Rice

Replica of Trojan asteroids fits in single atom

RICE (US) — Physicists have built an accurate model of part of the solar system inside a single atom of potassium.

In a paper published in the journal Physical Review Letters, the Rice University team showed they could cause an electron in an atom to orbit the nucleus in precisely the same way that Jupiter’s Trojan asteroids orbit the sun.

The findings uphold a prediction made in 1920 by famed Danish physicist Niels Bohr about the relationship between the then-new science of quantum mechanics and Isaac Newton’s tried-and-true laws of motion.

httpv://www.youtube.com/watch?v=t9sJ-H2hM88

“Bohr predicted that quantum mechanical descriptions of the physical world would, for systems of sufficient size, match the classical descriptions provided by Newtonian mechanics,” says lead researcher Barry Dunning, professor of physics at Rice.

“Bohr also described the conditions under which this correspondence could be observed. In particular, he says it should be seen in atoms with very high principal quantum numbers, which are exactly what we study in our laboratory.”

Bohr was a pioneer of quantum physics. His 1913 atomic model, which is still widely invoked today, postulated a small nucleus surrounded by electrons moving in well-defined orbits and shells.

The word “quantum” in quantum mechanics derives from the fact that these orbits can have only certain well-defined energies. Jumps between these orbits lead to absorption or emission of specific amounts of energy termed quanta. As an electron gains energy, its quantum number increases, and it jumps to higher orbits that circle ever farther from the nucleus.

In the new experiments, Rice graduate students Brendan Wyker and Shuzhen Ye began by using an ultraviolet laser to create a Rydberg atom. Rydberg atoms contain a highly excited electron with a very large quantum number. In the experiments, potassium atoms with quantum numbers between 300 and 600 were studied.

“In such excited states, the potassium atoms become hundreds of thousands of times larger than normal and approach the size of a period at the end of a sentence,” Dunning says. “Thus, they are good candidates to test Bohr’s prediction.”
He says comparing the classical and quantum descriptions of the electron orbits is complicated, in part because electrons exist as both particles and waves. To “locate” an electron, physicists calculate the likelihood of finding the electron at different locations at a given time.

These predictions are combined to create a “wave function” that describes all the places where the electron might be found. Normally, an electron’s wave function looks like a diffuse cloud that surrounds the atomic nucleus, because the electron might be found on any side of the nucleus at a given time.

Dunning and co-workers previously used a tailored sequence of electric field pulses to collapse the wave function of an electron in a Rydberg atom; this limited where it might be found to a localized, comma-shaped area called a “wave packet.” This localized wave packet orbited the nucleus of the atom much like a planet orbits the sun. But the effect lasted only for a brief period.

“We wanted to see if we could develop a way to use radio frequency waves to capture this localized electron and make it orbit the nucleus indefinitely without spreading out,” Ye says.

They succeeded by applying a radio frequency field that rotated around the nucleus itself. This field ensnared the localized electron and forced it to rotate in lockstep around the nucleus. A further electric field pulse was used to measure the final result by taking a snapshot of the wave packet and destroying the delicate Rydberg atom in the process.

After the experiment had been run tens of thousands of times, all the snapshots were combined to show that Bohr’s prediction was correct: The classical and quantum descriptions of the orbiting electron wave packets matched.

In fact, the classical description of the wave packet trapped by the rotating field parallels the classical physics that explains the behavior of Jupiter’s Trojan asteroids.

Jupiter’s 4,000-plus Trojan asteroids—so called because each is named for a hero of the Trojan wars—have the same orbit as Jupiter and are contained in comma-shaped clouds that look remarkably similar to the localized wave packets created in the experiments.

And, just as the wave packet in the atom is trapped by the combined electric field from the nucleus and the rotating wave, the Trojans are trapped by the combined gravitational field of the sun and orbiting Jupiter.

The researchers are now working on their next experiment: They’re attempting to localize two electrons and have them orbit the nucleus like two planets in different orbits.

“The level of control that we’re able to achieve in these atoms would have been unthinkable just a few years ago and has potential applications in, for example, quantum computing and in controlling chemical reactions using ultrafast lasers,” Dunning says.

The National Science Foundation, the Robert A. Welch Foundation, the Austrian Science Fund, and the Department of Energy funded the research. Paper co-authors include researchers from the Vienna University of Technology; Oak Ridge National Laboratory; and the University of Tennessee.

