Abstract

We investigate numerically the solution of Dirac equation and analytically the Klein-Gordon equation and discuss the relativistic motion of an electron wave packet in the presence of an intense static electric field. In contrast to the predictions of the (non-relativistic) Schrödinger theory, the spreading rate in the field’s polarization direction as well as in the transverse directions is reduced.

© 1998 Optical Society of America

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References

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  1. H.R. Reiss, “Relativistic Strong-Field Photoionization”, J. Opt. Soc. Am. B 7, 574 (1990).
    [Crossref]
  2. H.R. Reiss, “Theoretical Methods in Quantum Optics: S-Matrix and Keldysh Techniques for Strong-Field Processes”, Prog. Quantum Electron. 16, 1 (1992).
    [Crossref]
  3. M.D. Perry and G. Mourou, “Terawatt to Petawatt Subpicosecond Lasers”, Science 264, 917 (1994)
    [Crossref] [PubMed]
  4. K. Boyer and C.K. Rhodes, “Superstrong Coherent Multi-Electron Intense-Field Interaction”, J. Phys. B 27, L633 (1994)
    [Crossref]
  5. C.I. Moore, J.P. Knauer, and D.D. Meyerhofer, “Observation of the Transition from Thomson to Compton Scattering in Multiphoton Interactions with Low-Energy Electrons”, Phys. Rev. Lett. 74, 2439 (1995)
    [Crossref] [PubMed]
  6. P. Monot, T. Auguste, and J. L. Miquel, “Experimental Demonstration of Relativistic Self-Channeling for a Multiterawatt Laser Pulse in an Underdense Plasma”, Phys. Rev. Lett. 74, 2953 (1995).
    [Crossref] [PubMed]
  7. “Atoms in Intense Laser Fields”, ed. M. Gavrila (Academic Press, Boston1992).
  8. Q. Su, J.H. Eberly, and J. Javanainen, “Dynamics of Atomic Ionization Suppression and Electron Localization in an Intense High-Frequency Radiation Field”, Phys. Rev. Lett. 64, 862 (1990)
    [Crossref] [PubMed]
  9. M. Dšrr, R.M. Potvliege, and R. Shakeshaft, “Tunneling Ionization of Atomc Hydrogen by an Intense Low-Frequency Field”, Phys. Rev. Lett. 64, 2003 (1990)
    [Crossref]
  10. K.C. Kulander, K.J. Schafer, and J.L. Krause, “Dynamic Stabilization of Hydrogen in an Intense High Frequency, Pulsed Laser Field”, Phys. Rev. Lett. 66, 2601 (1991)
    [Crossref] [PubMed]
  11. M.P. De Boer, J. H. Hoogenraad, and R. B. Vrijen, “Indications of High-Intensity Adiabatic Stabilization in Neon”, Phys. Rev. Lett. 71, 3263 (1993)
    [Crossref] [PubMed]
  12. N.J van Druten, R. C. Constantinescu, and H. G. Muller, “Adiabatic Stabilization: Observation of the Surviving Population”, Phys. Rev. A 55, 622 (1997).
    [Crossref]
  13. J. Grochmalicki, Maciej A. Lewenstein, Martin Wilkens, and Kazimierz Rzazewski, “Beyond Above-Threshold Ionization: Ionization of an Atom by an Ultrashort Laser Pulse Above Atomic Intensity”, J. Opt. Soc. Am. B 7, 607 (1990)
    [Crossref]
  14. F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Field”, Phys. Rev. A 47,4464 (1993)
    [Crossref] [PubMed]
  15. F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Pulse. II”, Phys. Rev. A 48, 554 (1993)
    [Crossref] [PubMed]
  16. M. Horbatsch, “Magnetic Field Effects in High-Frequency Photoionization by Intense Laser Pulses”, Z. Phys. D 25, 305 (1993).
    [Crossref]
  17. D.P. Crawford and H.R. Reiss, “Stabilization in Relavistic Photoionization with Circular Polarized Light”, Phys. Rev. A 50, 1844 (1994)
    [Crossref] [PubMed]
  18. M. Protopapas, C.H. Keitel, and P.L. Knight, “Relativistic Mass Shift Effects in Adiabatic Intense Laser Field Stabilization of Atoms”, J. Phys. B 29, L591 (1996).
    [Crossref]
  19. J.H. Eberly, Prog. Opt.7, 359 ed. E. Wolf (Amsterdam: North-Holland1969);
    [Crossref]
  20. R. Grobe and M.V. Fedorov, “Packet Spreading, Stabilization, and Localization in Superstrong Fields”, Phys. Rev. Lett 68, 2592 (1992)
    [Crossref] [PubMed]
  21. Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields”, Laser Phys. (in press).
  22. J.D. Bjorken and S.D. Drell, “Relativistic Quantum Mechanics” (McGraw-Hill, New York1964).
  23. B.A. Smetanko, Q. Su, and R. Grobe, “FFT Based Split Operator Techniques for Solving the Dirac Equation”, Comm. Comp. Phys. (to be submitted).

