Fractional quantum mechanics
Fractional quantum mechanics
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DOI：
10.1103/physreve.62.3135
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发表时间：
2000-09-01
影响因子：
2.4
通讯作者：
Laskin, N
中科院分区：
文献类型：
--
作者：
Laskin, N
已经发展出一种量子物理学的路径积分方法。定义了列维飞行路径上的分数路径积分。结果表明，如果布朗轨迹的分形性导致标准量子力学和统计力学，那么列维路径的分形性就会导致分数量子力学和分数统计力学。通过我们的分数路径积分方法已经发展出分数量子力学和分数统计力学。已经发现薛定谔方程的分数形式推广。已经建立了非相对论量子力学粒子的能量和动量之间的关系。已经得到分数平面波函数的方程。我们使用福克斯H函数推导出了自由粒子量子力学核。已经建立了海森堡不确定性关系的分数形式推广。通过路径积分方法已经发展出分数统计力学。已经发现密度矩阵运动方程的分数形式推广。自由粒子的密度矩阵已经用福克斯H函数表示。我们还讨论了分数形式与著名的费曼路径积分方法在量子力学和统计力学方面的关系。
A path integral approach to quantum physics has been developed. Fractional path integrals over the paths of the Livy flights are defined. It is shown that if the fractality of the Brownian trajectories leads to standard quantum and statistical mechanics, then the fractality of the Levy paths leads to fractional quantum mechanics and fractional statistical mechanics. The fractional quantum and statistical mechanics have been developed via our fractional path integral approach. A fractional generalization of the Schrodinger equation has been found. A relationship between the energy and the momentum of the nonrelativistic quantum-mechanical particle has been established. The equation for the fractional plane wave function has been obtained. We have derived a free particle quantum-mechanical kernel using Fox's H function. A fractional generalization of the Heisenberg uncertainty relation has been established. Fractional statistical mechanics has been developed via the path integral approach. A fractional generalization of the motion equation for the density matrix has been found. The density matrix of a free particle has been expressed in terms of the Fox's H function. We also discuss the relationships between fractional and the well-known Feynman path integral approaches to quantum and statistical mechanics.