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姓名 周川昇(CHUAN-SHENG CHEW)  查詢紙本館藏   畢業系所 物理學系
論文名稱
(Wigner-Weyl′s transform and its contraction)
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摘要(中) 倘若我們選取 相干態(coherent state)為量子力學 希爾博特(Hilbert space) 向量空間的基地, 則我們發現其對應波函數(wave function)有與 韋恩-威爾變換(WignerWeyl’s transform) 相似的結構. 我們可以將 量力可量測(Obserable)在波函數的表示(representation)與 韋恩-威爾 上的表示視為僅為參數差1/2.另外,不論是量力的波函數表示,還是 韋恩-威爾 變換都有共容的古典極限.
摘要(英) We started with hbar = 1 in each different representation space said matrix representation on Hilbert space, realization as square integrable function and Wigner distribution. We introduced contraction parameter hbar by rescaled generator Q hat and P hat and consider their limitation i.e. hbar → 0 and showed comparable classical limit for each representation.

If coherent state has been chosen as base for our quantum Hilbert space, we found that correspond wave function (realization) have some star structure similar to Wigner-Weyl’s method and identify quantum observable algebra on both representation are equivalent up to a 1/2 factor.
關鍵字(中) ★ 周川昇 關鍵字(英)
論文目次 Contents

1 Introduction (1)

2 Quantum mechanic in Hilbert space (3)
2.1 Matrix representation (4)
2.2 Quantum dynamics (5)
2.3 Rrescaling and contraction consideration (5)
2.4 Appendix A (8)
2.4.1 Derivation for coherent state overlap (8)
2.4.2 Rearranged arbitrary operator polynomial into normal ordering (9)
2.4.3 Commutation relation for arbitrary ordering operator polynomial (10)
2.4.4 Limitation of matrix representation for normal ordering operator (13)

3 Wigner-Weyl’s transform (16)
3.1 Wigner-Weyl’s transform (17)
3.2 star product and Moyal’s bracket (18)
3.3 Average in distribution space (19)
3.4 Contraction in distribution space (20)
3.5 Appendix B (22)
3.5.1 Derivation explicit form for Wigner-Weyl’s transform (22)
3.5.2 Wigner-Weyl’s transform for rho_ab (23)
3.5.3 star cancelling (24)

4 Quantum mechanic realized in square integrable function (25)
4.1 Realization of quantum operator (25)
4.2 star product ? Moyal ’s bracket ? (26)
4.3 Transition amplitude (27)
4.4 Contraction in wave function space (28)
4.5 Appendix C (29)
4.5.1 Derive realization of canonical operator µ hat (29)
4.5.2 Homomorphism of defined map R(µ) (30)
4.5.3 Hermitian property for realization of canonical operator (30)
4.5.4 Weyl’s group action on function space (31)

5 Wigner-Weyl’s method as quantum realization on coherent state (33)
5.1 Weyl ordering polynomial as a fourier integration (34)
5.2 Bopp’s shifting and quantum observable (36)
5.3 star - twisted ⊗ : product correspond of fourier transform for noncommutative variables (36)
5.4 Transition amplitude (38)
5.5 Contraction limit (39)
5.6 Appendix D (41)
5.6.1 proof of the inverse fourier transformation F (41)
5.6.2 Derivation specific charateristic function g(µ)(41) 5.6.3 proof for tilde(F)[ (hat(F)^(−1)rho_ab(µ) ] (43)

Conclusion (44)

Bibliography (45)
參考文獻
[1] Herbert Goldstein “Classical Mechanics” third edition.

[2] J.J. Sakurai “Modern Quantum Mechaincs” revised eidition.

[3] John R. Klauder “Continuous representations and Path integrals, Revisited”,Volume 34 of the series NATO Advanced Study Institutes Series pp 5-38.

[4] John R. Klauder “A Modern Approach to Functional Integration”.

[5] A. M. Perelomov “Generalized coherent states and some of their applications”,1977 Sov. Phys. Usp. 20 703.

[6] William B. Case “Wigner functions and Weyl transforms for pedestrians”, Am.J. Phys. 76(10), October 2008.

[7] M. Hillery, R.F. O’Connell, M.O. Scully, E.P. Wigner “Distribution functions in physics: Fundamentals”,Physics Reports, volume 106, issue 3, April 1984, pages
121-167.

[8] J. E. Moyal “Quantum mechanics as a statistical theory”, Mathematical Proceedings of the Cambridge Philosophical Society, volume 45, issue 01, January
1949, pp 99 - 124.

[9] G. S. Agarwal and E. Wolf “Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. I. Mapping Theorems and Ordering of Functions of Noncommuting Operators”, Phys. Rev. D 2, 2161 (1970).45

[10] G. S. Agarwal and E. Wolf “Calculus for Functions of Noncommuting Operators and General Phase-Space Methods in Quantum Mechanics. II. Quantum Mechanics in Phase Space Phys. Rev. D 2, 2187 (1970).

[11] D. A. Dubin, Mark A. Hennings, T. B. Smith Mathematical Aspects of Weyl Quantization and Phase”, World Scientific, 2000.
指導教授 江祖永(Otto C.W. Kong) 審核日期 2017-5-16
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