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Deriving the path integral formulas from the Schrödinger equation.

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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]
Copyright of Physics Essays is the property of Physics Essays Publication and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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– Name: Abstract
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  Data: 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]
– Name: Abstract
  Label: Abstract (French)
  Group: Ab
  Data: 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]
– Name: AbstractSuppliedCopyright
  Label:
  Group: Ab
  Data: <i>Copyright of Physics Essays is the property of Physics Essays Publication and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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        Value: 10.4006/0836-1398-38.3.189
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      – Code: eng
        Text: English
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        PageCount: 6
        StartPage: 189
    Subjects:
      – SubjectFull: Schrödinger equation
        Type: general
      – SubjectFull: Path integrals
        Type: general
      – SubjectFull: Schroedinger, Erwin, 1887-1961
        Type: general
      – SubjectFull: Lagrangian mechanics
        Type: general
      – SubjectFull: Hamiltonian operator
        Type: general
      – SubjectFull: Green's functions
        Type: general
      – SubjectFull: Feynman integrals
        Type: general
      – SubjectFull: Quantum mechanics
        Type: general
      – SubjectFull: Feynman, Richard Phillips, 1918-1988
        Type: general
      – SubjectFull: Fourier transforms
        Type: general
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      – TitleFull: Deriving the path integral formulas from the Schrödinger equation.
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            – D: 01
              M: 09
              Text: Sep2025
              Type: published
              Y: 2025
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