New Neumerical Method to Calculate Time-Dependent Quantum Properties in Finite Temperature Based on the Heisenberg Equation of Motion

Affiliation(s)

Department of Materials Science and Engineering, Nagasaki University, Nagasaki, Japan.

Department of Materials Science and Engineering, Nagasaki University, Nagasaki, Japan.

ABSTRACT

For the purpose of computer calculation to evaluate time-dependent quantum properties in finite temperature, we propose new numerical method expressed in the forms of simultaneous differential equations. At first we derive the equation of motion in finite temperature, which is found to be same expression as Heisenberg equation of motion except for the c-number. Based on this equation, we construct numerical method to estimate time-dependent physical properties in finite temperature precisely without using analytical procedures such as Keldysh formalism. Since our approach is so simple and is based on the simultaneous differential equations including no terms related to self-energies, computer programming can be easily performed. It is possible to estimate exact time-dependent physical properties, providing that Hamiltonian of the system is taken to be a one-electron picture. Furthermore, we refer to the application to the many body problem and it is numerically possible to calculate physical properties using Hartree Fock approximation. Our numerical method can be applied to the case even when perturbative Hamiltonians are newly introduced or Hamiltonian shows complex time-dependent behavior. In this article, at first, we derive the equation of motion in finite temperature. Secondly, for the purpose of verification and of exhibiting the usefulness, we show the derivation of gap equation of superconductivity and of sum rule of electrical conductivity and the application to the many body problem. Finally we apply this method to these two cases: the first case is most simplified resonance charge transfer neutralization of an ion and the second is the same process but impurity potential is newly introduced as perturbative Hamiltonian. Through both cases, it is found that neutralization process is not so sensitive to temperature, however, impurity potential as small as 10 meV strongly influences the neutralization of ion.

For the purpose of computer calculation to evaluate time-dependent quantum properties in finite temperature, we propose new numerical method expressed in the forms of simultaneous differential equations. At first we derive the equation of motion in finite temperature, which is found to be same expression as Heisenberg equation of motion except for the c-number. Based on this equation, we construct numerical method to estimate time-dependent physical properties in finite temperature precisely without using analytical procedures such as Keldysh formalism. Since our approach is so simple and is based on the simultaneous differential equations including no terms related to self-energies, computer programming can be easily performed. It is possible to estimate exact time-dependent physical properties, providing that Hamiltonian of the system is taken to be a one-electron picture. Furthermore, we refer to the application to the many body problem and it is numerically possible to calculate physical properties using Hartree Fock approximation. Our numerical method can be applied to the case even when perturbative Hamiltonians are newly introduced or Hamiltonian shows complex time-dependent behavior. In this article, at first, we derive the equation of motion in finite temperature. Secondly, for the purpose of verification and of exhibiting the usefulness, we show the derivation of gap equation of superconductivity and of sum rule of electrical conductivity and the application to the many body problem. Finally we apply this method to these two cases: the first case is most simplified resonance charge transfer neutralization of an ion and the second is the same process but impurity potential is newly introduced as perturbative Hamiltonian. Through both cases, it is found that neutralization process is not so sensitive to temperature, however, impurity potential as small as 10 meV strongly influences the neutralization of ion.

Cite this paper

S. Kondo, "New Neumerical Method to Calculate Time-Dependent Quantum Properties in Finite Temperature Based on the Heisenberg Equation of Motion,"*Journal of Modern Physics*, Vol. 3 No. 10, 2012, pp. 1537-1549. doi: 10.4236/jmp.2012.310190.

S. Kondo, "New Neumerical Method to Calculate Time-Dependent Quantum Properties in Finite Temperature Based on the Heisenberg Equation of Motion,"

References

[1] S. Kondo and K. Yamada, “Analysis of Resonance Charge Transfer Neutralization on the Basis of Heisenberg Equations of Motion,” Progress of Theoretical Physics, Vol. 122, No. 3, 2009, pp. 713-734. doi:10.1143/PTP.122.713

[2] R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems,” Journal of the Physical Society of Japan, Vol. 12, No. 3, 1957, pp. 570-586. doi:10.1143/JPSJ.12.570

[3] R. Brako and D. M. Newns, “Charge Exchange in Atom-Surface Scattering: Thermal versus Quantum Mechanical Non-Adiabaticity,” Surface Science, Vol. 108, No. 2, 1981, pp. 253-270.