1997 (1)

N.J van Druten, R. C. Constantinescu, and H. G. Muller, “Adiabatic Stabilization: Observation of the Surviving Population”, Phys. Rev. A 55, 622 (1997).
[Crossref]

1996 (1)

M. Protopapas, C.H. Keitel, and P.L. Knight, “Relativistic Mass Shift Effects in Adiabatic Intense Laser Field Stabilization of Atoms”, J. Phys. B 29, L591 (1996).
[Crossref]

1995 (2)

C.I. Moore, J.P. Knauer, and D.D. Meyerhofer, “Observation of the Transition from Thomson to Compton Scattering in Multiphoton Interactions with Low-Energy Electrons”, Phys. Rev. Lett. 74, 2439 (1995)
[Crossref] [PubMed]

P. Monot, T. Auguste, and J. L. Miquel, “Experimental Demonstration of Relativistic Self-Channeling for a Multiterawatt Laser Pulse in an Underdense Plasma”, Phys. Rev. Lett. 74, 2953 (1995).
[Crossref] [PubMed]

1994 (3)

M.D. Perry and G. Mourou, “Terawatt to Petawatt Subpicosecond Lasers”, Science 264, 917 (1994)
[Crossref] [PubMed]

K. Boyer and C.K. Rhodes, “Superstrong Coherent Multi-Electron Intense-Field Interaction”, J. Phys. B 27, L633 (1994)
[Crossref]

D.P. Crawford and H.R. Reiss, “Stabilization in Relavistic Photoionization with Circular Polarized Light”, Phys. Rev. A 50, 1844 (1994)
[Crossref] [PubMed]

1993 (4)

F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Field”, Phys. Rev. A 47,4464 (1993)
[Crossref] [PubMed]

F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Pulse. II”, Phys. Rev. A 48, 554 (1993)
[Crossref] [PubMed]

M. Horbatsch, “Magnetic Field Effects in High-Frequency Photoionization by Intense Laser Pulses”, Z. Phys. D 25, 305 (1993).
[Crossref]

M.P. De Boer, J. H. Hoogenraad, and R. B. Vrijen, “Indications of High-Intensity Adiabatic Stabilization in Neon”, Phys. Rev. Lett. 71, 3263 (1993)
[Crossref] [PubMed]

1992 (2)

R. Grobe and M.V. Fedorov, “Packet Spreading, Stabilization, and Localization in Superstrong Fields”, Phys. Rev. Lett 68, 2592 (1992)
[Crossref] [PubMed]

H.R. Reiss, “Theoretical Methods in Quantum Optics: S-Matrix and Keldysh Techniques for Strong-Field Processes”, Prog. Quantum Electron. 16, 1 (1992).
[Crossref]

1991 (1)

K.C. Kulander, K.J. Schafer, and J.L. Krause, “Dynamic Stabilization of Hydrogen in an Intense High Frequency, Pulsed Laser Field”, Phys. Rev. Lett. 66, 2601 (1991)
[Crossref] [PubMed]

1990 (4)

J. Grochmalicki, Maciej A. Lewenstein, Martin Wilkens, and Kazimierz Rzazewski, “Beyond Above-Threshold Ionization: Ionization of an Atom by an Ultrashort Laser Pulse Above Atomic Intensity”, J. Opt. Soc. Am. B 7, 607 (1990)
[Crossref]

H.R. Reiss, “Relativistic Strong-Field Photoionization”, J. Opt. Soc. Am. B 7, 574 (1990).
[Crossref]

Q. Su, J.H. Eberly, and J. Javanainen, “Dynamics of Atomic Ionization Suppression and Electron Localization in an Intense High-Frequency Radiation Field”, Phys. Rev. Lett. 64, 862 (1990)
[Crossref] [PubMed]

M. Dšrr, R.M. Potvliege, and R. Shakeshaft, “Tunneling Ionization of Atomc Hydrogen by an Intense Low-Frequency Field”, Phys. Rev. Lett. 64, 2003 (1990)
[Crossref]

Auguste, T.

