### Abstract

A new approach for solving the time-dependent wave function in quantum scattering problem is presented. The conventional wave packet method, which directly solves the time-dependent Schrödinger equation, normally requires a large number of grid points since the Schrödinger picture wave function both travels and spreads in time. Also, since the Schrödinger picture wave function oscillates in time with frequency ω= E /ℏ, a very small time increment is required to integrate the Schrödinger equation, especially for high energy collisions. The new method presented in this paper transforms the Schrödinger picture wave function into the interaction picture and carries out the integration in it. The new approach is superior to conventional one in that (1) a smaller numerical grid is required due to the localized nature of the interaction picture wave function, since it is not a traveling wave and does not spread appreciably in coordinate space, and thus behaves like a bound state wave function. (2) The interaction picture wave function varies slowly with time and is essentially independent of energy, permitting the use of a large time increment in the numerical integration. Because of these two features in this new approach, we are able to integrate the time dependent wave function once and obtain an accurate S matrix over a wide range of energy efficiently.

Original language | English (US) |
---|---|

Pages (from-to) | 324-331 |

Number of pages | 8 |

Journal | The Journal of chemical physics |

Volume | 92 |

Issue number | 1 |

State | Published - 1990 |

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### ASJC Scopus subject areas

- Atomic and Molecular Physics, and Optics

### Cite this

**New method in time-dependent quantum scattering theory : Integrating the wave function in the interaction picture.** / Zhang, John.

Research output: Contribution to journal › Article

*The Journal of chemical physics*, vol. 92, no. 1, pp. 324-331.

}

TY - JOUR

T1 - New method in time-dependent quantum scattering theory

T2 - Integrating the wave function in the interaction picture

AU - Zhang, John

PY - 1990

Y1 - 1990

N2 - A new approach for solving the time-dependent wave function in quantum scattering problem is presented. The conventional wave packet method, which directly solves the time-dependent Schrödinger equation, normally requires a large number of grid points since the Schrödinger picture wave function both travels and spreads in time. Also, since the Schrödinger picture wave function oscillates in time with frequency ω= E /ℏ, a very small time increment is required to integrate the Schrödinger equation, especially for high energy collisions. The new method presented in this paper transforms the Schrödinger picture wave function into the interaction picture and carries out the integration in it. The new approach is superior to conventional one in that (1) a smaller numerical grid is required due to the localized nature of the interaction picture wave function, since it is not a traveling wave and does not spread appreciably in coordinate space, and thus behaves like a bound state wave function. (2) The interaction picture wave function varies slowly with time and is essentially independent of energy, permitting the use of a large time increment in the numerical integration. Because of these two features in this new approach, we are able to integrate the time dependent wave function once and obtain an accurate S matrix over a wide range of energy efficiently.

AB - A new approach for solving the time-dependent wave function in quantum scattering problem is presented. The conventional wave packet method, which directly solves the time-dependent Schrödinger equation, normally requires a large number of grid points since the Schrödinger picture wave function both travels and spreads in time. Also, since the Schrödinger picture wave function oscillates in time with frequency ω= E /ℏ, a very small time increment is required to integrate the Schrödinger equation, especially for high energy collisions. The new method presented in this paper transforms the Schrödinger picture wave function into the interaction picture and carries out the integration in it. The new approach is superior to conventional one in that (1) a smaller numerical grid is required due to the localized nature of the interaction picture wave function, since it is not a traveling wave and does not spread appreciably in coordinate space, and thus behaves like a bound state wave function. (2) The interaction picture wave function varies slowly with time and is essentially independent of energy, permitting the use of a large time increment in the numerical integration. Because of these two features in this new approach, we are able to integrate the time dependent wave function once and obtain an accurate S matrix over a wide range of energy efficiently.

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UR - http://www.scopus.com/inward/citedby.url?scp=36549103015&partnerID=8YFLogxK

M3 - Article

VL - 92

SP - 324

EP - 331

JO - Journal of Chemical Physics

JF - Journal of Chemical Physics

SN - 0021-9606

IS - 1

ER -