Spreading of a free wave packet

Lee, Hai‐Woong

doi:10.1119/1.13075

Cited by 18 publications

(14 citation statements)

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“…This distribution, first discussed by Wigner [16], and reviewed extensively in the research [17] and pedagogical [18] literature (and even in the context of wave packet spreading [19]), is as close as one can come to a quantum phase-space distribution, and while not directly measurable, can still be profitably used to illustrate any x − p correlations. For the standard minimum-uncertainty Gaussian wavefunctions defined by Eqs.…”

Section: Standard Minimum-uncertainty Gaussian Wave Packetsmentioning

confidence: 97%

“…We can then define similar quantities for the kinetic energy in the 'front' and/or 'back' halves of the wave packet, using x t as the measuring point, via For the standard Gaussian wave packet in Eq. (19), the local kinetic energy density is given by…”

Section: Standard Minimum-uncertainty Gaussian Wave Packetsmentioning

confidence: 99%

“…The time-development of a standard Gaussian wave packet, ψ (G ) (x, t), described by Eq (19),. with the modulus (solid) and real part (dashed) shown for t = 0 and t = 2t 0 on the top plot.…”

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confidence: 99%

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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

2005

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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 (Δxtion 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 0894-9875/05/1000-0455/0 455 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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Section: Standard Minimum-uncertainty Gaussian Wave Packetsmentioning

confidence: 97%

Section: Standard Minimum-uncertainty Gaussian Wave Packetsmentioning

confidence: 99%

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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

2005

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“…Of course, for any τ ∈ [0, 1] we can define "a Wigner τ -function" as a τ -symbol of a density operator (Definition 1 corresponds to the case τ = 1 2 ); for τ = 0 and τ = 1 such objects (but defined in a different way) were considered in the literature (see, for example, [9] and the references therein).…”

Section: Wigner Function Definition 1 a Wigner Function W T Correspmentioning

confidence: 99%

“…Remark 4. A paper [2] of Wigner himself contains Definition 2; Definition 4 can be found, for example, in [9]; Definition 3, in fact, exists in [11].…”

Section: Proposition 4 Definition 4 Of a Wigner Function Is Equivalementioning

confidence: 99%

Wigner Function and Diffusion in a Collision‐Free Medium of Quantum Particles

Козлов¹,

Smolyanov²

2007

Theory Probab. Appl.

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A quantum Poincaré model (realizing behavior of ideal gas of noninteracting quantum Bolztman particles) is introduced. We use the fact that the evolution of the Wigner function corresponding to a quantum system with a quadratic Hamiltonian coincides with the evolution of a probability distribution on a phase space of the Hamiltonian system, the quantization of which gives the quantum system under consideration.A quantum analogue of the Poincaré model realizing the irreversible behavior of ideal gas (continuous medium) of identical (pointlike) noninteracting free classical particles is described with the help of a Wigner function.In the original Poincaré model [1] (detailed analysis of this model can be found in [3]) the ideal gas of identical (pointlike) noninteracting free classical particles in a rectangular parallelepiped is considered. A density of particles or (which is the same) a density of ideal gas is identified with a probability density of a positions distribution of a free particle in a parallelepiped (other characteristics of ideal gas are not considered in the frameworks of the studied model). It is proved in [3] that when time tends to infinity (or to minus infinity) this probability distribution tends to uniform distribution (assuming that the initial density of distribution is a summable function of positions and velocities). Hence the density of ideal gas tends to uniform density when time infinitely increases that corresponds to the irreversibility of the evolution of such a gas (in fact, in [3] a motion on torus is considered and it is noticed that this situation is universal for nondegenerate completely integrable Hamiltonian systems). It is worth emphasizing that the density also tends to be uniform when time decreases to minus infinity as the equations describing the dynamic of a particle system are invariant with regard to time inversion.In the present paper it is shown that in the case of a noncompact configuration space, a probability distribution of positions for a free classical particle and a probability distribution of results of positions measurements for a free quantum particle both tend to uniform distribution. This means by definition that the ratio of probabilities that a particle appears in two different compact domains of the configuration space tends to a ratio of Lebesgue measures of these domains (again both in the case of both infinitely increasing and infinitely decreasing time) (cf. [4]).Applying this result to an ideal gas of identical noninteracting free particles, we assume that the density of such a gas is a density of a probability distribution of positions (in the quantum case again we mean results of position measurements) for * Received by the editors October 6, 2005.

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Propagation, Spreading, and Scattering

Duffey

1984

A Development of Quantum Mechanics

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Spreading of a free wave packet

Cited by 18 publications

References 0 publications

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

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

Wigner Function and Diffusion in a Collision‐Free Medium of Quantum Particles

Propagation, Spreading, and Scattering

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