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研究生： 周川昇
CHUAN-SHENG CHEW
論文名稱： Wigner-Weyl's transform and its contraction
指導教授： 江祖永
Otto C.W. Kong
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 物理學系
Department of Physics
論文出版年： 2017
畢業學年度： 105
語文別： 英文
論文頁數： 51
中文關鍵詞： 周川昇
相關次數： 點閱：13  下載：0 
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    • 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.

    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.

    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)

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