More news from Rice University: www.media.rice.edu/media/

chat21 Comments

You are free to share this article under the Creative Commons Attribution-NoDerivs 3.0 Unported license.

21 Comments

  1. don hall / bearcreekresearch

    Moore’s Law inline with ability to control / predict electron orbit … yep, I believe it is just the ticket!
    Now to downsize the equipment, upscale the power available, put financing on notice, and let the innovation begin!

  2. mk

    Trojan Hydrogen is the microscopic Haunebu http://discaircraft.greyfalcon.us/HAUNEBU.htm device, the flying dish which presumably used the Earth magnetic field and megawatts of currents inside the spherical tank of Mercury, the Marconi vortex dynamo to generate huge magnetic dipole around the craft and emulate antigravity and the giant gyroscope to stabilize.
    The Trojan electron rotating around is the current and the atom magnetic dipole
    can be used to control the atom flight in gradients of magnetic fields according to
    E(x)=- mu B(x) energy formula.

  3. mk

    Haunebu, the flying dishes of Germany

    Haunebu first could be a rigid airship like Zeppelin
    but being build as the flying dish.
    The ball on some line drawings could be simply metal
    ball filled with hydrogen. The metal ball with
    light gas becomes
    a baloon when the ratio of the metal thickness
    to radius of the ball is smaller then the ratio
    between the air density and the metal density.
    Some
    pictures from below show clearly that light cover
    was spanned on rigid skeleton. This implies it was light.
    When it was light is could be propelled ionicly i.e.
    first the air around was highly ionized by the reversed operation of Hans Coler free energy machine
    which can generate rotating magnetic field same as by
    the Tesla transformer, then the stream of ions
    was accelerated by the van der Graaf generator to generate
    the drag. The problem is this would require an enormous
    power since this kind of motors are inefficient and it
    would have to have nuclear reactor on board.
    Most likely it was a combination of tunnel hovercraft
    propulsion and the rigid airship or simply
    the helicopter and the balloon in one. Magnetic
    field from the impulser could be used only to orient it with respect to the
    north using earth magnetic field when the helicopter-airship setup was used to put it high in the stratosphere first and then the rocket engine was turned on.
    No one knows for what the Marconi dynamo was clearly but
    it is the electric motor working on induction.
    When the mercury is put in the rotating magnetic field
    it will induce currents in the mercury which further
    generate magnetic field which repel with the original, the mechanical rotation of the mercury occurs and the hydraulic torque which can be extracted by the turbine. This motor could be used further to
    power the dish size propeller around which is seen on some drawings to enhance the ionic drag.

    Also the basic idea to create antigravity with
    strong electromagnetic field is correct within
    non classified Physics.
    It is known that the charged particles like
    electrons or nuclei gain the mass when they are surrounded by the electromagnetic field.
    The mass gain of the electron for example is
    given by a simple formula dm=E^2 a^3 where
    E^2 is the electromagnetic energy density and
    and a is the size of the matter oscillation.
    It simply means that the Einstein mass gain is
    purely electromagnetic due to the surrounding
    photons. The Haunebu in principle could be
    as rigid as a tank. It would simply ionize the air around with ultra strong electromagnetic fields
    and further accelerate the air electrons and ions
    so strongly that they become heavy relativistic plasma, plasma denser than water which could
    push it up as a balloon but with rigid construction.
    To make the mass of the air plasma sufficient so
    the flying dish weights nothing it is roughly
    the same mass of Uranium that the dish weighs
    before starting so the energy problem occurs.
    Assuming the energy problem was solved somehow
    the hull would have to stand the relativistic plasma
    around and so beta gamma and alpha radiation.
    Assuming the basic electromagnetic-Archimedean mechanism survives in some more complex theory like Einstein once worked but the formula stays the same
    without matter, the force acting on Haunebu
    even far above the Planet and without the air or
    any matter like i a free space
    would be against the gravity and equal E^2 V g/c^2 = G M E^2/(c^2 R^2) V where
    E^2 is the electromagnetic energy density V is the Haunebu
    volume c is the speed of light, g is the local gravity acceleration and, R is the Earth radius, and M the Earth mass with G
    the gravitational constant.

    The working Haunebu model
    is quite simple to make since the normal radiation pressure is observable in the vacuum. It is a variation between the metal foil balloon with helium and the outer LC circuit of the Tesla transformer. The balloon may be spherical and it has
    the light inductance coil attached to it. The balloon must be tuned to the weight resonance
    i. e. must approximately neither fall or go up.

