Simple examples of position-momentum correlated Gaussian free-particle wave packets in one dimension with the general form of the time-dependent spread in position

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Abstract

We provide simple examples of closed-form Gaussian wave packet solutions of the free-particle Schrödinger equation in one dimension which exhibit the most general form of the time-dependent spread in position, namely (Δx t ) 2 = (Δx 0) 2 + At + (Δp 0) 2 t 2/m 2, where A ≡ 〈(x̂ - 〈x̂〉 0)(p̂ - 〈p̂〉 0) + (p̂ - 〈p̂〉 0) (x̂ - 〈x̂〉 0)〉 0 contains information on the position-momentum correlation structure of the initial wave packet. We exhibit straightforward examples corresponding to squeezed states, as well as quasi-classical cases, for which A < 0 so that the position spread can (at least initially) decrease in time because of such correlations. We discuss how the initial correlations in these examples can be dynamically generated (at least conceptually) in various bound state systems. Finally, we focus on providing different ways of visualizing the x - p correlations present in these cases, including the time-dependent distribution of kinetic energy and the use of the Wigner quasi-probability distribution. We discuss similar results, both for the time-dependent Δx t and special correlated solutions, for the case of a particle subject to a uniform force.

Original languageEnglish (US)
Pages (from-to)455-475
Number of pages21
JournalFoundations of Physics Letters
Volume18
Issue number5
DOIs
StatePublished - Aug 1 2005

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wave packets
momentum
kinetic energy

All Science Journal Classification (ASJC) codes

  • Physics and Astronomy(all)

Cite this

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title = "Simple examples of position-momentum correlated Gaussian free-particle wave packets in one dimension with the general form of the time-dependent spread in position",
abstract = "We provide simple examples of closed-form Gaussian wave packet solutions of the free-particle Schr{\"o}dinger equation in one dimension which exhibit the most general form of the time-dependent spread in position, namely (Δx t ) 2 = (Δx 0) 2 + At + (Δp 0) 2 t 2/m 2, where A ≡ 〈(x̂ - 〈x̂〉 0)(p̂ - 〈p̂〉 0) + (p̂ - 〈p̂〉 0) (x̂ - 〈x̂〉 0)〉 0 contains information on the position-momentum correlation structure of the initial wave packet. We exhibit straightforward examples corresponding to squeezed states, as well as quasi-classical cases, for which A < 0 so that the position spread can (at least initially) decrease in time because of such correlations. We discuss how the initial correlations in these examples can be dynamically generated (at least conceptually) in various bound state systems. Finally, we focus on providing different ways of visualizing the x - p correlations present in these cases, including the time-dependent distribution of kinetic energy and the use of the Wigner quasi-probability distribution. We discuss similar results, both for the time-dependent Δx t and special correlated solutions, for the case of a particle subject to a uniform force.",
author = "Robinett, {Richard Wallace} and Michael Doncheski and Bassett, {L. C.}",
year = "2005",
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T1 - Simple examples of position-momentum correlated Gaussian free-particle wave packets in one dimension with the general form of the time-dependent spread in position

AU - Robinett, Richard Wallace

AU - Doncheski, Michael

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PY - 2005/8/1

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N2 - We provide simple examples of closed-form Gaussian wave packet solutions of the free-particle Schrödinger equation in one dimension which exhibit the most general form of the time-dependent spread in position, namely (Δx t ) 2 = (Δx 0) 2 + At + (Δp 0) 2 t 2/m 2, where A ≡ 〈(x̂ - 〈x̂〉 0)(p̂ - 〈p̂〉 0) + (p̂ - 〈p̂〉 0) (x̂ - 〈x̂〉 0)〉 0 contains information on the position-momentum correlation structure of the initial wave packet. We exhibit straightforward examples corresponding to squeezed states, as well as quasi-classical cases, for which A < 0 so that the position spread can (at least initially) decrease in time because of such correlations. We discuss how the initial correlations in these examples can be dynamically generated (at least conceptually) in various bound state systems. Finally, we focus on providing different ways of visualizing the x - p correlations present in these cases, including the time-dependent distribution of kinetic energy and the use of the Wigner quasi-probability distribution. We discuss similar results, both for the time-dependent Δx t and special correlated solutions, for the case of a particle subject to a uniform force.

AB - We provide simple examples of closed-form Gaussian wave packet solutions of the free-particle Schrödinger equation in one dimension which exhibit the most general form of the time-dependent spread in position, namely (Δx t ) 2 = (Δx 0) 2 + At + (Δp 0) 2 t 2/m 2, where A ≡ 〈(x̂ - 〈x̂〉 0)(p̂ - 〈p̂〉 0) + (p̂ - 〈p̂〉 0) (x̂ - 〈x̂〉 0)〉 0 contains information on the position-momentum correlation structure of the initial wave packet. We exhibit straightforward examples corresponding to squeezed states, as well as quasi-classical cases, for which A < 0 so that the position spread can (at least initially) decrease in time because of such correlations. We discuss how the initial correlations in these examples can be dynamically generated (at least conceptually) in various bound state systems. Finally, we focus on providing different ways of visualizing the x - p correlations present in these cases, including the time-dependent distribution of kinetic energy and the use of the Wigner quasi-probability distribution. We discuss similar results, both for the time-dependent Δx t and special correlated solutions, for the case of a particle subject to a uniform force.

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