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USSR Computational Mathematics and Mathematical Physics

The Feynman-Schrödinger equation and the method of statistical perturbations

Iosif G. Dyadkin

Published Jan 1, 1968• 0 citations
Publisher
DOI
10.1016/0041-5553(68)90096-7

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In this paper we give an alternative treatment of the Schrodinger equation with the Morse potential, which based on the exact summation of the Feynman perturbation series in its original form. Using Fourier transform we establish a recurrence equation between terms of the perturbation series. Finally, by the inverse Fourier transform and some technical tools of the ordinary differential equations of the second order, we can compute the exact sum of the perturbation series which is the Green’s function of the problem.
Multi-component systems of the microscopic world, i. e. high-dimensional quantum systems, can be described by various formalisms, which are in the end equivalent. A formalism on the basis of a differential evolution equation, the Schrödinger equation, as well as a formalism on the basis of a path integral, the Feynman path integral, is possible, where this integral formalism represents nothing else but the method of Green’s function. By some kinds of transformation one can get other possible formalisms like the Heisenberg formalism or the interaction formalism. In this chapter in particular I want to look at the method of R. P. Feynman , and I want to present the extension of the introduced method of integral calculation (see chapter 4) to quantum systems. In this context a statistical basic function will be introduced, which contains all statistically relevant information of both quantum systems and non-microscopic systems (like laser systems or thermodynamic systems). Thus, the universality of the introduced concept of system theory will become evident, and therefore a pathway to quantum systems is given. Furthermore, I will again attach importance to the problem of interrelation between equations of the Schrödinger and the Fokker-Planck type.
This chapter discusses phase transitions and symmetry breaking and the mean field approach to condensation in Bose systems and looks at strong interactions in ‘normal’ Fermi systems. ‘Statistical field theory’ encompasses all the applications of quantum field theory to statistical physics. In the context of these notes, it is interpreted as the study of ‘statistical mechanics of quantum many particle systems, employing the methods of field theory’. The identification of the symmetry broken in a phase with long range order allows to construct simple ‘mean field’ theories which provide a qualitative description of the system. Field theoretic methods were those developed by Feynman, Dyson, Schwinger, and others for understanding QED. They were ‘field theoretic’ in that one had to go beyond quantum mechanics, and the Schrodinger equation, to set up a theory. Unlike quantum field theory, there is a Schrodinger equation governing many particle phenomena in condensed matter.