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

Research output: Contribution to journalArticle

30 Citations (Scopus)

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)=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.

Original languageEnglish (US)
Pages (from-to)1801-1813
Number of pages13
JournalJournal of Mathematical Physics
Volume41
Issue number4
DOIs
StatePublished - Jan 1 2000

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Wave Packet
wave packets
Power Law
square wells
Time Scales
oscillators
WKB Approximation
Wentzel-Kramer-Brillouin method
One-dimensional System
Autocorrelation Function
Energy Levels
Harmonic Oscillator
Quantum Systems
harmonic oscillators
autocorrelation
Limiting
energy levels
Exponent
exponents
Arbitrary

All Science Journal Classification (ASJC) codes

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

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

In: Journal of Mathematical Physics, Vol. 41, No. 4, 01.01.2000, p. 1801-1813.

Research output: Contribution to journalArticle

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PY - 2000/1/1

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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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