The quantum bouncer by the path integral method
作者:
D. A. Goodings,
T. Szeredi,
期刊:
American Journal of Physics
(AIP Available online 1991)
卷期:
Volume 59,
issue 10
页码: 924-930
ISSN:0002-9505
年代: 1991
DOI:10.1119/1.16673
出版商: American Association of Physics Teachers
关键词: FEYNMAN PATH INTEGRAL;QUANTUM MECHANICS;WKB APPROXIMATION;ORBITS;EIGENVALUES
数据来源: AIP
摘要:
The path integral formulation of quantum mechanics in the semiclassical or WKB approximation provides a physically intuitive way of relating a classical system to its quantum analog. A fruitful way of studying quantum chaos is based upon applying the Gutzwiller periodic orbit sum rule, a result derived by the path integral method in the WKB approximation. This provides some motivation for learning about path integral techniques. In this paper a pedagogical example of the path integral formalism is presented in the hope of conveying the basic physical and mathematical concepts. The ‘‘quantum bouncer’’ is studied—the quantum version of a particle moving in one dimension above a perfectly reflecting surface while subject to a constant force directed toward the surface. The classical counterpart of this system is a ball bouncing on a floor in a constant gravitational field, collisions with the floor being assumed to be elastic. Path integration is used to derive the energy eigenvalues and eigenfunctions of the quantum bouncer in the WKB or semiclassical approximation. The results are shown to be the same as those obtained by solving the Schrödinger equation in the same approximation.
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