and the energy E(1)=E1+d e (Trojan wavepacket) and |X2>=(|100> - |211>)/2^0.5 and E(2)=E1-d e (anty-Trojan wavepacket), d=128/243 a0. The first eigenstate is of the type 1 + Exp (i phi) while the other 1 - Exp (i phi) for any fixed r. Therefore the first eigenstate density is like cos(phi) localized around the angle phi=0 while the second is Pi-shifted. The radial localization is due to the overlap between two hydrogen states in r. Very clear banana-like shape of the packet can be obtained in two-level approximation when one elevates the excited n=2 component of the state by small detuning from the energy difference so the relative radial contributions from both states are equal at overlap between a0 and 4a0. One may therefore notice that the different energies of Trojan and anty-Trojan wave packet is also the Autler-Townes splitting. For the larger and larger n (w=E_(n+1)-E_n, l=m=n-1) wavepackets sharply focus relatively to the atom size as the 1/n effect.

Simple man theory of Trojan wavepacket

Trojan wavepacket viewer

I therefore engineered the smallest electric motor one can build with dynamic fields design (with a single atom) !!!

In normal electric motor (DC) magnetic field changes around rotor adjusting the torque to maximum for given phase (using the switch called commutator )

In inductive or dynamic magnetic field electric motor (AC) the magnetic field of the stator rotates around rotor with steady magnet (no commutator is nessesary)

In my motor the rotor is a displaced electric charge (dipole) not a magnet (it is actually the electron on circular Kepler orbit) and the electric field is rotating instead of magnetic (the stator is the nucleus) - you can also consider it as simplest example of electrostatic propulsion on atomic scale

Like in classical macroscopic electric motor the bearing is not frictionless but the friction is with the electromagnetic vacuum modes - is radiative

Mechanical model - electrostatic Coulomb force extords the torque instead of magnetic forces - first designed by Physicists Benjamin Franklin and Johann Christian Poggendorf

You can build a hybrid at home and power it with your TV screen !!! or Van de Graaff generator (DC version)

I spent 6 hours tunning mine so be patient ... - focuse on bearing smootness (Marklin car bearing seems the best if you have time beyond ball pen bearing) and the sharpness of commutator needles (just needles)

(Or you can call it photon wind power plant if you like)

The tricky point is that with the quantum matter it is not so simple - the rotor is really liquid and like a droplet of water exists only within a fine parameter range (sizes of the droplet versus the electric field strength). (see the theory above).

The macroscopic applicable electrostatic motor whould have something between the hard drive technology, high voltage capacitor technology and the display technology - the volumetric multi-cylindrical capacitor arrays in almost superfluid dielectric with thousands of electronically switched sub-sectors and hundreds of sub-surfaces should be the best.

Old US patents of electrostatic motors

and now more complicated engine - Langumuir states are physical in crossed fields !!! APS

Making Trojan wavepacket (or quantum motors) costs 500000 dollars but it generates "visible" Hawking radiation in muonic Hydrogen

Since is accelerated with 10^25 g there

This how the atoms get the kinetic energy during Coulomb explosion so the cold fussion occurs in the Deuterium clusters micro-fog

Two-electron pairs orbiting in strong magetic field (or electron-positron pairs in CP field). As the wave functions do not overlap the recombination process can be frozen indefinitely

Like the model trains initailly localized wave packets will move on periodic trajectories like on a track spreading fast on unstable trajectories and not dispersing on locally stable trajectories. While all trajectories are points in a certain instatenous rotating frame and the WKB wave functions spike singularly around the turning points of zero velocity there must be localized eignstates there around to phase interfere. The locally rotating frame augmented to the trajectory tangent circle may be used to study adiabatic stability. I am looking for various iteresting, for example 8-shaped trajectories which can support on-average nonspreading Trojan-like wave packets. Triangle, square and general polygon trajectories are possible in atoms in combination magnetic and electromagnetic fields for one and more electrons.

Interested in selected exotic eigenstates, I am currently working on Monte Carlo genetic algorithm implementation of fixed node method. The nodal constrains are tossed randomly and than the genetic operations are performed on best nodal populations to produce better energy estimate of selected excited states and so on.

