Abstract

In Bohr’s original planetary model of the atom the electron moves along orbits of special geometric simplicity. While wave mechanics precludes the idea that a physical path could be ascribed to the electron, a classical or planetary atom can still be envisaged in which the electronic wavepacket neither spreads nor disperses as its center moves along the Kepler orbit, and this orbit is confined to a single plane in space. We show theoretically how an electronic wavepacket may be localized in this fashion in a similar way to ion confinement in a Penning trap. Because external fields are needed to keep the packet confined, a more fitting analogy than a planetary orbit is the motion of a charged dust grain in one of the rings of a giant planet such as Saturn.

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References

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  1. E. Rutherford, Phil. Mag. 21, 669 (1911).
    [CrossRef]
  2. E. Nagaoka, "Kinetics of a system of particles illustrating the line and the band spectrum and the phenomena of radioactivity", Phil. Mag. 7, 445-455 (1904).
    [CrossRef]
  3. N. Bohr, Phil. Mag. 26, 1-25 (1913); ibid., 476-502; ibid., 857-875.
    [CrossRef]
  4. M. Born, The Mechanics of the Atom, (republished by F. Ungar, New York, 1960) (translated by J.W. Fisher); pp. 130-241.
  5. G. Raithel, M. Fauth, H. Walther, " Atoms in strong crossed electric and magnetic fields: Evidence for states with large electric-dipole moments", Phys. Rev. A 47 , 419-440 (1991).
    [CrossRef]
  6. H. Dehmelt, "Nobel Prize Lecture", Rev. Mod. Phys. 62, 525-531 (1992).
    [CrossRef]
  7. J. A. Burns, L. Schaer, "Orbital evolution of circumplanetary dust by resonant charge variations", Nature 337, 340-343 (1989).
    [CrossRef]
  8. L. Schaer, L. and Burns, "Charged dust in planetary magnetospheres: Hamiltonian dynamics and numerical simulations for highly charged grains", J. Geophys. Res. 99, 17211-17223 (1994).
    [CrossRef]
  9. T. F. Gallagher, Rydberg Atoms, (Cambridge University Press, Cambridge, 1994).
    [CrossRef]
  10. M. Nauenberg, "Quantum wave packets on Kepler elliptic orbits", Phys. Rev. A 40, 1133-1136 (1989).
    [CrossRef] [PubMed]
  11. A. O. Barut, B. W. Xu, " Non-spreading coherent states riding on Kepler orbits", Helv. Phys. Act. 66, 712-720 (1993).
  12. J. A. Yeazell, C. R. Stroud, "Observation of fractional revivals in the evolution of a Rydberg atomic wave packet", Phys. Rev. A 43, 5153-5156 (1991).
    [CrossRef] [PubMed]
  13. P. Fu, T. J. Scholz, J. M. Hettema, T. F. Gallagher, "Ionization of Rydberg atoms by circularly polarized microwave field", Phys. Rev. Lett. 64, 511-514 (1990).
    [CrossRef] [PubMed]
  14. C. H. Cheng, C. Y. Lee, T. F. Gallagher, "Production of circular Rydberg states with circularly polarized microwave fields", Phys. Rev. Lett. 73, 3078-3081 (1994).
    [CrossRef] [PubMed]
  15. M. Nauenberg, Comment on "Ionization of Rydberg states by a circularly polarized microwave field", Phys. Rev. Lett. 64, 2731 (1990).
    [CrossRef] [PubMed]
  16. M. Nauenberg, "Canonical Kepler map", Europhys. Lett. 13, 611-616 (1990).
    [CrossRef]
  17. P. Kappertz, M. Nauenberg, "Circularly polarized microwave ionization of hydrogen", Phys. Rev. A 47, 4749-4755 (1993).