[4] R. Brako and D. M. Newns, “Theory of Electronic Processes in Atom Scattering from Surfaces,” Reports on Progress in Physics, Vol. 52, No. 6, 1989, pp. 655-698. doi:10.1088/0034-4885/52/6/001

[5] W. Bloss and D. Hone, “Theory of Charge Exchange Scattering from Surfaces,” Surface Science, Vol. 72, No. 2, 1978, pp. 277-297.

[6] H. Winter and R. Zimny, Proceedings of XV ICPEAC, Elsevier, Brighton, 1987.

[7] R. Brako, H. Winter and D. M. Newns, “Anisotropic Excitation of Hydrogen 2p after Grazing Ion-Surface Scattering,” Europhysics Letters, Vol. 7, No. 3, 1988, pp. 213-218. doi:10.1209/0295-5075/7/3/005

[8] D. M. Newns, “Self-Consistent Model of Hydrogen Chemisorption,” Physical Review, Vol. 178, No. 3, 1969, pp. 1123-1135. doi:10.1103/PhysRev.178.1123

[9] L. V. Keldysh, “Diagram Technique for Nonequilibrium,” Soviet Physics JETP, Vol. 20, No. 4, 1965, pp. 1018-1026.

[10] T. Sakai, M. Sakaue, H. Kasai and A. Okiji, “Theory of Dynamics of Electron Wave Packets in Time-Resolved Two-Photon Photoemission via Image States,” Applied Surface Science, Vol. 169-170, 2001, pp. 57-62. doi:10.1016/S0169-4332(00)00633-4

[11] E. A. García, C. G. Pascual, P. G. Bolcatto, M. C. G. Passeggi and E. C. Goldberg, “Ion Fractions in the Scattering of Hydrogen on Different Reconstructed Silicon Surfaces,” Surface Science, Vol. 600, No. 10, 2006, pp. 2195-2206. doi:10.1016/j.susc.2006.03.008

[12] M. Krawiec, M. Jalochowski and M. Kisiel, “High Resolution Scanning Tunneling Spectroscopy of Ultrathin Pb on Si(111)-(6 × 6) Substrate,” Surface Science, Vol. 600, No. 8, 2006, pp. 1641-1645.

[13] T. Mii and K. Makoshi, “An Interpolation Formula for Electron Transfer in Atom-Surface Collision and Its Validity,” Surface Science, Vol. 363, No. 1-3, 1996, pp. 145- 149.

[14] E. A. García, M. A. Romero, C. González and E. C. Goldberg, “Neutralization of Li+ Ions Scattered by the Cu(100) and (111) Surfaces: A Localized Picture of the Atom-Surface Interaction,” Surface Science, Vol. 603, No. 4, 2009, pp. 97-605. doi:10.1016/j.susc.2008.12.022

[15] A. Sindona and G. Falcone, “Evidences of a Double Resonant Ionization Mechanism in Sputtering of Metals,” Surface Science, Vol. 529, No. 3, 2003, pp. 471-489. doi:10.1016/S0039-6028(03)00334-0

[16] A. Sindona, P. Riccardi, S. Maletta and G. Falcone, “Double Resonant Neutralization in Hyperthermal Energy Alkali Ion Scattering at Clean Metal Surfaces,” Nuclear Instruments and Method in Physics Research B, Vol. 267, No. 4, 2009, pp. 578-583. doi:10.1016/j.nimb.2008.11.053

[17] K. W. Sulston and F. O. Goodman, “Electronic Bath approach to Thermal Effects in Ion-Surface Scattering,” The Journal of Chemical Physics, Vol. 112, No. 5, 2000, p. 2486.

