### Abstract

We consider the numerical approximation of quantum mechanical wave packets by a finite difference method for the Schrödinger equation. We discuss known results for the solutions of the equations for N coupled harmonic oscillators and separation of variables solutions of finite difference equations for the heat and wave equations. We find separation of variables solutions of the Schrödinger finite difference equations with the same spatial part as the above solutions. We compare approximating a stationary state of the Schrödinger equation via four approaches: full finite difference equations for the Schrödinger and wave equations, and individual separation of variables solutions for each set of difference equations. We approximate the Fourier integral representation of a Gaussian wave packet by a Riemann sum, thus writing the wave packet as a finite superposition of plane waves. We then approximate the wave packet using the Schrödinger and wave finite difference equations.

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

Title of host publication | Advances in Analysis |

Subtitle of host publication | Problems of Integration |

Publisher | Nova Science Publishers, Inc. |

Pages | 283-305 |

Number of pages | 23 |

ISBN (Electronic) | 9781536117196 |

ISBN (Print) | 9781612091303 |

State | Published - Jan 1 2012 |

### Fingerprint

### Keywords

- Finite differences
- Oscillators
- Schrodinger
- Wave equation

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Analysis: Problems of Integration*(pp. 283-305). Nova Science Publishers, Inc..

**Finite difference approximation of quantum mechanical wave packets.** / Kime, Katherine A.

Research output: Chapter in Book/Report/Conference proceeding › Chapter

*Advances in Analysis: Problems of Integration.*Nova Science Publishers, Inc., pp. 283-305.

}

TY - CHAP

T1 - Finite difference approximation of quantum mechanical wave packets

AU - Kime, Katherine A.

PY - 2012/1/1

Y1 - 2012/1/1

N2 - We consider the numerical approximation of quantum mechanical wave packets by a finite difference method for the Schrödinger equation. We discuss known results for the solutions of the equations for N coupled harmonic oscillators and separation of variables solutions of finite difference equations for the heat and wave equations. We find separation of variables solutions of the Schrödinger finite difference equations with the same spatial part as the above solutions. We compare approximating a stationary state of the Schrödinger equation via four approaches: full finite difference equations for the Schrödinger and wave equations, and individual separation of variables solutions for each set of difference equations. We approximate the Fourier integral representation of a Gaussian wave packet by a Riemann sum, thus writing the wave packet as a finite superposition of plane waves. We then approximate the wave packet using the Schrödinger and wave finite difference equations.

AB - We consider the numerical approximation of quantum mechanical wave packets by a finite difference method for the Schrödinger equation. We discuss known results for the solutions of the equations for N coupled harmonic oscillators and separation of variables solutions of finite difference equations for the heat and wave equations. We find separation of variables solutions of the Schrödinger finite difference equations with the same spatial part as the above solutions. We compare approximating a stationary state of the Schrödinger equation via four approaches: full finite difference equations for the Schrödinger and wave equations, and individual separation of variables solutions for each set of difference equations. We approximate the Fourier integral representation of a Gaussian wave packet by a Riemann sum, thus writing the wave packet as a finite superposition of plane waves. We then approximate the wave packet using the Schrödinger and wave finite difference equations.

KW - Finite differences

KW - Oscillators

KW - Schrodinger

KW - Wave equation

UR - http://www.scopus.com/inward/record.url?scp=84892069089&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84892069089&partnerID=8YFLogxK

M3 - Chapter

AN - SCOPUS:84892069089

SN - 9781612091303

SP - 283

EP - 305

BT - Advances in Analysis

PB - Nova Science Publishers, Inc.

ER -