P. Monot, T. Auguste, and J. L. Miquel, “Experimental Demonstration of Relativistic Self-Channeling for a Multiterawatt Laser Pulse in an Underdense Plasma”, Phys. Rev. Lett. 74, 2953 (1995).
[Crossref] [PubMed]

Bjorken, J.D.

J.D. Bjorken and S.D. Drell, “Relativistic Quantum Mechanics” (McGraw-Hill, New York1964).

Boyer, K.

K. Boyer and C.K. Rhodes, “Superstrong Coherent Multi-Electron Intense-Field Interaction”, J. Phys. B 27, L633 (1994)
[Crossref]

Constantinescu, R. C.

N.J van Druten, R. C. Constantinescu, and H. G. Muller, “Adiabatic Stabilization: Observation of the Surviving Population”, Phys. Rev. A 55, 622 (1997).
[Crossref]

Crawford, D.P.

D.P. Crawford and H.R. Reiss, “Stabilization in Relavistic Photoionization with Circular Polarized Light”, Phys. Rev. A 50, 1844 (1994)
[Crossref] [PubMed]

De Boer, M.P.

M.P. De Boer, J. H. Hoogenraad, and R. B. Vrijen, “Indications of High-Intensity Adiabatic Stabilization in Neon”, Phys. Rev. Lett. 71, 3263 (1993)
[Crossref] [PubMed]

Drell, S.D.

J.D. Bjorken and S.D. Drell, “Relativistic Quantum Mechanics” (McGraw-Hill, New York1964).

Dšrr, M.

M. Dšrr, R.M. Potvliege, and R. Shakeshaft, “Tunneling Ionization of Atomc Hydrogen by an Intense Low-Frequency Field”, Phys. Rev. Lett. 64, 2003 (1990)
[Crossref]

Eberly, J.H.

Q. Su, J.H. Eberly, and J. Javanainen, “Dynamics of Atomic Ionization Suppression and Electron Localization in an Intense High-Frequency Radiation Field”, Phys. Rev. Lett. 64, 862 (1990)
[Crossref] [PubMed]

J.H. Eberly, Prog. Opt.7, 359 ed. E. Wolf (Amsterdam: North-Holland1969);
[Crossref]

Faisal, F.H.M.

F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Field”, Phys. Rev. A 47,4464 (1993)
[Crossref] [PubMed]

F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Pulse. II”, Phys. Rev. A 48, 554 (1993)
[Crossref] [PubMed]

Fedorov, M.V.

R. Grobe and M.V. Fedorov, “Packet Spreading, Stabilization, and Localization in Superstrong Fields”, Phys. Rev. Lett 68, 2592 (1992)
[Crossref] [PubMed]

Grobe, R.

R. Grobe and M.V. Fedorov, “Packet Spreading, Stabilization, and Localization in Superstrong Fields”, Phys. Rev. Lett 68, 2592 (1992)
[Crossref] [PubMed]

Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields”, Laser Phys. (in press).

B.A. Smetanko, Q. Su, and R. Grobe, “FFT Based Split Operator Techniques for Solving the Dirac Equation”, Comm. Comp. Phys. (to be submitted).

Grochmalicki, J.

Hoogenraad, J. H.

M.P. De Boer, J. H. Hoogenraad, and R. B. Vrijen, “Indications of High-Intensity Adiabatic Stabilization in Neon”, Phys. Rev. Lett. 71, 3263 (1993)
[Crossref] [PubMed]

Horbatsch, M.

M. Horbatsch, “Magnetic Field Effects in High-Frequency Photoionization by Intense Laser Pulses”, Z. Phys. D 25, 305 (1993).
[Crossref]

Javanainen, J.

Q. Su, J.H. Eberly, and J. Javanainen, “Dynamics of Atomic Ionization Suppression and Electron Localization in an Intense High-Frequency Radiation Field”, Phys. Rev. Lett. 64, 862 (1990)
[Crossref] [PubMed]

Keitel, C.H.

M. Protopapas, C.H. Keitel, and P.L. Knight, “Relativistic Mass Shift Effects in Adiabatic Intense Laser Field Stabilization of Atoms”, J. Phys. B 29, L591 (1996).
[Crossref]

Knauer, J.P.