You can also use it with tossed-node imaginary-time split operator method

Studying the time dependent free dynamics of the Trojan wave packets after the sudden field turn-off I have recently discovered that the full quantum revival occurs in any quantum system with the rational energies and therefore asymptotically in near-ifininity in any quantum system. Thus it is equivalent to the Poincare revival. While the normal revival considerations expand the spectra up to be harmonic or quadratic as for the quantum well for the Trojan wave packet the validity of expansion is equivalent to the expansion truncation. Depending on the number of states involved in the time evolution expansion the exact revival time of the Trojan wave packet becomes the Poincare when it is as long as non-physical on the Earth laboratory time scale.

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My hot topics for today:

Quantum theory of dielectric constant of organic materials

Inertial confinement of solid hydrogen

Theory of dielectric constant of Bose condensate

Superradiance of Rydberg gases in quantum cavities

Car-Parinello simulations of egzotermic reactions

Cellular automata by quantum dots arrays

Physics of Magnetars

Fussion in atomic clusters in strong fields

Quantum monodromy in helium in crossed fields

Spontaneous light amplification through superradiant emission of radiation (SLASER)

Non-fixed node Monte Carlo with genetic algorithm

Bose-Hubbart theory of color Bose and mixed gases

Nonlinear acoustic subsceptibilities of Bose Gases

Beta electron beams from on-wave acceleration

Detection of heavy mases with mesoscopic SQUIDs

Unruh-Davies effect from quantum entanglement

Ionization Kondo effect in Rydberg gases

Mesoscopic superconductivity

Split-operator methods for spinors and wave function of photon in dielectric media

Contraction-of-propagator methods for protein folding

Quantum computing with quantum dots

Trojan wavepackets with Wannier excitons

Spontaneous order in systems with negative mass

Superradiance with mouonic hydrogen with cyclotronic motion

Spacial control of dielectric constant

Optical phase solvers of Schrodinger equations

Quantum entropy ordering with Trojan wavepackets

Ticking of quantum states

Quantum entropy fluctuations in systems with anomalous spectra

Chaotic oscillations of quantum states in rotating systems

Channeling in multi-center scattering

Model Born-Infeld theories with electromagnetic vacuum collapse

Bohm Hydrodynamical Quantum Mechanics and real time Diffussion Monte Carlo

Dicke superradiant phase transition as supersolid formation in Bose-Einstein condensate with oscillatory interactions: uniform(normal) versus oscillatory (superradiant) phase

Cactoo fractals

Configuration existence theorem for N electrons in magnetic and circularly polarized fields: The maximum number of configurations may be the product of all differential foldings (maximum number of times the ZVS gradient manifold is cut by even lower dimensional arbitrary plane to disjoined sets): There may be at least 2^180=1532495540865888858358347027150309183618739122183602176 maximum number of configurations of electrons corresponding to carbon C60 (N=60) assuming the lowest nontrivial folding 2 (parabola-like). To see this theorem working in 3D one may consider a rose and two planes as some differential ZVS gradient manifolds. The rose has clearly high folding and then when cut again perpendicularly is cuts to a lot of points.

Concentration of primes: 469!-1 was once found prime and 469! - 1 -4362, 469! -1 -4902, 469! - 1 -5406, 469! -1 -5652, 469!-1-8580, 469! -1 + 1130, 469! - 1 + 2084, 469! - 1 + 2592, 469! - 1 + 6662, 469! -1 + 6998 are all prime !

11^1008 + 998672782 was once found prime and so IntegerPart[2(11^1008 + 9986727 82)^2 - 1)^0.5] + 3695 is.

Exact revival of the wave function of mathematical Hydrogen: The arbitrary wave function (not only Gaussian wavepacket and approximately) will recover exactly for the system with the energies En=-R/n^2 after the time T=2 Pi hbar Lm /R where Lm is the lowest common multiple of squares of all quantum numbers involved in the wave function. When first 120 states are involved this is about 10^79 years !!!

