### Abstract

We theoretically studied the time-developing ballistic phenomena of a single-electron confined in a half-ellipse infinite-potential wall by solving the time-dependent Schrödinger equation numerically. We also solved the corresponding Newton equation in order to compare the classical results with the quantum ones, and extracted the quantum features. The ellipse-shaped potential wall completely reflects an electron and causes the focusing ratio of unity in the classical limit. The dispersion of the wave packet of an electron, however, weakens this characteristic nature, and reduces the focusing ratio from unity. Because the dispersion also lets an electron arrive at the collector indistinctly, we define the effective arrival time by finding inflections in the time-dependent profile of the probability density at the collector. Based on the second-derivation technique, we further determine the quantum arrival time (QAT) at which the intrusion of the wave packet occurs dominantly. The comparison of this QAT with the classical arrival time (CAT) determines whether the corresponding ballistic propagation can be discussed on the basis of the quantum consideration or the classical prediction. We further studied how the change in the half-ellipse potential wall shape affects the ballistic phenomena through the change in the ellipticity γ, the system size L and the dispersion degree σ of the wave packet. Using the ellipse-shaped infinite-potential wall, the application of the magnetic field causes irrational cyclotron motion assisted by the ellipse potential, in addition to the rational cyclotron motions. The numerical solution of the time-dependent Schrödinger equation determines the unique cyclotron motion whose peculiarity is caused by the dispersion of the wave packet and is rarely predicted by the classical limit.

Original language | English |
---|---|

Pages (from-to) | 4252-4268 |

Number of pages | 17 |

Journal | Japanese Journal of Applied Physics, Part 1: Regular Papers and Short Notes and Review Papers |

Volume | 44 |

Issue number | 6 A |

DOIs | |

Publication status | Published - 2005 Jun |

### Fingerprint

### Keywords

- Ballistic phenomena
- Classical arrival time
- Cyclotron motion
- Half-ellipse confined room
- Quantum arrival time
- Time-dependent Schrödinger equation
- Wave packet

### ASJC Scopus subject areas

- Physics and Astronomy (miscellaneous)

### Cite this

*Japanese Journal of Applied Physics, Part 1: Regular Papers and Short Notes and Review Papers*,

*44*(6 A), 4252-4268. https://doi.org/10.1143/JJAP.44.4252

**Time-dependent ballistic phenomena of electron injected into half-ellipse confined room.** / Koiso, Takuji; Muraguchi, Masakazu; Takeda, Kyozaburo; Watanabe, Naoki.

Research output: Contribution to journal › Article

*Japanese Journal of Applied Physics, Part 1: Regular Papers and Short Notes and Review Papers*, vol. 44, no. 6 A, pp. 4252-4268. https://doi.org/10.1143/JJAP.44.4252

}

TY - JOUR

T1 - Time-dependent ballistic phenomena of electron injected into half-ellipse confined room

AU - Koiso, Takuji

AU - Muraguchi, Masakazu

AU - Takeda, Kyozaburo

AU - Watanabe, Naoki

PY - 2005/6

Y1 - 2005/6

N2 - We theoretically studied the time-developing ballistic phenomena of a single-electron confined in a half-ellipse infinite-potential wall by solving the time-dependent Schrödinger equation numerically. We also solved the corresponding Newton equation in order to compare the classical results with the quantum ones, and extracted the quantum features. The ellipse-shaped potential wall completely reflects an electron and causes the focusing ratio of unity in the classical limit. The dispersion of the wave packet of an electron, however, weakens this characteristic nature, and reduces the focusing ratio from unity. Because the dispersion also lets an electron arrive at the collector indistinctly, we define the effective arrival time by finding inflections in the time-dependent profile of the probability density at the collector. Based on the second-derivation technique, we further determine the quantum arrival time (QAT) at which the intrusion of the wave packet occurs dominantly. The comparison of this QAT with the classical arrival time (CAT) determines whether the corresponding ballistic propagation can be discussed on the basis of the quantum consideration or the classical prediction. We further studied how the change in the half-ellipse potential wall shape affects the ballistic phenomena through the change in the ellipticity γ, the system size L and the dispersion degree σ of the wave packet. Using the ellipse-shaped infinite-potential wall, the application of the magnetic field causes irrational cyclotron motion assisted by the ellipse potential, in addition to the rational cyclotron motions. The numerical solution of the time-dependent Schrödinger equation determines the unique cyclotron motion whose peculiarity is caused by the dispersion of the wave packet and is rarely predicted by the classical limit.

AB - We theoretically studied the time-developing ballistic phenomena of a single-electron confined in a half-ellipse infinite-potential wall by solving the time-dependent Schrödinger equation numerically. We also solved the corresponding Newton equation in order to compare the classical results with the quantum ones, and extracted the quantum features. The ellipse-shaped potential wall completely reflects an electron and causes the focusing ratio of unity in the classical limit. The dispersion of the wave packet of an electron, however, weakens this characteristic nature, and reduces the focusing ratio from unity. Because the dispersion also lets an electron arrive at the collector indistinctly, we define the effective arrival time by finding inflections in the time-dependent profile of the probability density at the collector. Based on the second-derivation technique, we further determine the quantum arrival time (QAT) at which the intrusion of the wave packet occurs dominantly. The comparison of this QAT with the classical arrival time (CAT) determines whether the corresponding ballistic propagation can be discussed on the basis of the quantum consideration or the classical prediction. We further studied how the change in the half-ellipse potential wall shape affects the ballistic phenomena through the change in the ellipticity γ, the system size L and the dispersion degree σ of the wave packet. Using the ellipse-shaped infinite-potential wall, the application of the magnetic field causes irrational cyclotron motion assisted by the ellipse potential, in addition to the rational cyclotron motions. The numerical solution of the time-dependent Schrödinger equation determines the unique cyclotron motion whose peculiarity is caused by the dispersion of the wave packet and is rarely predicted by the classical limit.

KW - Ballistic phenomena

KW - Classical arrival time

KW - Cyclotron motion

KW - Half-ellipse confined room

KW - Quantum arrival time

KW - Time-dependent Schrödinger equation

KW - Wave packet

UR - http://www.scopus.com/inward/record.url?scp=23944451079&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=23944451079&partnerID=8YFLogxK

U2 - 10.1143/JJAP.44.4252

DO - 10.1143/JJAP.44.4252

M3 - Article

VL - 44

SP - 4252

EP - 4268

JO - Japanese Journal of Applied Physics, Part 1: Regular Papers & Short Notes

JF - Japanese Journal of Applied Physics, Part 1: Regular Papers & Short Notes

SN - 0021-4922

IS - 6 A

ER -