### Abstract

The harmonic oscillator and the infinite square well are two of the simplest one-dimensional quantum systems which exhibit wave packet revivals (trivially so in the case of the oscillator.) These two potentials can be thought of as special cases of the general one-dimensional power-law potential given by V_{(k)}(x)=V_{0}|xla|^{k} (with k = 2 and k→∞ for the oscillator and square well, respectively.) Using an autocorrelation function approach and the WKB approximation for the quantized energy levels in such potentials, we exhibit numerical evidence for wave packet revivals in the case of arbitrary k>0 which agree well with more analytic results. We derive expressions for the revival and collapse time scales in terms of the physical quantities of the system as well as the wave packet parameters. We find that both times scale with the power-law exponent k as \(k+2)l(k-2)\. In this way, we can explicitly exhibit the approach to the two familiar limiting cases. We also briefly consider the case of a "half" well where an infinite wall in added at the origin.

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

Pages (from-to) | 1801-1813 |

Number of pages | 13 |

Journal | Journal of Mathematical Physics |

Volume | 41 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 2000 |

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### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Mathematical Physics

### Cite this

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*Journal of Mathematical Physics*, vol. 41, no. 4, pp. 1801-1813. https://doi.org/10.1063/1.533213

**Wave packet revivals and quasirevivals in one-dimensional power law potentials.** / Robinett, Richard Wallace.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Wave packet revivals and quasirevivals in one-dimensional power law potentials

AU - Robinett, Richard Wallace

PY - 2000/1/1

Y1 - 2000/1/1

N2 - The harmonic oscillator and the infinite square well are two of the simplest one-dimensional quantum systems which exhibit wave packet revivals (trivially so in the case of the oscillator.) These two potentials can be thought of as special cases of the general one-dimensional power-law potential given by V(k)(x)=V0|xla|k (with k = 2 and k→∞ for the oscillator and square well, respectively.) Using an autocorrelation function approach and the WKB approximation for the quantized energy levels in such potentials, we exhibit numerical evidence for wave packet revivals in the case of arbitrary k>0 which agree well with more analytic results. We derive expressions for the revival and collapse time scales in terms of the physical quantities of the system as well as the wave packet parameters. We find that both times scale with the power-law exponent k as \(k+2)l(k-2)\. In this way, we can explicitly exhibit the approach to the two familiar limiting cases. We also briefly consider the case of a "half" well where an infinite wall in added at the origin.

AB - The harmonic oscillator and the infinite square well are two of the simplest one-dimensional quantum systems which exhibit wave packet revivals (trivially so in the case of the oscillator.) These two potentials can be thought of as special cases of the general one-dimensional power-law potential given by V(k)(x)=V0|xla|k (with k = 2 and k→∞ for the oscillator and square well, respectively.) Using an autocorrelation function approach and the WKB approximation for the quantized energy levels in such potentials, we exhibit numerical evidence for wave packet revivals in the case of arbitrary k>0 which agree well with more analytic results. We derive expressions for the revival and collapse time scales in terms of the physical quantities of the system as well as the wave packet parameters. We find that both times scale with the power-law exponent k as \(k+2)l(k-2)\. In this way, we can explicitly exhibit the approach to the two familiar limiting cases. We also briefly consider the case of a "half" well where an infinite wall in added at the origin.

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U2 - 10.1063/1.533213

DO - 10.1063/1.533213

M3 - Article

VL - 41

SP - 1801

EP - 1813

JO - Journal of Mathematical Physics

JF - Journal of Mathematical Physics

SN - 0022-2488

IS - 4

ER -