    While the Tesla transformer is turned up near and
    with the frequency of the balloon LC 2 Pi (1/LC)^0.5
    the balloon will intercept the energy also radiate and fly up.

    One may notice that even if Germany could produce
    real Tachyons with the negative mass to repel with Earth as one an conclude from the
    name of the main engine the Tachyonator, the energy order of such
    device would be the same. The mass of the particles
    generated by Haunebu of the negative mass by
    the deexcitation of the vacuum and the relative
    energy -mc^2 to the normal vacuum would be equal to the
    energy emitted from for example the 1/2 contribution to the
    all electromagnetic field oscillators as in
    quantum Feynman theory.
    This energy would be the same
    that needed in the conventional theory without Tachyons
    to create electromagnetic fields so dense that the
    electromagnetic Archimedes principle is applied to Haunebu in the Earth gravity field to generate normal buoyancy by such fields and the effective
    relativistic mass density equivalent to this field
    is such that the mass of burned Uranium to generate them
    would be the same that the mass of Haunebu and much above normal atomic bombs. This negative mass would
    have first cancel with the positive Haunebu mass
    and has also the energy -Mc^2.
    The later however is the true effect since photons
    really fall in the gravity fields and their pressure
    on Haunebu is different above and below for the
    fixed densities.

    Working die Glocke, one of the central elements on some line drawings, chained
    in the heavy concrete testing ring original movie suggests however that the
    main antigravity was due to the normal nuclear
    reactor inside die Glocke simply creating the virtual helicopter
    blades from the air pressure for the tunnel propulsion as one can see a
    clear radiation inside and the helicopter – like rotation
    and instability of the rotor: http://http://www.youtube.com/watch?v=ENqMezb8otw&feature=fvsr . Running with the open reactor Haunebu was clearly not environmental friendly.

    Haunebu could be also magnetic Helicopter from principle. Yet another and more realistic interpretation of the Haunebu technical drawings is possible, simply the magnetic helicopter using the Earth magnetic field, the field aligning the magnetic needle to show the North. Die Glocke could be than an electric or even Diesel motor to power the blades and the elecric power generator. Let us imagine that the helicopter blades are conducting and the current inside them is alternating and phased to the point when they are mostly perpendicular to the Earth surface magnetic field line when the Lorentz force on the current is maximum. But it again requires too giant currents of the magnitude of Tera Amperes through the blade so and superconductors if any to put 120 meters dish up interacting with the Earth field so weak so the Haunebu with this mechanism would be a balloon and the magnetic force would be only to steer it. With this interpretation the basic design as the magnetic helicopter is realistic but it again had to be also the balloon on weight resonance. Maybe the drawings are early and show good principle but with the lack of realistic parameters.
    To provide thrust at least supporting the levitation
    in Earth magnetic field only as weak as 10^-5 Tesla of the flying dish craft of the weight 10 tons or 100 000 Newtons and assuming at least 100 meters of looped semi-superconducting wire it requires from definition of Tesla providing Newton of Lorentz force from 1 m of the wire: 100 000 / 100 / 10 ^ -5 = 10^7 Amperes of the current which would constantly emit 10 MW of I*U heat if only under 1 V voltage. But the craft must accelate fast so should be really 100 times more or 1000 MW which is the whole big power station. Directly over the Earth magnetic poles the Haunebu could even use nonrotating fields but there the supporting component of the field would be even 1000 times smaller which is another factor. For the normal rocket the power of the rocket engine maybe 1-200 MW (2000 m/s of the trust gas velocity per Newton weight) so it would be giant military design most likely nuclear and supeconducting that way. Superconductors of this size are not possible but this could be also the tank of Mercury used as simulating superconductor in Marconi dynamo with induction currents to try to produce so giant current inside the liquid Mercury with no much heat. Die Glocke, the mysterious device seen on line drawings rotating the Mercury to induce currents would have to have such giant powers.

    With that interpretation Marconi vortex dynamo (MVD)
    powered by Die Glocke would be the giant electric
    power generator with the main goal not to extract
    that power out but to generate the giant magnetic dipole
    around Haunebu interacting with the dipole of the Earth field to repel with it and generate antygravity.
    The problem is that at the equator where the Earth
    magnetic field is parallel to Earth horizon it would
    no work that way at all and would try to turn the Haunebu over. The small version of the MVD would be the one electron, proton or neutron both spinning and having the magnetic moment which can be trapped and float in gradients of magnetic fields. Die Glocke would have to be at least the 1000 MW or at least of 10 big modern rockets nuclear power device.

