Wigner quasi-probability distribution for the infinite square well: Energy eigenstates and time-dependent wave packets

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Abstract

We calculate and visualize the Wigner quasi-probability distribution for the position and momentum, P W (n)(x, p), for the energy eigenstates of the infinite square well. We evaluate the time-dependent Wigner distribution, P w(x,p;t), for Gaussian wave packet solutions of this system, and illustrate the short-term semi-classical time dependence and the longer-term revival and fractional revival behavior. Our results indicate how the Wigner distribution can be used to examine the highly correlated dynamical position-momentum structure of quantum states. In particular, this tool provides an excellent way of demonstrating the patterns of highly correlated Schrödinger-cat-like "mini-packets" which appear at fractional multiples of the exact revival time.

Original languageEnglish (US)
Pages (from-to)1183-1192
Number of pages10
JournalAmerican Journal of Physics
Volume72
Issue number9
DOIs
StatePublished - Sep 1 2004

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square wells
wave packets
eigenvectors
momentum
cats
time dependence
energy

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)

Cite this

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title = "Wigner quasi-probability distribution for the infinite square well: Energy eigenstates and time-dependent wave packets",
abstract = "We calculate and visualize the Wigner quasi-probability distribution for the position and momentum, P W (n)(x, p), for the energy eigenstates of the infinite square well. We evaluate the time-dependent Wigner distribution, P w(x,p;t), for Gaussian wave packet solutions of this system, and illustrate the short-term semi-classical time dependence and the longer-term revival and fractional revival behavior. Our results indicate how the Wigner distribution can be used to examine the highly correlated dynamical position-momentum structure of quantum states. In particular, this tool provides an excellent way of demonstrating the patterns of highly correlated Schr{\"o}dinger-cat-like {"}mini-packets{"} which appear at fractional multiples of the exact revival time.",
author = "M. Belloni and Doncheski, {M. A.} and Robinett, {R. W.}",
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T2 - Energy eigenstates and time-dependent wave packets

AU - Belloni, M.

AU - Doncheski, M. A.

AU - Robinett, R. W.

PY - 2004/9/1

Y1 - 2004/9/1

N2 - We calculate and visualize the Wigner quasi-probability distribution for the position and momentum, P W (n)(x, p), for the energy eigenstates of the infinite square well. We evaluate the time-dependent Wigner distribution, P w(x,p;t), for Gaussian wave packet solutions of this system, and illustrate the short-term semi-classical time dependence and the longer-term revival and fractional revival behavior. Our results indicate how the Wigner distribution can be used to examine the highly correlated dynamical position-momentum structure of quantum states. In particular, this tool provides an excellent way of demonstrating the patterns of highly correlated Schrödinger-cat-like "mini-packets" which appear at fractional multiples of the exact revival time.

AB - We calculate and visualize the Wigner quasi-probability distribution for the position and momentum, P W (n)(x, p), for the energy eigenstates of the infinite square well. We evaluate the time-dependent Wigner distribution, P w(x,p;t), for Gaussian wave packet solutions of this system, and illustrate the short-term semi-classical time dependence and the longer-term revival and fractional revival behavior. Our results indicate how the Wigner distribution can be used to examine the highly correlated dynamical position-momentum structure of quantum states. In particular, this tool provides an excellent way of demonstrating the patterns of highly correlated Schrödinger-cat-like "mini-packets" which appear at fractional multiples of the exact revival time.

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