Bibliographic Details
| Title: |
Deriving the path integral formulas from the Schrödinger equation. |
| Authors: |
Wang, Huai-Yu1 wanghuaiyu@mail.tsinghua.edu.cn |
| Source: |
Physics Essays. Sep2025, Vol. 38 Issue 3, p189-194. 6p. |
| Subjects: |
Schrödinger equation, Path integrals, Schroedinger, Erwin, 1887-1961, Lagrangian mechanics, Hamiltonian operator, Green's functions, Feynman integrals, Quantum mechanics, Feynman, Richard Phillips, 1918-1988, Fourier transforms |
| Abstract (English): |
This paper derives Feynman's path integral formula from the Schrödinger equation. To this aim, the solution of the Schrödinger equation is expressed by the initial condition and Green's function. This expression indicates that a wave function at the initial moment propagates to the final state through all possible spatial paths. The iteration of the expression embodies the spirit of the path integral. The Fourier transform of the Green's function is utilized, which helps to convert the operator Hamiltonian into a numerical one. Then, the latter is spatially discretized, and finally the Hamiltonian is transformed into a Lagrangian. We point out the reason why the Schrödinger equation can be deduced in turn from the path integral formula with a certain approximation. [ABSTRACT FROM AUTHOR] |
| Abstract (French): |
Cet article dérive la formule de l'intégrale de chemin de Feynman à partir de l'équation de Schrödinger. À cette fin, la solution de l'équation de Schrödinger est exprimée par la condition initiale et la fonction de Green. Cette expression indique qu'une fonction d'onde à l'instant initial se propage vers l'état final à travers tous les chemins spatiaux possibles. L'itération de l'expression incarne l'esprit de l'intégrale de chemin. La transformée de Fourier de la fonction de Green est utilisée, ce qui aide à convertir l'hamiltonien opérateur en un hamiltonien numérique. Ce dernier est ensuite discrétisé spatialement, et finalement l'hamiltonien est transformé en un lagrangien. Nous soulignons la raison pour laquelle l'équation de Schrödinger peut être déduite à son tour de la formule de l'intégrale de chemin avec une certaine approximation. [ABSTRACT FROM AUTHOR] |
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| Database: |
Engineering Source |