    However the Van de Graaf generator is mentioned
    as well which suggests also usage of huge voltages. It appears at least from the principle to be possible to use mechanical currents in the Earth magnetic field to cause flying craft levitation,
    for example scaled balls well known from toy school
    electrostatic machines which would rotate parallel
    to the Earth surface and the plane of the Earth magnetic field where at most places they are contained.
    The symmetric pair of such balls rotating
    again would cause the twist and the Haunebu to flip if
    they have the same charges but if they had opposite charges the net force would be lifting. The voltage between them would have to oscillate and maximize where their linear velocity would be perpendicular to Earth field magnetic lines since otherwise the lifting force would oscillate and average to 0.
    This could require the Tesla generator powered by the
    static voltage van der Graaf. But it again requires giant numbers for voltages and powers.
    The capacitance of perfectly conducting sphere
    is C = 4 Pi Epsilon_0 R where the electric
    susceptibility of the vacuum is very small
    Epsilon_0 = 8.85 ^-12 F/m. To get mechanical current
    of balls charges of 10 000 000 Ampers and the proper
    Lorentz force to lift heavy object for ball of the radius 10 m and rotating at mechanical 10 kHz of modern electric motor and around
    some 100 meters path it would take 10^10 = 10 000 MV to put one Coulomb of charge on the ball and generate
    10 000 Ampers that way and 1000 more i.e 10 000 GV and 10 kHz oscillatory voltage.

  4. mk

    Trojan wave packet by mathematics is the Landau state within the electrodynamics
    allowing the vector potential (and so fields or two-fields) to be really complex not just in the sense of electronics to calculate absorption etc. This leads to the concept or Twoparticles on the classical level
    when the equations of particle motion together with the Maxwell equations describe the motion
    of fully correlated pairs of particles but not just one.

  5. mk

    Trojan wave packet by mathematics is the Landau state within the electrodynamics
    allowing the vector potential (and so fields or two-fields) to be really complex not just in the sense of electronics to calculate absorption etc. This leads to the concept or Twoparticles on the classical level
    when the equations of particle motion together with the Maxwell equations describe the motion
    of fully correlated pairs of particles but not just one.
    I hope that more effects like Quantum Hall Effect either on one or collective excitonic Trojan wavepackets
    in semiconductors will be possible to observe.

  6. mk

    It is thruly wonderful that the Bohr atom popular as a topic in home countries of Trojan Packet original
    inventors, Klar in Germany (1989) and Ivo Bialynicki-Birula in Poland (1994) was finally made in the lab but
    without the stress of discrete frustrating orbit radius n^2 and angular momentum n hbar quantization.

  7. mk

    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 Schroedinger 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 the angular momentum hbar l= m w r_0^2/hbar and approximately right energie.

  8. mk

    The theory of Trojan wave packets as a
    Rabi problem http://en.wikipedia.org/wiki/Rabi_problem is as follows:
    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. 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.

  9. mk

    The realistic theory of creating Tachyons or particles with negative gravitational
    mass appears also realistic withing improvement of general theory of relativity
    allowing gravity fields with non-vanishing rotations i.e. the magnetic gravity
    fields generated by currents of mass. The gravitational Lorentz force in maximum vector
    configuration will be given by

    F = dm/dt G/c^2 M v /r g

    G is the gravitational constant from Newton law.
    The first is the Ciolkowski factor of generating the normal rocket
    thrust from the momentum ejection dm/dt * v , the second shows
    how weak the effect and is the ratio of the hyphotetical self iteraction energy
    at the distance from the mass conductor to the relativistic rest energy
    or
    G M*M/r divided by M c^2

    The samle factor comes from the fact that the gravitational waves should also move
    with the speed of light and should enter that way the wave equation for them,
    The theory immediately implies anty-particle in the sense of not the electric
    charge but in sense of the gravitational mass which would be negative.
    Would would generate the pair is quantum graviton with the energy
    of 2 m c^2 in analogy to electron positron pair.
    From symmetry anty-g-particles would attract each other like the normal matter
    but they would repel with the normal matter and the Earth.
    Thule tachyonator could be filled with this any-g-matter permanently like
    the baloon with the gas lighter then air. It however would have to use
    rocket engines to land and dock since regenerating so many tachyons would be
    again as expensive energetically as Mc^2.
    The gravitation magnetism was discovered recently in superconductor experiments however much stronger that this theory.
    How weak the gravitational Lorentz force is really if it existed ? If the mass current
    was that of the Amazon river some 100 000 cubic meter of water per second and the electrons were speeding as much as they can at the speed of light to maximize the force the electron beam would deflect as much in the old TV cathode ray tube due to the normal gravity (below 0.1 mm) on a giant distance of 1000 kilometers !!!

