arXiv quant-ph/97100101997-10quant-ph

Schroedinger方程的非线性相位修正

Nonlinear Phase Modification of the Schroedinger Equation

Waldemar Puszkarz

备注：: Latex, 21 pages, extended and slightly modified, new references added

摘要：

提出了薛定谔方程的一种非线性修正，其中薛定谔方程的拉格朗日密度通过$Δ^{m}\ln (Ψ^{*}/Ψ)$乘以$Ψ^{*}Ψ$中的多项式项来扩展.这通过波函数$Ψ=R\exp (iS)$的相位$S$而不是振幅$R$以伽利略不变方式引入齐次非线性。从这个一般方案中，我们选择以某种合理的方式定义的最简单的极小模型。这个模型提供了一种最简单的方法来修改量子力学的玻姆公式，以便允许对量子势的领先相位贡献和对概率流的领先量子贡献，从而消除玻姆原始公式中存在的不对称性。它保留了薛定谔方程的大部分物理相关性质，包括量子力学系统的定态。它可以被认为是扩展对象的非线性量子力学的最简单模型，也出现在所提出的一般方案中。将该模型推广到$n$粒子，并研究了复合系统的可分性问题。注意到除了强可分扩张之外，还存在弱可分扩张。一般的修改计划在更广泛的频谱的非线性修改的薛定谔方程的地方进行了讨论。指出它所产生的模型具有唯一的能量定义，即场论能量泛函与量子力学能量泛函一致。它被发现，拉格朗日为它的最简单的变体表示拉格朗日为限制版本的Doebner-戈尔丁修改这个方程。

英文摘要：

A nonlinear modification of the Schrödinger equation is proposed in which the Lagrangian density for the Schrödinger equation is extended by terms polynomial in $Δ^{m}\ln (Ψ^{*}/Ψ)$ multiplied by $Ψ^{*}Ψ$. This introduces a homogeneous nonlinearity in a Galilean invariant manner through the phase $S$ rather than the amplitude $R$ of the wave function $Ψ=R\exp (iS)$. From this general scheme we choose the simplest minimal model defined in some reasonable way. The model in question offers the simplest way to modify the Bohm formulation of quantum mechanics so as to allow a leading phase contribution to the quantum potential and a leading quantum contribution to the probability current removing asymmetries present in Bohm's original formulation. It preserves most of physically relevant properties of the Schrödinger equation including stationary states of quantum-mechanical systems. It can be thought of as the simplest model of nonlinear quantum mechanics of extended objects among other such models that also emerge within the general scheme proposed. The extensions of this model to $n$ particles and the question of separability of compound systems are studied. It is noted that there exists a weakly separable extension in addition to a strongly separable one. The place of the general modification scheme in a broader spectrum of nonlinear modifications of the Schrödinger equation is discussed. It is pointed out that the models it gives rise to have a unique definition of energy in that the field-theoretical energy functional coincides with the quantum-mechanical one. It is found that the Lagrangian for its simplest variant represents the Lagrangian for a restricted version of the Doebner-Goldin modification of this equation.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文