[1] S. Kondo and K. Yamada, “Analysis of Resonance Charge Transfer Neutralization on the Basis of Heisenberg Equations of Motion,” Progress of Theoretical Physics, Vol. 122, No. 3, 2009, pp. 713-734. doi:10.1143/PTP.122.713

[2] R. Kubo, “Statistical-Mechanical Theory of Irreversible Processes. I. General Theory and Simple Applications to Magnetic and Conduction Problems,” Journal of the Physical Society of Japan, Vol. 12, No. 3, 1957, pp. 570-586. doi:10.1143/JPSJ.12.570

[3] R. Brako and D. M. Newns, “Charge Exchange in Atom-Surface Scattering: Thermal versus Quantum Mechanical Non-Adiabaticity,” Surface Science, Vol. 108, No. 2, 1981, pp. 253-270.

[4] R. Brako and D. M. Newns, “Theory of Electronic Processes in Atom Scattering from Surfaces,” Reports on Progress in Physics, Vol. 52, No. 6, 1989, pp. 655-698. doi:10.1088/0034-4885/52/6/001

[5] W. Bloss and D. Hone, “Theory of Charge Exchange Scattering from Surfaces,” Surface Science, Vol. 72, No. 2, 1978, pp. 277-297.

[6] H. Winter and R. Zimny, Proceedings of XV ICPEAC, Elsevier, Brighton, 1987.

[7] R. Brako, H. Winter and D. M. Newns, “Anisotropic Excitation of Hydrogen 2p after Grazing Ion-Surface Scattering,” Europhysics Letters, Vol. 7, No. 3, 1988, pp. 213-218. doi:10.1209/0295-5075/7/3/005

[8] D. M. Newns, “Self-Consistent Model of Hydrogen Chemisorption,” Physical Review, Vol. 178, No. 3, 1969, pp. 1123-1135. doi:10.1103/PhysRev.178.1123

[9] L. V. Keldysh, “Diagram Technique for Nonequilibrium,” Soviet Physics JETP, Vol. 20, No. 4, 1965, pp. 1018-1026.

[10] T. Sakai, M. Sakaue, H. Kasai and A. Okiji, “Theory of Dynamics of Electron Wave Packets in Time-Resolved Two-Photon Photoemission via Image States,” Applied Surface Science, Vol. 169-170, 2001, pp. 57-62. doi:10.1016/S0169-4332(00)00633-4

[11] E. A. García, C. G. Pascual, P. G. Bolcatto, M. C. G. Passeggi and E. C. Goldberg, “Ion Fractions in the Scattering of Hydrogen on Different Reconstructed Silicon Surfaces,” Surface Science, Vol. 600, No. 10, 2006, pp. 2195-2206. doi:10.1016/j.susc.2006.03.008

[12] M. Krawiec, M. Jalochowski and M. Kisiel, “High Resolution Scanning Tunneling Spectroscopy of Ultrathin Pb on Si(111)-(6 × 6) Substrate,” Surface Science, Vol. 600, No. 8, 2006, pp. 1641-1645.

[13] T. Mii and K. Makoshi, “An Interpolation Formula for Electron Transfer in Atom-Surface Collision and Its Validity,” Surface Science, Vol. 363, No. 1-3, 1996, pp. 145- 149.

[14] E. A. García, M. A. Romero, C. González and E. C. Goldberg, “Neutralization of Li+ Ions Scattered by the Cu(100) and (111) Surfaces: A Localized Picture of the Atom-Surface Interaction,” Surface Science, Vol. 603, No. 4, 2009, pp. 97-605. doi:10.1016/j.susc.2008.12.022

[15] A. Sindona and G. Falcone, “Evidences of a Double Resonant Ionization Mechanism in Sputtering of Metals,” Surface Science, Vol. 529, No. 3, 2003, pp. 471-489. doi:10.1016/S0039-6028(03)00334-0

[16] A. Sindona, P. Riccardi, S. Maletta and G. Falcone, “Double Resonant Neutralization in Hyperthermal Energy Alkali Ion Scattering at Clean Metal Surfaces,” Nuclear Instruments and Method in Physics Research B, Vol. 267, No. 4, 2009, pp. 578-583. doi:10.1016/j.nimb.2008.11.053

[17] K. W. Sulston and F. O. Goodman, “Electronic Bath approach to Thermal Effects in Ion-Surface Scattering,” The Journal of Chemical Physics, Vol. 112, No. 5, 2000, p. 2486.