Kähler Product and Symmetry Data in Quantum Mechanics

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姓名

劉為尹(Wei-Yin Liu) 查詢紙本館藏

畢業系所

物理學系

論文名稱

(Kähler Product and Symmetry Data in Quantum Mechanics)

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摘要(中)

一般我們會使用希爾伯特空間中的向量代表量子力學中的物理狀態，但實際上非簡併的空間是映射希爾伯特空間。希爾伯特空間與映射希爾伯特空間都是凱勒流形。我們可以從薛丁格方程式與漢彌爾頓方程式看出凱勒流形在物理上的重要性。
凱勒積介紹了漢彌爾頓函數與量子物理中運算子之間的同構關係。對稱資料由在流形上某點的漢彌爾頓方程式及其導數組成，可以做為運算子的′值′的候選。

摘要(英)

We usually consider a vector in Hilbert space to represent a physical state in quantum mechanics but the nondegenerate space is projective Hilbert space. Hilbert space and projective Hilbert space are both Kähler manifolds. We can show the importance of Kähler manifold to physics by the connection to Schrödinger equation and Hamilton’s equation.
Kähler product introduce an isomorphism between Hamiltonian functions and operators in quantum physics. Symmetry data is a candidate of the “value” of the operator, which contains the Hamiltonian function and its derivatives at a point on the manifolds.

關鍵字(中)

★ 量子力學
★ 凱勒流形

關鍵字(英)

★ Quantum mechanics
★ Kähler Manifold

論文目次

一、 Introduction ............................ 1
二、 Kähler manifold ........................ 5
2.1 Complex structure . .................. 5
2.2 Riemannian structure . ................ 6
2.3 Symplectic structure . ................. 7
2.4 Hamiltonian mechanics . ............... 7
2.4.1 Hamilton′s equation . ................. 7
2.4.2 Hamiltonian vector field . ............... 9
2.5 Metrics in real and complex coordinates . ...... 9
三、 Physical space in quantum mechanics ........... 11
3.1 Schrödinger equation . ................. 11
3.2 Projective Hilbert space P . ............. 12
3.2.1 Metric of P . ...................... 12
3.2.2 Hamiltonian vector field on P . ............ 13
3.2.3 Killing reduction . ................... 14
四、 Kähler product.................. 15
五、 Symmetry data ......................... 17
5.1 Covariant derivative .................. 17
5.1.1 Koszul form . ..................... 18
5.1.2 The covariant derivative of a Killing vector field 19
5.2 Symmetry data in H and P . ............. 21
六、 Summary and conclusion ............................ 25
參考文獻.................................. 27
附錄一 ................................. 29
A.1 The relationship between X and X . ......... 29
A.2 Coordinate transformation of vector and covector
between z and w . ................... 34
A.3 Calculation details of symmetry data . ....... 35
A.4 Creation,annihilation and number operators . .... 36

參考文獻

[1] Chruscinski, D.and Jamiolkowski,A.(2004).Geometric Phases in Classical and Quantum Mechanics,Birkhäuser.
[2] T.A. Schilling,Geometry of quantum mechanics,doctoral thesis (The Pennsylvania State University 1996);A.Ashtekar,A.,Schilling,
T.A., Geometrical Formulation of Quantum Mechanics,in A.Harvey (ed.), On Einstein′s Path,Springer,p.23(1998),[gr-qc/9706069].
[3] R. Cirelli,A.Manià,and L.Pizzocchero,Quantum Mechanics as an infnite-Dimensional Hamiltonian System with Uncertainty Structure. PartI,J.Math.Phys.31(1990)2891-2897.
[4] W. Ballmann,(2006).Lectures on Kähler Manifolds,European
Mathematical Society.
[5] A. Moroianu, (2010).Lectures on Kähler Geometry,CambridgeUniversityPress.
[6] D. N.Kupeli,Singular Semi-Riemannian Geometry-Springer,(1996).Centre for Mathematics and Computer Science,Amsterdam,The Netherlands.
[7] R. M.Wald,General Relativity, (1984).University of Chicago Press.
[8] O.C. Stoica,On Singular Semi-Riemannian Manifolds,aXriv:
1105.0201
[9] J. J.Sakurai,Jim J.Napolitano,Modern Quantum Mechanics(2nd Edition). Addison-Wesley.
[10] D. Xiao,M.C.Chang,and Q.Niu,Berry phase effects on electronic properties,Rev.Mod.Phys.82,1959(2010).

指導教授

江祖永(Otto C. W. Kong)

審核日期

2019-3-27

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