C.I. Moore, J.P. Knauer, and D.D. Meyerhofer, “Observation of the Transition from Thomson to Compton Scattering in Multiphoton Interactions with Low-Energy Electrons”, Phys. Rev. Lett. 74, 2439 (1995)
[Crossref] [PubMed]

Knight, P.L.

M. Protopapas, C.H. Keitel, and P.L. Knight, “Relativistic Mass Shift Effects in Adiabatic Intense Laser Field Stabilization of Atoms”, J. Phys. B 29, L591 (1996).
[Crossref]

Krause, J.L.

K.C. Kulander, K.J. Schafer, and J.L. Krause, “Dynamic Stabilization of Hydrogen in an Intense High Frequency, Pulsed Laser Field”, Phys. Rev. Lett. 66, 2601 (1991)
[Crossref] [PubMed]

Kulander, K.C.

K.C. Kulander, K.J. Schafer, and J.L. Krause, “Dynamic Stabilization of Hydrogen in an Intense High Frequency, Pulsed Laser Field”, Phys. Rev. Lett. 66, 2601 (1991)
[Crossref] [PubMed]

Lewenstein, Maciej A.

Meyerhofer, D.D.

C.I. Moore, J.P. Knauer, and D.D. Meyerhofer, “Observation of the Transition from Thomson to Compton Scattering in Multiphoton Interactions with Low-Energy Electrons”, Phys. Rev. Lett. 74, 2439 (1995)
[Crossref] [PubMed]

Miquel, J. L.

P. Monot, T. Auguste, and J. L. Miquel, “Experimental Demonstration of Relativistic Self-Channeling for a Multiterawatt Laser Pulse in an Underdense Plasma”, Phys. Rev. Lett. 74, 2953 (1995).
[Crossref] [PubMed]

Monot, P.

P. Monot, T. Auguste, and J. L. Miquel, “Experimental Demonstration of Relativistic Self-Channeling for a Multiterawatt Laser Pulse in an Underdense Plasma”, Phys. Rev. Lett. 74, 2953 (1995).
[Crossref] [PubMed]

Moore, C.I.

C.I. Moore, J.P. Knauer, and D.D. Meyerhofer, “Observation of the Transition from Thomson to Compton Scattering in Multiphoton Interactions with Low-Energy Electrons”, Phys. Rev. Lett. 74, 2439 (1995)
[Crossref] [PubMed]

Mourou, G.

M.D. Perry and G. Mourou, “Terawatt to Petawatt Subpicosecond Lasers”, Science 264, 917 (1994)
[Crossref] [PubMed]

Muller, H. G.

N.J van Druten, R. C. Constantinescu, and H. G. Muller, “Adiabatic Stabilization: Observation of the Surviving Population”, Phys. Rev. A 55, 622 (1997).
[Crossref]

Perry, M.D.

M.D. Perry and G. Mourou, “Terawatt to Petawatt Subpicosecond Lasers”, Science 264, 917 (1994)
[Crossref] [PubMed]

Potvliege, R.M.

M. Dšrr, R.M. Potvliege, and R. Shakeshaft, “Tunneling Ionization of Atomc Hydrogen by an Intense Low-Frequency Field”, Phys. Rev. Lett. 64, 2003 (1990)
[Crossref]

Protopapas, M.

M. Protopapas, C.H. Keitel, and P.L. Knight, “Relativistic Mass Shift Effects in Adiabatic Intense Laser Field Stabilization of Atoms”, J. Phys. B 29, L591 (1996).
[Crossref]

Radozycki, T.

F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Pulse. II”, Phys. Rev. A 48, 554 (1993)
[Crossref] [PubMed]

F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Field”, Phys. Rev. A 47,4464 (1993)
[Crossref] [PubMed]

Reiss, H.R.

D.P. Crawford and H.R. Reiss, “Stabilization in Relavistic Photoionization with Circular Polarized Light”, Phys. Rev. A 50, 1844 (1994)
[Crossref] [PubMed]

H.R. Reiss, “Theoretical Methods in Quantum Optics: S-Matrix and Keldysh Techniques for Strong-Field Processes”, Prog. Quantum Electron. 16, 1 (1992).
[Crossref]

H.R. Reiss, “Relativistic Strong-Field Photoionization”, J. Opt. Soc. Am. B 7, 574 (1990).
[Crossref]

Rhodes, C.K.

K. Boyer and C.K. Rhodes, “Superstrong Coherent Multi-Electron Intense-Field Interaction”, J. Phys. B 27, L633 (1994)
[Crossref]

Rzazewski, Kazimierz

Schafer, K.J.