  10. mk

    The Trojan wave packet itself can detect the gravitoelectromagnetic force when two counter-rotating
    packets in linearly polarized electromagnetic field are interfering and the phase difference is
    accumulating over long period of time http://pra.aps.org/abstract/PRA/v57/i3/p2239_1 .
    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 !!!

  11. mk

    To cause the d= 1 mm deflection of the electron beam in the conventional cathode ray tube with
    the normal length l= 1m by the gravitoelectromagnetic force by the Amazon river flowing 10^8 kg/s it would have to operate on ultracol infinitesimal U= (me / 8 e) (dm/dt G/(2Pi r c^2)^2 l^4/d^2 = 10^-32 V Voltage and for the larger below !!!

  12. mk

    For huge quantum numbers the Trojan atom therefore turns out to be a single electron cathode
    ray tube so sensitive to detect the gravitoelectromagnetic force if existed and voltages below 10^-32 V with the nucleus as anode it will never reach and itself as the cathode where the anode-cathode voltage is rotating.

  13. mk

    It is also possible that Xerium 525 the hypothetical
    energy Ambrosia powering die Glocke http://www.unsolvedrealm.com/2011/01/12/the-nazi-bell-anti-gravity-discovered-show-me/ where the positron
    and electrons in so called Trojan states.
    They require rotating electromagnetic fields
    to keep electron and positron pairs frozen
    and not recombining in so called Trojan wave
    packet states http://www.amazon.es/Trojan-Wave-Packe t-Surhone-Lambert-M/dp/6130460 66X where their wave functions do not
    overlap at all which prevents recombination and
    the relativistic energy release. Positron propelled
    Haunebu could really fly since the energy release
    from positronic “Xerium” would be of few hundred of atomic bombs.
    Otto Schuman could in principle discover Trojan wave packets as classified even
    on Z-3 computer
    as the Nazi scientist and at least line drawings of Haunebu
    propelled by the Trojan positronium pair could be real: http://www.patentstorm.us/patents/7709819/description.html

  14. mk

    This scenario applied for Deuterium nuclei but in the opposite sense that the mutual orbit
    will be of the order of the wave packet spread can be used to provide the theory of the cold nuclear
    fusion. The double-star system mutual Deuterium-Deuterium circular Trojan
    orbits in ultra strong magnetic filed will cause the fusion rate corresponding the muon
    catalysis for the packets moving near the speed of light and the field strength one
    10^6 Teslas million of times less that this in neutron stars called magnetars.

  15. mk

    If die Glocke was the fussion reactor on the other hand first in addition to
    CP fields to generate Deuterium nuclei Trojan wave packets it would have to produce giant
    magnetic fields of the order o millions of Tesla but the current civilian Earth labs can produce barely
    1000 and in explosion spikes.

  16. mk

    What appears to be most interesting about Trojan
    wavepackets that they seem to be alive. One could
    even consider them as the simplest one electron life forms maybe
    even formed in the early universe full of light and the
    hydrogen. True or not they formally fulfill the definition
    of live: they are stable under slowly varying environment
    like changing the field strength and the frequency and
    the “multiply” when close to each other when the dipole
    moment of one will cause the hydrogen in neighbouring
    circular states to form the Trojan by the dipole induction. They grow under the light from totally spread
    to well focused and localized so the are like the plants.
    While propagating they exhibit internal jello-like
    shaky motion due to the influence of
    infinity of states crossing the Trojan lines with
    weaker and weaker avoided crossings. It appears
    that the life cannot originated on the planet without
    its rotation so the chemical Hamiltonians have
    the infinitesimal rotational corrections to cause
    the irregular spaghetti looking energy lines in
    the parameter spaces and
    stimulate life leading chemical attempts and this
    where the Trojan wave packets live in strongly rotating frame. Besides the simply look like alive Paramecium caudatum and their Kirlian spectrum which in the
    vacuum is spontaneous emission is strongly altered and suppressed. They are life or not Trojan wave packets
    are alive !!!

