Free Particle and Path Integral

Yuan Tong-quan

The solution of a point free particle on a circle by means of path integral is presented in this article and the method of path integral is compared with time evolution operator.It is found that the method of path integral is easier.

嘉铭 罗

本文研究的是量子力学中关于自由粒子运动的路径积分问题，通过将测度理论中的方法引入对自由粒子运动路径的描述，从而构建不同测度意义下路径积分量之间的关系。 In this paper, the path integral of free particle motion in quantum mechanics is studied. By introducing the method of measure theory into the description of free particle motion path, therefore, the relationship between the path product components in different measurement sense is constructed.

Christian Grosche, Frank Steiner

Jianping Dong, Han Geng

Fractional quantum mechanics in the non-relativistic case is described by the space-fractional Schrödinger equation containing the fractional Riesz operator, which is a non-local operator. The non-locality makes it difficult to solve the fractional Schrödinger equation in the local potentials. In this paper, we study the solution of the fractional Schrödinger equation of a free particle moving on a circle by use of the Lévy path integrals approach. We present the Lévy path integrals propagator of a particle on circle and then use it to get the wave functions and energy eigenvalues of the free particle case. In addition, the Laplace transform, energy-time transform, and momentum representation of the free particle kernel are also obtained. The results of this paper contain the ones in standard quantum mechanics as special cases. The problem of a particle on circle is a fundamental problem of path integrals with topological constraints in quantum systems, and the method of Lévy path integrals can be generalized to study more complex local potentials.