Generation of Langmuir Trojan states in Helium: First the Trojan wavepacket is generated from one electron leaving He+ ion core, the plane of motion is adiabatically shifted by the static Stark field turn on parallel to magnetic field and then the second Trojan wavepacket from Helium He+ ion is generated but to much smaller off resonant orbit not much to influence the first one. Later "two color" CP field is adiabatically adjusted to one frequency while turning off the asymmetric Stark and turning on the magnetic field.

Barnett-Einsten-de Hass (rotationally induced) quantum Hall effect: When Trojan wavepackets are build from Wannier excitons and confined in quasi two dimensions the rotation of the sample will induce alternating Hall voltage along the thickness due to normal ridid body rotator response in this direction. When the number of Trojan excitons is quantized the Hall voltage will.

Detection of gravitoelectromagnetic F= v*dm/dt*(Gm^2/r)/(2 Pi m c^2) Lorentz force with gravitational Aharonov-Bohm oscillations in Trojan wavepackets. For Trojan atom placed on the fast satellite orbiting Earth within about 1 hour in the Trojan Michelson-Morley experiment the possible gravitoelectromagnetic flux due to the Earth existence will shift the electron interference pattern for n=600 by one full fringe during 1 year !!!

Trojan Hydrogen as Josephson Junction through Quantum Phase Model (QPM)

"Compressed sping" phase transition in Trojan elliptical harmonic two-electronquantum dots

Coherent tunneling between two Trojan atoms in two ultrashort delta laser pulses

Trojan wave packets from electron gas in the Coulomb field of higly charged ions in ultrastrong CP laser field

Uncertainty relations for excited Trojan wave packets

Exact revival of the wave function of arbitrary quantum system with rational energies. The quantum wave function will recover exactly for the system with the energies En=C Nn/Mn after the time T=2 Pi hbar Lm /(C Ld) where Lm is the lowest common multiple of all denominators in fractions in energies involved in the wave function and Ld the greatest common divisor of the numerators (most probably 1 for long sequences with large primes). Since the set of the rational numbers is so called dense in the set of real numbers (for any irrational number there is the rational one arbitrary close to it) each quantum system revives exacly on theoretical computer with truncated accurancy since any number there is integer/10^n. Example 1) Jaynes-Cummings Model: Consider the energies of Jaynes-Cummings model En=(n-1/2)omega +- (1/2)[(omega-Omega)^2 + 4 g^2 n]^(1/2). To make energies rational (integer here) let us assume the state is localized in n and spanned around the principal quantum number n0. The energy Taylor expansion around the leading state n0 is En= (n-1/2)omega + Const + g^2/[(omega-Omega)^2 + 4 g^2 n0]^(1/2)(n-n0). While the coefficient multiplying (n-n0) is small we can assume omega is also the integer multiple of it. While only even or odd ns are involved (the eigenstates mix the consecutive photon numbers) we get the revival time (to revive the state up to the phase factor) T= Pi*[(omega-Omega)^2 + 4 g^2 n0]^(1/2)/g^2 Example 2) Hydrogen atom for shorter times: Consider energies of hydrogen atom En=-1/2n^2. Let us assume that the quantum state is localized around the principal quantum number n0 (Gaussian-like in n around n0) and expand En=-1/2n0^2 + 1/n0^3*(n-n0) -3/2n0^4*(n-n0)^2 to make it as varable integer multiplying the common rational. Since the coefficient multiplying the quadratic term is small to the Kepler frequency 1/n0^3, the Kepler frequency 1/n0^3 can be assumed the integer multiple of it. Without the loss of generality we can assume expansion in n-n0 as even. Therefore the revival time is 2*Pi*n0^4/3.

Superrevivals in Jaynes-Cummings model: Mapping the revival dynamics of the Trojan wave packets after the sudden field turn-off onto the Jaynes-Cummings model the revivals in J-C models are full rotations of the spreading Trojan wave packet while the full revivals of Trojan wavepacket are superrevivals in Jaynes-Cummings model after the time T= Pi*[(omega-Omega)^2 + 4 g^2 n0]^(3/2)/g^4 when the normal revivals will to reshape back to the original decay and around. The superrevival time is about n0/g^2 longer then the revival time !