    [CrossRef] [PubMed]
  18. V. Szebehely, Theory of orbits: The restricted problem of three bodies, (Academic, New York and London, 1967).
  19. D. Farrelly and T. Uzer, "Ionization mechanism of Rydberg atoms in a circulary polarized microwave field",Phys. Rev. Lett. 74, 1720-1723 (1995).
    [CrossRef] [PubMed]
  20. I. Bialynicki-Birula, M. Kalinski, J. H. Eberly, "Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons", Phys. Rev. Lett. 73, 1777-1780 (1994).
    [CrossRef] [PubMed]
  21. M. Kalinski, J. H. Eberly, I. Bialynicki-Birula,"Numerical observation of stable field supported Rydberg wave packets", Phys. Rev. A 52, 2460-2463 (1995).
    [CrossRef]
  22. M. Kalinski, J. H. Eberly, "Trojan wave packets: Mathieu theory and generation from circular states", Phys. Rev. A 53, 1715-1724 (1996).
    [CrossRef]
  23. M. Kalinski, J. H. Eberly,"New states of hydrogen in a circulary polarized microwave field", Phys. Rev. Lett. 77, 2420-2423 (1995).
    [CrossRef]
  24. D. Farrelly, E. Lee, T. Uzer, Comment on "Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons", Phys. Rev. Lett. 75, 972 (1995).
    [CrossRef] [PubMed]
  25. I. Bialynicki-Birula, M. Kalinski, J. H. Eberly, Reply to Ref. 22, Phys. Rev. Lett. 75, 973 (1995).
    [CrossRef] [PubMed]
  26. A. Buchleitner, These de doctorat, Universite Pierre et Marie Currie, Paris, 1993 (unpublished).
  27. A. Buchleitner, D. Delande, "Nondispersive electronic wave packets in multiphoton processes", Phys. Rev. Lett. 75, 1487-1490 (1995).
    [CrossRef] [PubMed]
  28. J. Zakrzewski, D. Delande, A. Buchleitner, "Nonspreading electronic wave packets and conductance uctuations", Phys. Rev. Lett. 75, 4015-4018 (1995).
    [CrossRef] [PubMed]
  29. D. Delande, J. Zakrzewski, A. Buchleitner, " A wave packet can be a stationary state", Europhys. Lett. 32, 107-112 (1995).
    [CrossRef]
  30. D. Farrelly, E. Lee, T. Uzer, "Magnetic field stabilization of Rydberg wavepackets in a circulary polarized microwave field", Phys. Lett. A 204, 359-372 (1995).
    [CrossRef]
  31. E. Lee, A. F. Brunello, D. Farrelly, "A single atom Quasi-Penning trap", Phys. Rev. Lett. 75, 3641-3644 (1995).
    [CrossRef] [PubMed]
  32. A. F. Brunello, D. Farrelly, T. Uzer, "Nonstationary, nondispersive wave packets in a Rydberg atom", Phys. Rev. Lett. 76, 2874-2877 (1996).
    [CrossRef] [PubMed]
  33. E. Lee, A. F. Brunello, D. Farrelly, "Coherent states in a Rydberg atom: Classical mechanics", Phys. Rev. A 55, 2203-2221 (1997).
    [CrossRef]
  34. C. Cerjan, E. Lee, D. Farrelly, T. Uzer, "Coherent states in a Rydberg atom: Quantum mechanics", Phys. Rev. A 55, 2222-2231 (1997).
    [CrossRef]
  35. K. Hornberger, A. Buchleitner, "Spontaneous decay of nondispersive wave packets", (to be published).
  36. Z. Bialynicki-Birula, I. Bialynicki-Birula, "Radiative decay of Trojan wave packets", (to be published).
  37. G. W. Hill, Am. J. Math.1, 5-128, (1878).
    [CrossRef]
  38. R. Greenberg, D. R. Davis, "Stability at potential maxima: The L4 and L5 points of the Restricted Three-Body Problem", Am. J. Phys.46, 1068-1070, (1978).
    [CrossRef]
  39. A. Deprit, in The Big Bang and George Lemaitre, edt. A. Berger, (Reidel, Dordrecht, 1984), pp. 151-180.

