Quantum Systems Foundations of the Feynman Integral

Pavel Exner

Pavel Exner

Jaime Nava, Juan Ferret, Владик Крейнович +3 more

To describe physics properly, we need to take into account quantum effects. Thus, for every non-quantum physical theory, we must come up with an appropriate quantum theory. A traditional approach is to replace all the scalars in the classical description of this theory by the corresponding operators. The problem with the above approach is that due to non-commutativity of the quantum operators, two mathematically equivalent formulations of the classical theory can lead to different (non-equivalent) quantum theories. An alternative quantization approach that directly transforms the non-quantum action functional into the appropriate quantum theory, was indeed proposed by the Nobelist Richard Feynman, under the name of path integration. Feynman path integration is not just a foundational idea, it is actually an efficient computing tool (Feynman diagrams).\nFrom the pragmatic viewpoint, Feynman path integral is a great success. However, from the foundational viewpoint, we still face an important question: why the Feynman's path integration formula? In this paper, we provide a natural explanation for Feynman's path integration formula.

L. Q. English, Ron Winters

The simple harmonic oscillator plays a prominent role in most undergraduate quantum mechanics courses. The study of this system using path integrals can serve to introduce a formulation of quantum mechanics which is usually considered beyond the scope of most undergraduate courses. However, given the current interest in the interpretation and foundations of quantum mechanics, nonstandard approaches such as Feynman’s path integral formalism can be helpful in developing insights into the structure of quantum mechanics. In this paper we evaluate the path integration appearing in Feynman’s treatment in a natural and direct manner utilizing a symbolic computational program. This approach makes the use of the path integral formulation of quantum mechanics accessible to most undergraduate physics majors. As a by-product of our approach, we find a representation of the reciprocal of the sinc function, sinc (x)≡sin(x)/(x), in terms of an infinite product of partial approximates of a continued fraction. We have not found this representation in the literature.

Sergio A Albeverio, Raphael J. Høegh-Krohn

Mathieu Beau

In this article we present an analytic solution of the famous problem of diffraction and interference of electrons through one and two slits (for simplicity, only the one-dimensional case is considered). In addition to exact formulas, we exhibit various approximations of the electron distribution which facilitate the interpretation of the results. Our derivation is based on the Feynman path integral formula and this work could therefore also serve as an interesting pedagogical introduction to Feynman’s formulation of quantum mechanics for university students dealing with the foundations of quantum mechanics. 1 ha l-0