### Abstract

For the creation operator a† and the annihilation operator a of a harmonic oscillator, we consider Weyl ordering expression of (a†a) ^{n} and obtain a new symmetric expression of Weyl ordering w.r.t. a†a ≡ N and aa† = N + 1 where N is the number operator. Moreover, we interpret intertwining formulas of various orderings in view of the difference theory. Then we find that the noncommutative parameter corresponds to the increment of the difference operator w.r.t. variable N. Therefore, quantum (noncommutative) calculations of harmonic oscillators are done by classical (commutative) ones of the number operator by using the difference theory. As a by-product, nontrivial relations including the Stirling number of the first kind are also obtained.

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

Pages (from-to) | 827-840 |

Number of pages | 14 |

Journal | Modern Physics Letters A |

Volume | 19 |

Issue number | 11 |

DOIs | |

Publication status | Published - 2004 Apr 10 |

Externally published | Yes |

### Fingerprint

### Keywords

- Difference operator
- Harmonic oscillator
- Quantization
- Stirling number
- Weyl ordering

### ASJC Scopus subject areas

- Physics and Astronomy(all)
- Statistical and Nonlinear Physics
- Mathematical Physics
- Nuclear and High Energy Physics

### Cite this

*Modern Physics Letters A*,

*19*(11), 827-840. https://doi.org/10.1142/S021773230401374X

**A new symmetric expression of Weyl ordering.** / Fujii, Kazuyuki; Suzuki, Tatsuo.

Research output: Contribution to journal › Article

*Modern Physics Letters A*, vol. 19, no. 11, pp. 827-840. https://doi.org/10.1142/S021773230401374X

}

TY - JOUR

T1 - A new symmetric expression of Weyl ordering

AU - Fujii, Kazuyuki

AU - Suzuki, Tatsuo

PY - 2004/4/10

Y1 - 2004/4/10

N2 - For the creation operator a† and the annihilation operator a of a harmonic oscillator, we consider Weyl ordering expression of (a†a) n and obtain a new symmetric expression of Weyl ordering w.r.t. a†a ≡ N and aa† = N + 1 where N is the number operator. Moreover, we interpret intertwining formulas of various orderings in view of the difference theory. Then we find that the noncommutative parameter corresponds to the increment of the difference operator w.r.t. variable N. Therefore, quantum (noncommutative) calculations of harmonic oscillators are done by classical (commutative) ones of the number operator by using the difference theory. As a by-product, nontrivial relations including the Stirling number of the first kind are also obtained.

AB - For the creation operator a† and the annihilation operator a of a harmonic oscillator, we consider Weyl ordering expression of (a†a) n and obtain a new symmetric expression of Weyl ordering w.r.t. a†a ≡ N and aa† = N + 1 where N is the number operator. Moreover, we interpret intertwining formulas of various orderings in view of the difference theory. Then we find that the noncommutative parameter corresponds to the increment of the difference operator w.r.t. variable N. Therefore, quantum (noncommutative) calculations of harmonic oscillators are done by classical (commutative) ones of the number operator by using the difference theory. As a by-product, nontrivial relations including the Stirling number of the first kind are also obtained.

KW - Difference operator

KW - Harmonic oscillator

KW - Quantization

KW - Stirling number

KW - Weyl ordering

UR - http://www.scopus.com/inward/record.url?scp=2142713819&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=2142713819&partnerID=8YFLogxK

U2 - 10.1142/S021773230401374X

DO - 10.1142/S021773230401374X

M3 - Article

VL - 19

SP - 827

EP - 840

JO - Modern Physics Letters A

JF - Modern Physics Letters A

SN - 0217-7323

IS - 11

ER -