Other (39)

E. Rutherford, Phil. Mag. 21, 669 (1911).
[CrossRef]

E. Nagaoka, "Kinetics of a system of particles illustrating the line and the band spectrum and the phenomena of radioactivity", Phil. Mag. 7, 445-455 (1904).
[CrossRef]

N. Bohr, Phil. Mag. 26, 1-25 (1913); ibid., 476-502; ibid., 857-875.
[CrossRef]

M. Born, The Mechanics of the Atom, (republished by F. Ungar, New York, 1960) (translated by J.W. Fisher); pp. 130-241.

G. Raithel, M. Fauth, H. Walther, " Atoms in strong crossed electric and magnetic fields: Evidence for states with large electric-dipole moments", Phys. Rev. A 47 , 419-440 (1991).
[CrossRef]

H. Dehmelt, "Nobel Prize Lecture", Rev. Mod. Phys. 62, 525-531 (1992).
[CrossRef]

J. A. Burns, L. Schaer, "Orbital evolution of circumplanetary dust by resonant charge variations", Nature 337, 340-343 (1989).
[CrossRef]

L. Schaer, L. and Burns, "Charged dust in planetary magnetospheres: Hamiltonian dynamics and numerical simulations for highly charged grains", J. Geophys. Res. 99, 17211-17223 (1994).
[CrossRef]

T. F. Gallagher, Rydberg Atoms, (Cambridge University Press, Cambridge, 1994).
[CrossRef]

M. Nauenberg, "Quantum wave packets on Kepler elliptic orbits", Phys. Rev. A 40, 1133-1136 (1989).
[CrossRef] [PubMed]

A. O. Barut, B. W. Xu, " Non-spreading coherent states riding on Kepler orbits", Helv. Phys. Act. 66, 712-720 (1993).

J. A. Yeazell, C. R. Stroud, "Observation of fractional revivals in the evolution of a Rydberg atomic wave packet", Phys. Rev. A 43, 5153-5156 (1991).
[CrossRef] [PubMed]

P. Fu, T. J. Scholz, J. M. Hettema, T. F. Gallagher, "Ionization of Rydberg atoms by circularly polarized microwave field", Phys. Rev. Lett. 64, 511-514 (1990).
[CrossRef] [PubMed]

C. H. Cheng, C. Y. Lee, T. F. Gallagher, "Production of circular Rydberg states with circularly polarized microwave fields", Phys. Rev. Lett. 73, 3078-3081 (1994).
[CrossRef] [PubMed]

M. Nauenberg, Comment on "Ionization of Rydberg states by a circularly polarized microwave field", Phys. Rev. Lett. 64, 2731 (1990).
[CrossRef] [PubMed]

M. Nauenberg, "Canonical Kepler map", Europhys. Lett. 13, 611-616 (1990).
[CrossRef]

P. Kappertz, M. Nauenberg, "Circularly polarized microwave ionization of hydrogen", Phys. Rev. A 47, 4749-4755 (1993).
[CrossRef] [PubMed]

V. Szebehely, Theory of orbits: The restricted problem of three bodies, (Academic, New York and London, 1967).

D. Farrelly and T. Uzer, "Ionization mechanism of Rydberg atoms in a circulary polarized microwave field",Phys. Rev. Lett. 74, 1720-1723 (1995).
[CrossRef] [PubMed]

I. Bialynicki-Birula, M. Kalinski, J. H. Eberly, "Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons", Phys. Rev. Lett. 73, 1777-1780 (1994).
[CrossRef] [PubMed]

M. Kalinski, J. H. Eberly, I. Bialynicki-Birula,"Numerical observation of stable field supported Rydberg wave packets", Phys. Rev. A 52, 2460-2463 (1995).
[CrossRef]

M. Kalinski, J. H. Eberly, "Trojan wave packets: Mathieu theory and generation from circular states", Phys. Rev. A 53, 1715-1724 (1996).
[CrossRef]

M. Kalinski, J. H. Eberly,"New states of hydrogen in a circulary polarized microwave field", Phys. Rev. Lett. 77, 2420-2423 (1995).
[CrossRef]

D. Farrelly, E. Lee, T. Uzer, Comment on "Lagrange equilibrium points in celestial mechanics and nonspreading wave packets for strongly driven Rydberg electrons", Phys. Rev. Lett. 75, 972 (1995).
[CrossRef] [PubMed]

I. Bialynicki-Birula, M. Kalinski, J. H. Eberly, Reply to Ref. 22, Phys. Rev. Lett. 75, 973 (1995).
[CrossRef] [PubMed]

A. Buchleitner, These de doctorat, Universite Pierre et Marie Currie, Paris, 1993 (unpublished).

A. Buchleitner, D. Delande, "Nondispersive electronic wave packets in multiphoton processes", Phys. Rev. Lett. 75, 1487-1490 (1995).
[CrossRef] [PubMed]

J. Zakrzewski, D. Delande, A. Buchleitner, "Nonspreading electronic wave packets and conductance uctuations", Phys. Rev. Lett. 75, 4015-4018 (1995).
[CrossRef] [PubMed]