Exact superfast revivals of special quantum states in Jaynes-Cummings model on resonance: Let on resonance En =(n-1/2)omega + Const +- g n^(1/2) and omega is integer multiple of g then any quantum state such that n = kn^2 revives exactly after the time Trev= 2*Pi/g when Trev* g* 2*Pi*n^(1/2) = 2*Pi*kn . Specifically the incomplete coherent states with the number state expansion holes and n components quantum numbers being the square will revive fast and exactly. If kn are localized around large kn0 then kn^2 \approx kn0^2 + 2*k0*(kn-kn0) and n are approximately even or odd with the constant gap.

Looking for arbitrary primes using the exact revival of arbitrary quantum system with rational energies: If En= C Nn/Mm (Nn=1, Mn=n^2 for Hydrogen) has Mn as the power of prime then the superposition of N such consecutive levels has the autocorrelation function always below 1 during the time 0 and the exact revival time 2 Pi hbar Mp1*Mp2...MpN/C (lowest common multiple of primes is their product) so if all first N of primes (really those up to MpN+1/2) is known the next bigger N-th+1 can be found from no reaching 1 condition (or is MpN+1 a prime ?).

Phase shifted exact revivals of the wave function. Let En=C/n^2 and the wave fuction is the superposition of such n that n=10^kn. Let knmax is the maksimum of kn in the wave function expansion. Lm=10^2 knmax and the exact full revival occurs after time T=2 Pi Lm/C but since for the each integer 10^2 kmax/3, 10^2 kmax/3=3333...3 + 1/3 the phase shifted exact revival such that Psi(T)=Exp(i 2 Pi/3)Psi(0) happens already after the shorter time T=2 Pi Lm /3/C.

Arbitrary state preparation with the exact revival of the wave function: Providing that the amplitudes of the wave function are first prepared the arbitrary phases can be reached only by waiting for the sufficient time of the quantum evolution as the time of the full revival with the energies renormalized by the inverses of the rational phases up to full multiples of 2 Pi or the wave function phases sweep the whole asymptotically infinitely dimensional phase hypercube modulo 2 Pi for the arbitrarily long time.

New method of Trojan Wavepacket generation: 1) Arbitrary Gaussian distribution of the circular states is generated around the resonant state. 2) Free no-field evolution is allowed till the Trojan wave packet is self-formed. 3) The parameter matching CP field is turned on instantaneously in phase.

Quantum gates with two Trojan atoms coupled by dipole-dipole interaction with Trojan and anty-Trojan states as 0 or 1.

Dynamic ferroelectricity and antyferroelectricity in the system of interacting Trojan atoms.

Dirichlet problem for the d'Alembert wave equation with one periodically oscillating wall: Let u(x,t) is the solution of the wave equation and u(0,t)=0 on one rigit wall at x=0 and further u(q(t),t)=0 with q(t)=a+b*sin(w*t). Let f(x) = x + 2*a + 2*sin(w*(x+a)). Then u(x,t)=g(f(f(f(...(t-x))))-g(f(f(f...(t+x)))) with the finite multiple Feigenbaum f composition is slow on convergence to the fixed point or differs a little on the next interation and is the approximate solution when g(x+2*a) = g(x) i.e. g is periodic with 2*a or exact for u(h(t),t)=0 with h(t) periodic and close to q(t).

Quantum scars in infinite stadium potential well as quantum carpets for one-dimensional Klein-Gordon equation with the imaginary time and time-dependent semi-circle boundaries u(q(t),t)=0 with q(t)=+-[a+b*$in(t/b)], $in(t)=[1-(1-t)^2]^0.5. The scar energy is the square of the mass energy of the pseudo-relativistic particle at rest. The scar reflection from the parallel stadium edges is the Klein paradox with the particle probality coming out from everywhere in the whole volume (or the one-dimensional space) due to the imaginary time or propagation velocity.

Kondo temperature suppression of ionization of Trojan Hydrogen as Superradiant Phase Transition.