K.C. Kulander, K.J. Schafer, and J.L. Krause, “Dynamic Stabilization of Hydrogen in an Intense High Frequency, Pulsed Laser Field”, Phys. Rev. Lett. 66, 2601 (1991)
[Crossref] [PubMed]

Shakeshaft, R.

M. Dšrr, R.M. Potvliege, and R. Shakeshaft, “Tunneling Ionization of Atomc Hydrogen by an Intense Low-Frequency Field”, Phys. Rev. Lett. 64, 2003 (1990)
[Crossref]

Smetanko, B.A.

Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields”, Laser Phys. (in press).

B.A. Smetanko, Q. Su, and R. Grobe, “FFT Based Split Operator Techniques for Solving the Dirac Equation”, Comm. Comp. Phys. (to be submitted).

Su, Q.

Q. Su, J.H. Eberly, and J. Javanainen, “Dynamics of Atomic Ionization Suppression and Electron Localization in an Intense High-Frequency Radiation Field”, Phys. Rev. Lett. 64, 862 (1990)
[Crossref] [PubMed]

B.A. Smetanko, Q. Su, and R. Grobe, “FFT Based Split Operator Techniques for Solving the Dirac Equation”, Comm. Comp. Phys. (to be submitted).

Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields”, Laser Phys. (in press).

van Druten, N.J

N.J van Druten, R. C. Constantinescu, and H. G. Muller, “Adiabatic Stabilization: Observation of the Surviving Population”, Phys. Rev. A 55, 622 (1997).
[Crossref]

Vrijen, R. B.

M.P. De Boer, J. H. Hoogenraad, and R. B. Vrijen, “Indications of High-Intensity Adiabatic Stabilization in Neon”, Phys. Rev. Lett. 71, 3263 (1993)
[Crossref] [PubMed]

Wilkens, Martin

J. Opt. Soc. Am. B (2)

J. Phys. B (2)

K. Boyer and C.K. Rhodes, “Superstrong Coherent Multi-Electron Intense-Field Interaction”, J. Phys. B 27, L633 (1994)
[Crossref]

M. Protopapas, C.H. Keitel, and P.L. Knight, “Relativistic Mass Shift Effects in Adiabatic Intense Laser Field Stabilization of Atoms”, J. Phys. B 29, L591 (1996).
[Crossref]

Laser Phys. (1)

Q. Su, B.A. Smetanko, and R. Grobe, “Wave packet motion in relativistic electric fields”, Laser Phys. (in press).

Phys. Rev. A (4)

D.P. Crawford and H.R. Reiss, “Stabilization in Relavistic Photoionization with Circular Polarized Light”, Phys. Rev. A 50, 1844 (1994)
[Crossref] [PubMed]

N.J van Druten, R. C. Constantinescu, and H. G. Muller, “Adiabatic Stabilization: Observation of the Surviving Population”, Phys. Rev. A 55, 622 (1997).
[Crossref]

F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Field”, Phys. Rev. A 47,4464 (1993)
[Crossref] [PubMed]

F.H.M. Faisal and T. Radozycki, “Three-Dimensional Relativistic Model of a Bound Particle in an Intense Laser Pulse. II”, Phys. Rev. A 48, 554 (1993)
[Crossref] [PubMed]

Phys. Rev. Lett (1)

R. Grobe and M.V. Fedorov, “Packet Spreading, Stabilization, and Localization in Superstrong Fields”, Phys. Rev. Lett 68, 2592 (1992)
[Crossref] [PubMed]

Phys. Rev. Lett. (6)

C.I. Moore, J.P. Knauer, and D.D. Meyerhofer, “Observation of the Transition from Thomson to Compton Scattering in Multiphoton Interactions with Low-Energy Electrons”, Phys. Rev. Lett. 74, 2439 (1995)
[Crossref] [PubMed]

P. Monot, T. Auguste, and J. L. Miquel, “Experimental Demonstration of Relativistic Self-Channeling for a Multiterawatt Laser Pulse in an Underdense Plasma”, Phys. Rev. Lett. 74, 2953 (1995).
[Crossref] [PubMed]

Q. Su, J.H. Eberly, and J. Javanainen, “Dynamics of Atomic Ionization Suppression and Electron Localization in an Intense High-Frequency Radiation Field”, Phys. Rev. Lett. 64, 862 (1990)
[Crossref] [PubMed]