D. Delande, J. Zakrzewski, A. Buchleitner, " A wave packet can be a stationary state", Europhys. Lett. 32, 107-112 (1995).
[CrossRef]

D. Farrelly, E. Lee, T. Uzer, "Magnetic field stabilization of Rydberg wavepackets in a circulary polarized microwave field", Phys. Lett. A 204, 359-372 (1995).
[CrossRef]

E. Lee, A. F. Brunello, D. Farrelly, "A single atom Quasi-Penning trap", Phys. Rev. Lett. 75, 3641-3644 (1995).
[CrossRef] [PubMed]

A. F. Brunello, D. Farrelly, T. Uzer, "Nonstationary, nondispersive wave packets in a Rydberg atom", Phys. Rev. Lett. 76, 2874-2877 (1996).
[CrossRef] [PubMed]

E. Lee, A. F. Brunello, D. Farrelly, "Coherent states in a Rydberg atom: Classical mechanics", Phys. Rev. A 55, 2203-2221 (1997).
[CrossRef]

C. Cerjan, E. Lee, D. Farrelly, T. Uzer, "Coherent states in a Rydberg atom: Quantum mechanics", Phys. Rev. A 55, 2222-2231 (1997).
[CrossRef]

K. Hornberger, A. Buchleitner, "Spontaneous decay of nondispersive wave packets", (to be published).

Z. Bialynicki-Birula, I. Bialynicki-Birula, "Radiative decay of Trojan wave packets", (to be published).

G. W. Hill, Am. J. Math.1, 5-128, (1878).
[CrossRef]

R. Greenberg, D. R. Davis, "Stability at potential maxima: The L4 and L5 points of the Restricted Three-Body Problem", Am. J. Phys.46, 1068-1070, (1978).
[CrossRef]

A. Deprit, in The Big Bang and George Lemaitre, edt. A. Berger, (Reidel, Dordrecht, 1984), pp. 151-180.

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Figures (4)

Fig. 1.
Fig. 1.

Effective potential (V) with ωc = 3.46T, ωf =50 GHz, and F = 2000V/cm. Energy and distance are in atomic units (a.u.) A section (y = z = 0) through the potential is shown. Also plotted is the harmonic approximation (Vho ) to the potential and the probability density (∣Ψ∣2) of the corresponding vacuum state, which in the laboratory frame constitutes our wave packet. For snapshots of its progress on its orbit, see Fig.3 of Ref. 34.

Fig. 2.
Fig. 2.

Level curves of the potential together with contours (at 0.25, 0.5, 0.75, 0.95) of the vacuum coherent state as obtained by Taylor expansion about the minimum. The parameters are the same as Fig. 1. The outer minimum exists provided F > F c = 3 [ ω f ( ω c ω f ) ] 2 / 3 4 : 3 the well depth, its distance from the nucleus and the width of the barrier all depend sensitively on the fields used, providing considerable flexibility in the selection of appropriate experimental parameters.

Fig. 3:
Fig. 3:

Combined Poincaré surface of sections for 10 classical trajectories obtained by integrating Hamilton’s equations for Eq. (3) with ϵ = 1, Ω = 1/2, (for which value the velocity dependent forces are eliminated) and �� = -2.1.

Fig. 4:
Fig. 4:

Combined Poincaré surface of sections for 75 classical trajectories obtained by integrating Hamilton’s equations for Eq. (3) with ϵ = 0.9, Ω = 0.65, and �� = -1.85. The points generated by each trajectory have been assigned a different color. This allows one to pick out, e.g., groups of islands related by single resonance.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

H = p ˜ 2 2 1 r + ω c 2 ( x p y y p x ) + ω c 2 8 ( x 2 + y 2 ) F ( x cos ω f t + y sin ω f t ) .
H = K = p ˜ 2 2 1 r ( ω f ω c 2 ) ( x p y y p x ) + ω c 2 8 ( x 2 + y 2 ) F x
r ' = ω c 2 3 r , p ' = ω c 1 3 p .
H = 𝐾 = 1 2 ( p x 2 + p y 2 ) 1 r ( Ω 1 2 ) ( x p y y p x ) + 1 8 ( x 2 + y 2 ) ϵ x
V = H ( x ˙ 2 + y ˙ 2 ) 2 = 1 r + ω f ( ω c ω f ) 2 ( x 2 + y 2 ) F x

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