Trojan wave packet as the superradiant electron state of the Jaynes-Cummings model after Holstein-Primakoff transformation for two levels

Quantum Hall effect and annihilation suppression in electron-positron gas

Cold fussion of magnetically stabilized two-nuclei Trojan states of Deuterium

Slaloming of Trojan wave packet of Deuterium nucleus in Palladium crystal lattice

Ionization probability dependent interractive intelligent grid renormalization method for strong field ionization problems # => []. Each time step the probability to find an electron is calculated in the outer near-boundary spatial region. When the critical is reached the size of the spacial grid step is doubled and the wave function is rewritten from the previous grid each second grid point being again far from boundaries.

Trojan stability paradigm in harmonic two-electron quantum dots in electric and magnetic fields emulating rotation

Soft chaos induced shape life of Trojan wave packets due to Henon-Heiles like corrections and randomness of "irrelevant" Stark-Zeeman energy levels

Dynamic ferrolectricity of Trojan hydrogen atoms on 2D honeycomb lattice (Elok > 0)

Nonspreading wave packets of near-luminal electrons in free space. According the relativistic energy-momentum relation E^2= p^2 c^2 + m0^2 c^4 the dispersion relations E(k) for free electron and positron form the avoided crossing with the energy gap being twice the rest mass energy of the electron m0 c^2 or the energy of the pair creation. As the result they become linear and photon-like in utra-reltivistic limit k -> Infinity and Trojan-like wavepackets exist in free space moving with almost the speed of light. They can be generated from Trojan wave packets (focuser) by ultra-strong phased electric field delta kick applied to Trojan electron (accelerator).

Effective Hartree-Einstein equation for the relativistic electrons: p^2/(grad S(x)^2/4 c^2 + 4 m0^2)^(1/2) + V(xt) + m0 c^2 = i d/dt, Psi(x) = |Psi(x)|exp[iS(x)]

Rotational "Third force" on Trojan wave packets: In Electrodynamics there is other than the Coulomb Force and the Lorentz Force force acting on particle without the electric charge but with the dipole magnetic moment and with the spin (e.g. on neutrons) in the crossed combination of the electric and magnetic fields. Because the free evolving spin with the magnetic moment undergoes the precession according to Bloch equations i.e. the end of its vector axis moves around the circle and its tilt angle with respect to the magnetic field vector is constant it also generates the magnetic field which is time dependent. This field is generating the magnetic flux which is variable in time and according to the Faraday's induction law and the Special Theory of the Relativity it is seen in the laboratory frame as the combination of two fields, the magnetic and the electric one. Now if such spin is placed between the plates of the charged capacitor the variable in time magnetic flux from the dipole motion will induce forces acting on rigidly mounted electric charges. According to the III Newton's principle of dynamics about the reciprocity of force action this force also acts backwards with the same strength on the magnetic dipole. In the extreme condition of the precession angle of 90 degrees this force is therefore in the direction perpendicular to the electric field and parallel to the magnetic field. Such capacitor however is equivalent to the constant electric field generated by its charged plates. In general case of the electric and magnetic field orientation the third force can be expressed as: FIII=1/mc E x (m x B). The similar effect will be for Trojan wave packet when the electron spin is precessing under Eistein-de Haas-Barnett interaction in the rotating frame.

Intrinsic Trojan-like i C x y entanglement in quantum systems and spin 1-space coordinateentanglement in the wave function of photon

Violation of angular momentum Bell inequalities for two-atom entangled Trojan cat states

Electron free fall in the nuclear electric field and strong magnetic field: Trojan-like wave packets on Gryzinski-like trajectories in linearly polarized electromagnetic field with variable polarization plane.

Langmuir Trojan-like electron configurations on triangular, square and regular polygon- trajectories in strong magnetic and circularly polarized fields

Direct determination of the vacuum impedance from the existence of Trojan like wavepackets on regular poligon trajectories in the magnetic field

High harmonic generation from Trojan-like electrons in strong magnetic field on polygon trajectories due to near-infinite vertex acceleration

Writing with the electron nondispersing wavepacket orbit in strong time dependent magnetic and electromagnetic fields

Detection of Lense-Thirring effect with Trojan atoms: Above the Earth poles the antygravity experienced by the Trojan atom with the the electron rotating in the Earth spin direction is equall to (2/5) n alpha^2*/(omega_Earth/omega_Bohr) g where omega_Earth is the 2*Pi(3600*24) Hz Earth rotation while omega_Bohr is the first Bohr orbit frequency and while (omega_Earth/omega_Bohr) is measurable it is readily measurable for ultra-Rydberg atoms with n = 1/alpha^2 = 137^2.

