arXiv quant-ph/00050262000-05quant-ph

重新审视薛定谔：代数方法

Schrodinger revisited: an algebraic approach

M. R. Brown, B. J. Hiley

备注：: Gaussian wave packet example corrected, and minor typographical and phrasing errors rectified

摘要：

从量子Liouville方程出发，将密度算符分别记为算符代数的左、右理想中元素的乘积，发现薛定谔图像可以用密度算符和相位算符的两种表示独立的代数形式表示.这些形式分别是连续性方程，其中涉及的交换子的哈密顿与密度算子，和一个方程的时间发展的相位算子，其中涉及的反交换子的哈密顿与这个密度算子。我们发现，后者的方程起着两个重要的作用：（i）它表示的能量守恒在一个系统中的能量是很好的定义和（ii）它提供了一个简单的方法来评估规范的变化发生在Aharonov-Bohm，Aharonov-Casher，和Berry相位效应。这两个算子（即纯代数）方程也允许我们重新检查玻姆解释，表明它实际上是可以构建玻姆解释的表示以外的x-表示。我们讨论的玻姆解释的意义，根据这些新的结果在非交换结构，这使我们能够澄清它的关系，标准的量子力学。

英文摘要：

Starting with the quantum Liouville equation, we write the density operator as the product of elements respectively in the left and right ideals of an operator algebra and find that the Schrodinger picture may be expressed through two representation independent algebraic forms in terms of the density and phase operators. These forms are respectively the continuity equation, which involves the commutator of the Hamiltonian with the density operator, and an equation for the time development of the phase operator that involves the anti-commutator of the Hamiltonian with this density operator. We show that this latter equation plays two important roles: (i) it expresses the conservation of energy in a system where energy is well defined and (ii) it provides a simple way to evaluate the gauge changes that occur in the Aharonov-Bohm, the Aharonov-Casher, and Berry phase effects. Both these operator (i.e. purely algebraic) equations also allow us to re-examine the Bohm interpretation, showing that it is in fact possible to construct Bohm interpretations in representations other than the x-representation. We discuss the meaning of the Bohm interpretation in the light of these new results in terms of non-commutative structures and this enables us to clarify its relation to standard quantum mechanics.

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