M. Dšrr, R.M. Potvliege, and R. Shakeshaft, “Tunneling Ionization of Atomc Hydrogen by an Intense Low-Frequency Field”, Phys. Rev. Lett. 64, 2003 (1990)
[Crossref]

K.C. Kulander, K.J. Schafer, and J.L. Krause, “Dynamic Stabilization of Hydrogen in an Intense High Frequency, Pulsed Laser Field”, Phys. Rev. Lett. 66, 2601 (1991)
[Crossref] [PubMed]

M.P. De Boer, J. H. Hoogenraad, and R. B. Vrijen, “Indications of High-Intensity Adiabatic Stabilization in Neon”, Phys. Rev. Lett. 71, 3263 (1993)
[Crossref] [PubMed]

Prog. Quantum Electron. (1)

H.R. Reiss, “Theoretical Methods in Quantum Optics: S-Matrix and Keldysh Techniques for Strong-Field Processes”, Prog. Quantum Electron. 16, 1 (1992).
[Crossref]

Science (1)

M.D. Perry and G. Mourou, “Terawatt to Petawatt Subpicosecond Lasers”, Science 264, 917 (1994)
[Crossref] [PubMed]

Z. Phys. D (1)

M. Horbatsch, “Magnetic Field Effects in High-Frequency Photoionization by Intense Laser Pulses”, Z. Phys. D 25, 305 (1993).
[Crossref]

Other (4)

J.H. Eberly, Prog. Opt.7, 359 ed. E. Wolf (Amsterdam: North-Holland1969);
[Crossref]

“Atoms in Intense Laser Fields”, ed. M. Gavrila (Academic Press, Boston1992).

J.D. Bjorken and S.D. Drell, “Relativistic Quantum Mechanics” (McGraw-Hill, New York1964).

B.A. Smetanko, Q. Su, and R. Grobe, “FFT Based Split Operator Techniques for Solving the Dirac Equation”, Comm. Comp. Phys. (to be submitted).

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Figures (2)

Fig. 1.
Fig. 1.

The graphs show the temporal growth pattern of the spatial width obtained from Eqs. (6) Δx(t), Δy(t) and Δz(t) together with the non-relativistic width ΔxNR(t). Superimposed on the graphs for Δx(t), Δy(t) and Δz(t) are the width determined from the time-dependent wave function solution obtained from the full Dirac equation Eq. (2) (dash lines). The two graphs are indistinguishable. [E=1000 a.u., initial quantum state as in Eq. (3) with σ=Δx(t=0)=0.1 a.u.]

Fig. 2.
Fig. 2.

Displayed are the spatial probability distributions P(x, t) ≠ ∑i∫dydz∣Ψi(x, y, z, t)∣2 and P(z, t) ≠ ∑i∫dxdy∣Ψi(x, y, z, t)∣2 in the x- and z direction at time t=0.1 a.u. For comparison, the dashed lines show the corresponding distributions obtained from the non-relativistic Schrödinger time evolution. The initial wave packet was centered initially at r=(-3, 0, 0) for better graphical clarity. The non-relativistic wave packet has moved to x(t=0.1a.u.) =-3a.u. + Et2/2= 2.a.u. [Same parameters as in Fig. 1]

Equations (10)

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

A ( t ) = c E t = cEt e x
i ħ t Ψ r t = mc α ( p q c A ( t ) ) + β m c 2 Ψ r t
Ψ ( r , t = 0 ) = [ 2 π σ 2 ] 3 4 ( exp [ ( r 2 σ ) 2 ] , 0,0,0 )
H = m 2 c 4 + c 2 ( p q c A ) 2
x ( t ) = x + 1 qE m 2 c 4 + c 2 ( p + q E t ) 2 1 qE m 2 c 4 + c 2 p 2
y ( t ) = y + c p y qE ln { p x + qEt + m 2 c 4 + c 2 ( p + q E t ) 2 p x + m 2 c 4 + c 2 p 2 }
z ( t ) = z + c p z qE ln { p x + qEt + m 2 c 4 + c 2 ( p + q E t ) 2 p x + m 2 c 4 + c 2 p 2 }
Δ x 2 = ( x x ) 2
Δ x 2 ( t ) = Δ x 2 + 1 q 2 E 2 [ c p x 2 + p 2 + m 2 c 4 ] 1 q 2 E 2 c 2 p 2 + m 2 c 4 2
Δ x 2 ( t ) = Δ x 2 + c 2 p x 2 q 2 E 2

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