Cavendishexperiment for sigle atoms: Single electron cat states in the Hydrogen Rydberg pseudo-molecule to detect electron-proton gravity through gravitational Stark effect on Trojan wave packet

Delta pulse ionization current field effect single atom transistor (FET) using Trojan-anty Trojan dynamics

Direct exact violation of the Second Law of Theromdynamics by the spontaneous lowering of the quantum vom Neumann entropy in the ultra-long field-free time evolution of Trojan wave packets

Internal coordinate exitations of Trojan wave packet as the Hawking radiation

Attosecond Transient Absorption by Trojan Wave Packets

Intensity dependent dielectric constant theory of incoherent Trojan wave packets: Assuming the Gaussian density operator around the central state within the Trojan energy manifold of circular states the energy balance between the C.P. field absorption and the spontaneous emission predicts the existence of the ethernal localized density Trojan wave packets in the decohering thermal reservoir even when the off diagonal elements of the density matrix are zero.

Microwave superconductivity of Trojan electron-positron pairs at finite temperatures

Trojan Positronium formation during electron-positron Bhabha scattering in the laser field: The first step towards the Trojan matter-antymatter production is non-recombinating positronium in the Trojan wave packet state. In the first experiment the cathode positron anode ray tube providing low energetic beams of electrons from cathode and positrons from anode covered by the positrons producing radionuclide are aligned to collide in the center between te cathode and the anode. While the collision point is subjected to the microwave or the laser circularly polarized field finely tuned to the beam velocities and lepton beam densities the Trojan positronium formation should occure to stop the matter and antymatter currents to pass through the collision point and reduce the annihilation and the gamma radiation.

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Quantum kaleidoscops with BEC in 1D optical lattice with oscillatory interaction

Ergodicity of quantum phase in three state model

Trojan wave packets (or Bohr atom if someone cares less about quantization - Rutherford-Nagaoka atom) observed in the lab:

I originally discovered Trojan wave packets as follows: The Hamiltonian in two dimensions in circular coordinates is: H=-hbar^2/2m(d^2/r^2 dphi^2 + 1/r d/dr + d^2/dr^2) - w (hbar/i)(d/dphi) - 1/r - e r Cos(phi) . First I assumed that the wave function has a large angular momentum phase simply Psi -> Psi Exp(i l phi). The new wave function Psi fulfills the Schrodinger equation with the Hamiltonian H= -hbar^2/2m d^2/dr^2 +hbar^2/2m (l^2/r^2) - 1/r - hbar^2/2 m r^2 d^2/dphi^2 -i l hbar^2/m r^2 d/ dphi - w (hbar/i)(d/dphi) - e r Cos(phi). This Hamiltonian is still nonseparable so I applied Hartree approximation to the coordinate as it was of 1D particle. Simply H = H_r + H_phi, H_r= -hbar^2/2m d^2/dr^2 +hbar^2/2m (l^2/r^2) - 1/r and H_phi = - hbar^2/2 m r_0^2 d^2/dphi^2 - e r_0 Cos(phi) where selfconsistently Hartree averages were used r_0=Av(r), Av(d/dphi)=0, Av(d/dr)=0 to decouple to separability. The wave function is therefore a product of an approximate Gaussian in r localized around r_0 from locally harmonic normal centrifugal potential + Coulomb potential and the Mathieu function in phi. Trojan wave packet is here the localized inverted pendulum state and the anty-Trojan the ground state. This is opposite to the more exact Mathieu theory, the spectrum is inverted and of the pendulum mass 1 but not -1/3 and the exact classical stability of equilibrium points is opposite but it predicts both packets right where they are at phi=0 and phi=Pi points having theangular momentum hbar l= m w r_0^2/hbar and approximately right energies. The natural consequence is the existence of double-Trojan states in Helium and the mutual stabilization of two parallel Trojan trajectories in hoop earrings configuration: