Abstract

We present classical and quantum studies of the scattering dynamics of Rydberg electron wave packets from the electronic core of alkali atoms. In quantum systems an ideal state for studying such effects is an angularly localized wave packet in which the primary effect of the scattering is to cause precession. The scattering is enhanced by the application of an external dc electric field. We calculate and animate the field-induced dynamics of both hydrogenic and alkali wave packets and compare them to classical atomic models. We find that in alkali systems the scattered wavefunction can be divided into two components: one whose nearly hydrogenic behavior is due to quantum interference near the core, and another which exhibits the orbital precession found in classical models of nonhydrogenic atoms.

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References

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  1. M. Nauenburg, C. R. Stroud. Jr. and J. Yeazell, "The classical limit of an atom," Sci. Am. 270, 24-29 (1994).
  2. C. Raman, T. C. Weinacht and P. H. Bucksbaum, "Stark wave packets viewed with half-cycle pulses," Phys. Rev. A 55, R3995-R3998 (1997).
    [CrossRef]
  3. D. W. Schumacher, B. J. Lyons, and T. F. Gallagher, "Wave packets in perturbed Rydberg systems," Phys. Rev. Lett. 78, 4359-4362 (1997).
    [CrossRef]
  4. B. Hupper, J. Main, and G. Wunner, "Nonhydrogenic Rydberg atoms in a magnetic field: A rigorous semiclassical approach," Phys. Rev. A 53, 744-759 (1966).
    [CrossRef]
  5. P. A. Dando, T. S. Monteiro, D. Delande, and K. T. Taylor, "Role of core-scattered closed orbits in nonhydrogenic atoms," Phys. Rev. A , 54, 127-138 (1996).
    [CrossRef] [PubMed]
  6. M. Courtney, N. Spellmeyer, H. Jiao, and D. Kleppner, "Classical, semiclassical and quantum dynamics in the lithium Stark spectra," Phys. Rev. A , 51, 3604-3620 (1995); M. Courtney and D. Kleppner, "Core-induced chaos in diamagnetic lithium," Phys. Rev. A 53, 178-191 (1996).
    [CrossRef] [PubMed]
  7. J. Yeazell and C. R. Stroud, Jr., "Rydberg-atomwave packets localized in the angular variables," Phys. Rev. A 35, 2806-2809 (1987); "Observation of spatially localized atomic electron wave packets," Phys. Rev. Lett. 60, 1494-1497 (1988).
    [CrossRef] [PubMed]
  8. R. Bluhm and V. A. Kostelecky, "Long-term evolution and revival structure of Rydberg wave packets for hydrogen and alkali-metal atoms," Phys. Rev. A , 51, 4767-86 (1995).
    [CrossRef] [PubMed]
  9. J. Gay, D. Delande, and A. Bommier, "Atomic quantum states with maximum localization on classical elliptical orbits," Phys. Rev. A 39, 6587 (1989).
    [CrossRef] [PubMed]
  10. N. Bohr, "On the quantum theory of line spectra," in Niels Bohr, Collected Works, editedbyJ. Nielsen (North-Holland Pub. Co., New York, 1976), Vol. 3.
  11. T. P. Hezel, C. E. Burkhardt, M. Ciocca, and J. J. Leventhal, "Classical view of the Stark effect in hydrogen atoms," Am. J. Phys. 60, 324-328 (1992).
    [CrossRef]
  12. A. Hooker, C. H. Greene, and W. Clark, "Classical examination of the Stark effect in hydrogen," Phys. Rev. A 55, 4609-4612 (1997).
    [CrossRef]
  13. J. Finkelstein, Nonrelativistic Mechanics, (W. A. Benjamin, Inc., Reading, MA, 1973) 298.
  14. Goldstein, Classical Mechanics, (Addison-Wesley, Reading, MA, 1981) 509.
  15. P. Hezel, C. E. Burkhardt, M. Ciocca, and J. J. Leventhal, "Classical view of the properties of Rydberg atoms: Application of the correspondence principle," Am. J. Phys. 60, 329-335 (1992).
    [CrossRef]
  16. Pascale, R. E. Olson and C.O. Reinhold, "State-selective capture in collisions between ions and ground- and excited-state alkali-metal atoms," Phys. Rev. A 42, 5305-5314 (1990).
    [CrossRef] [PubMed]
  17. N. Bardsley, "Pseudopotentials in atomic and molecular physics," Case Studies in Atomic Physics, 4 299-368 (1974).
  18. L. Zimmerman, M. G. Littman, M. M. Kash, and D. Kleppner, "Stark structure of the Rydberg states of alkali-metal atoms," Phys. Rev. A 20, 2251-2275 (1979).
    [CrossRef]

Other (18)

M. Nauenburg, C. R. Stroud. Jr. and J. Yeazell, "The classical limit of an atom," Sci. Am. 270, 24-29 (1994).

C. Raman, T. C. Weinacht and P. H. Bucksbaum, "Stark wave packets viewed with half-cycle pulses," Phys. Rev. A 55, R3995-R3998 (1997).
[CrossRef]

D. W. Schumacher, B. J. Lyons, and T. F. Gallagher, "Wave packets in perturbed Rydberg systems," Phys. Rev. Lett. 78, 4359-4362 (1997).
[CrossRef]

B. Hupper, J. Main, and G. Wunner, "Nonhydrogenic Rydberg atoms in a magnetic field: A rigorous semiclassical approach," Phys. Rev. A 53, 744-759 (1966).
[CrossRef]

P. A. Dando, T. S. Monteiro, D. Delande, and K. T. Taylor, "Role of core-scattered closed orbits in nonhydrogenic atoms," Phys. Rev. A , 54, 127-138 (1996).
[CrossRef] [PubMed]

M. Courtney, N. Spellmeyer, H. Jiao, and D. Kleppner, "Classical, semiclassical and quantum dynamics in the lithium Stark spectra," Phys. Rev. A , 51, 3604-3620 (1995); M. Courtney and D. Kleppner, "Core-induced chaos in diamagnetic lithium," Phys. Rev. A 53, 178-191 (1996).
[CrossRef] [PubMed]

J. Yeazell and C. R. Stroud, Jr., "Rydberg-atomwave packets localized in the angular variables," Phys. Rev. A 35, 2806-2809 (1987); "Observation of spatially localized atomic electron wave packets," Phys. Rev. Lett. 60, 1494-1497 (1988).
[CrossRef] [PubMed]

R. Bluhm and V. A. Kostelecky, "Long-term evolution and revival structure of Rydberg wave packets for hydrogen and alkali-metal atoms," Phys. Rev. A , 51, 4767-86 (1995).
[CrossRef] [PubMed]

J. Gay, D. Delande, and A. Bommier, "Atomic quantum states with maximum localization on classical elliptical orbits," Phys. Rev. A 39, 6587 (1989).
[CrossRef] [PubMed]

N. Bohr, "On the quantum theory of line spectra," in Niels Bohr, Collected Works, editedbyJ. Nielsen (North-Holland Pub. Co., New York, 1976), Vol. 3.

T. P. Hezel, C. E. Burkhardt, M. Ciocca, and J. J. Leventhal, "Classical view of the Stark effect in hydrogen atoms," Am. J. Phys. 60, 324-328 (1992).
[CrossRef]

A. Hooker, C. H. Greene, and W. Clark, "Classical examination of the Stark effect in hydrogen," Phys. Rev. A 55, 4609-4612 (1997).
[CrossRef]

J. Finkelstein, Nonrelativistic Mechanics, (W. A. Benjamin, Inc., Reading, MA, 1973) 298.

Goldstein, Classical Mechanics, (Addison-Wesley, Reading, MA, 1981) 509.