  17. mk

    Nonspreading wave packets of free moving near-luminal electrons
    are also possible withing the relativistic quantum mechanics
    in free space:

    In contrast to the nonrelativistic Schrödinger equation the Klein-Gordon equation predicts similar to Trojan wave packets and solitons nonspreading wave packets of the electron in free space.

    Let us construct in one dimension a general solution of the Klein-Gordon equation summing up the plane waves with the envelope

    phi(x,t) = \int dk f(k) e^{ik x-i\omega_k t}

    and let assume that the envelope f
    is well localized around some k_0
    and therefore it is for example the Gaussian function in k

    f(k)=g(k-k_0) = A e^{-(k-k_0)^2 /\delta k^2 }

    i.e. the significant contribution to the sum is brought only by waves with the wave vectors k from
    the neighborhood of the k_0.

    In the ultra-relativistic limit we can therefore assume

    \hbar k \approx \hbar k_0 >> m_0 c

    i.e that the kinetic energy is much larger than the rest one
    and then

    \omega_k \approx \pm c k

    For this wave packet the Klein-Gordon equation therefore effectively becomes simply the wave equation without the mass:

    \frac{\partial{}^2}{\partial x^2}\phi(x,t)-\frac{1}{c^2}\frac{\partial{}^2}{\partial t^2}\phi(x,t) = 0

    With the general non-dispersing solutions

    :\phi(x,t) = \Gamma(x \pm vt)
    v \approx c

    For the Gaussian envelope we therefore get as the solution nonspreading Gaussian wave packets:

    \phi(x,t) = B e^{i k_0 (x \pm vt) } e^{-(x \pm v t)/ \delta x^2}.

    It is the disappearance of spreading which is characteristic for the non-relativistic quantum mechanics
    when width of the \delta is constantly growing (or initially decreasing to grow later) during its steady motion.

    Therefore Trojan wave packets ionized coherently in-phase with
    the ultra strong ultra-short half-cycle pulse or the train of pulses
    and accelerated like the spaceship with orbit-tangent engine burst
    allow to produce no-dispersing near-luminal wave packets in
    free space which are Gausson or solitons but without the
    nonlinearity.

  18. mk

    Trojan wave packet
    written in basis of
    eigenstates shows why we all should be Vegan:

    Why we should not eat ourselves in NEXT: The quantum theory of the Universe predicts the immortality
    and the reincarnation:

    The theory of soul can be predicted already even without physical and the chemical details of brain
    functioning and construction. Let us take the Trojan Wave Packet http://www.amazon.es/Trojan-Wave-Packet…/dp/613046066X and suddenly remove the creating field. It is both a being and the soul in one
    in this example. The Hydrogen wave function spanned by the eigenstates have the time
    propagating phases Exp(i t/n^2). Therefore after a time being proportional to the lowest common multiple of n^2
    they will all return to 1 and the quantum states will exactly revive. Now if the wave function of the
    universe has rational energies while spanned it will also exactly revive and so the consciousness.
    But it the complicated evolution of the universe it may revive not precisely because of the cut-off
    and reality of the true energies so it may settle down approximately on the animal brain and reincarnate
    as well.

  19. mk

    Trojan wave packet written in basis of eigenstates shows why we all should be Vegan:

    Why we should not eat ourselves in NEXT: The quantum theory of the Universe predicts the immortality and the reincarnation:

    The theory of soul can be predicted already even without physical and the chemical details of brain
    functioning and construction. Let us take the Trojan Wave Packet http://www.amazon.es/Trojan-Wave-Packet…/dp/613046066X and suddenly remove the creating field. It is both a being and the soul in one in this example. The Hydrogen wave function spanned by the eigenstates have the time
    propagating phases Exp(i t/n^2). Therefore after a time being proportional to the lowest common multiple of n^2
    they will all return to 1 and the quantum states will exactly revive. Now if the wave function of the
    universe has rational energies while spanned it will also exactly revive and so the consciousness.
    But it the complicated evolution of the universe it may revive not precisely because of the cut-off
    and reality of the true energies so it may settle down approximately on the animal brain and reincarnate
    as well.

  20. mk

    Simplifying Trojan wave packet without the field looks like a ghost and it behaves like ghost. While it is generated and the field is tuned off it shows up at super-midnight when all phases are at the super-twelve with multiple hands of he component states pointing there.

  21. mk

    Simplifying Trojan wave packet without the field looks like a ghost and it behaves like ghost. While it is generated and the field is tuned off it shows up at super-midnight when all phases are at the super-twelve with multiple hands of he component states complex phases pointing there.

We respect your privacy.