arXiv quant-ph/01120802001-12quant-phmath-phmath.MPnlin.SI

量子力学和经典力学中的刘维尔不变性

Liouville invariance in quantum and classical mechanics

Alec Maassen van den Brink, A. M. Zagoskin

备注：: REVTeX 4, 14 p.; 1 built-in + 1 PS figure; N.B. `Alec' is my first name, `Maassen van den Brink' my family name; Pls delete the `2000' postfix to `amsmath' on l. 2 of the ms file, it's for arXiv/LANL only; v2: main text subdivided into sections, small textual changes for clarity; v3: published version with reordered sections, and figure + historical refs added
发表信息：: Quantum Information Processing_1_, 55 (2002)

摘要：

量子力学的密度矩阵和海森堡公式--对于幺正演化--直接从薛定谔方程导出。然而，相应的演化算子，Liouvillian L=i[.，H]，不需要限于哈密顿量H的那些。这是由于L只涉及本征能_差_，即使能量本身不是简并的，本征能_差_也可以简并。值得注意的是，这种可能性在文献中很少被提及，也从未被更普遍地追求。我们考虑一个涉及介观约瑟夫森器件的例子，但分析只假设熟悉基本的量子力学。随后，这种L对称性被证明更广泛地出现，特别是在经典力学中。简要讨论了对称性与耗散系统和量子信息处理的关系。

英文摘要：

The density-matrix and Heisenberg formulations of quantum mechanics follow--for unitary evolution--directy from the Schr"odinger equation. Nevertheless, the symmetries of the corresponding evolution operator, the Liouvillian L=i[.,H], need not be limited to those of the Hamiltonian H. This is due to L only involving eigenenergy_differences_, which can be degenerate even if the energies themselves are not. Remarkably, this possibility has rarely been mentioned in the literature, and never pursued more generally. We consider an example involving mesoscopic Josephson devices, but the analysis only assumes familiarity with basic quantum mechanics. Subsequently, such _L-symmetries_ are shown to occur more widely, in particular also in classical mechanics. The symmetry's relevance to dissipative systems and quantum-information processing is briefly discussed.

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