P. Hezel, C. E. Burkhardt, M. Ciocca, and J. J. Leventhal, "Classical view of the properties of Rydberg atoms: Application of the correspondence principle," Am. J. Phys. 60, 329-335 (1992).
[CrossRef]

Pascale, R. E. Olson and C.O. Reinhold, "State-selective capture in collisions between ions and ground- and excited-state alkali-metal atoms," Phys. Rev. A 42, 5305-5314 (1990).
[CrossRef] [PubMed]

N. Bardsley, "Pseudopotentials in atomic and molecular physics," Case Studies in Atomic Physics, 4 299-368 (1974).

L. Zimmerman, M. G. Littman, M. M. Kash, and D. Kleppner, "Stark structure of the Rydberg states of alkali-metal atoms," Phys. Rev. A 20, 2251-2275 (1979).
[CrossRef]

Supplementary Material (2)

» Media 1: MOV (465 KB)     
» Media 2: MOV (531 KB)     

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Figures (6)

Figure 1:
Figure 1:

Numerical simulations of a classical electron trajectory showing Stark oscillations in a dc electric field. The electron is initially in a circular orbit in the xy-plane with n = 15 and Lz = |L| = 15. An 800 V/cm dc electric field is in the x-direction and the resulting Stark period Ts is approximately 84 Kepler periods. The first few orbits are colored red for clarity and the trajectory is followed for slightly less than half of a Stark period. a) The trajectory for a pure Coulomb potential. The electron backscatters in the linear obit and nearly retraces the original trajectory in the opposite direction. b) The trajectory in the sodium model potential. Note the precession of the highly eccentric orbit.

Figure 2:
Figure 2:

The classical dynamics of the angular momentum for the trajectories shown in Fig. 1. The evolution is followed for two complete Stark periods. The dashed curve is hydrogen and the solid curve is for the sodium model potential. Note that the sodium curve has a shorter Stark period because the scattering occurs at a non-zero value of the angular momentum.

Figure 3:
Figure 3:

An animation comparing the quantum Stark dynamics in hydrogen and sodium. In both cases the electron is initially in a circular state in the xy plane with n = 15 and 〈Lz 〉 = 14 with an 800 V/cm dc electric field in the x-direction. a) In hydrogen the quantum behavior is identical to the classical Stark evolution shown in Fig. 1a. b) The behavior in sodium is more complex. Note how part of the wave function behaves hydrogenically while the remainder shows the rapid precession of the non-Coulombic classical orbits in Fig. 1b. [Media 1]

Figure 4:
Figure 4:

The quantum mechanical evolution of the expectation value of the z-component of the angular momentum for the wavefunction dynamics shown in the animations (Figs. 3 and 6). The evolution is followed for one Stark period. a) 〈Lz (t)〉 for the states shown in Fig. 3. The dashed curve is hydrogen and the solid curve is sodium. b) The evolution for core-scattered wave packets shown in Fig. 6. The dashed is the wave packet given initially by m < 0 states and the solid curve is the wave packet given initially by m ≥ 0 states.

Figure 5:
Figure 5:

The ratio of the scattered wavefunction (m > 0) to the unscattered wavefunction (m ≤ 0) is plotted as a function of the Stark period. Classically for larger Stark periods the electron spends more time in the low angular momentum orbit. Quantum mechanically this extended period of scattering leads to an increase in the scattered fraction of the wavefunction.

Figure 6:
Figure 6:

An animation comparing the quantum Stark dynamics of the core-scattered wave packet (initially m ≥ 0 states) and the “hydrogenic” wave packet (initially m < 0 states). In both cases the dc electric field strength is 800 V/cm in the x-direction. a) This wave packet isolates the effect of core scattering and its behavior is very similar to the classical trajectory in Fig. 1b. b) Interference near the core minimizes core scattering and leads to nearly hydrogenic behavior. [Media 2]

Equations (5)

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H = p 2 2 + V ( r ) + r · E .
A ˙ T K = 3 2 L T K × E ( t ) ,
L ˙ T K = 3 n 2 2 A T K × E ( t ) ,
V ( r ) = 1 r [ 1 + 10 ( 1 + ar + br 2 ) e cr ] 1 2 α d r 2 ( r 3 + r c 3 ) 2 1 2 α′ q r 3 ( r 3 + r c 3 ) 3 ,
A ˙ = 1 r dr L × r .

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