## Abstract

We introduce a systematic approach to represent Leibniz’s nth-order differential operator d^{n} as the ratio of an infinite product of infinitesimal difference operators to an infinitesimal parameter. Because every difference operator can be expressed as a difference of two shift operators that translate the argument of a function by finite amounts, Leibniz’s differential operator d^{n} is eventually expressed as the infinite product of infinitesimal binomial operators consisting of the shift operators. We apply this strategy to demonstrate the derivation of the translation or time-evolution operators in quantum mechanics. This fills the logical gap in most textbooks on quantum mechanics that usually omit explicit derivations. Our approach could be employed in general physics or classical mechanics classes with which one can solve the equation of motion without prior knowledge of differential equations.

Original language | English |
---|---|

Pages (from-to) | 513-519 |

Number of pages | 7 |

Journal | Journal of the Korean Physical Society |

Volume | 83 |

Issue number | 7 |

DOIs | |

Publication status | Published - 2023 Oct |

### Bibliographical note

Funding Information:As members of the Korea Pragmatist Organization for Physics Education (KPOP), the authors would like to thank to the remaining members of KPOP for useful discussions. This work is supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) under Contract No. NRF-2020R1A2C3009918. The work is also supported in part by the National Research Foundation of Korea (NRF) under the BK21 FOUR program at Korea University, Initiative for science frontiers on upcoming challenges.

Funding Information:

As members of the Korea Pragmatist Organization for Physics Education (KPOP E \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {E}}$$\end{document} ), the authors would like to thank to the remaining members of KPOP E \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathscr {E}}$$\end{document} for useful discussions. This work is supported in part by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) under Contract No. NRF-2020R1A2C3009918. The work is also supported in part by the National Research Foundation of Korea (NRF) under the BK21 FOUR program at Korea University, Initiative for science frontiers on upcoming challenges.

Publisher Copyright:

© 2023, The Author(s).

## Keywords

- Difference operator
- Differential equation
- Differential operator
- Time evolution
- Translation

## ASJC Scopus subject areas

- General Physics and Astronomy

## Fingerprint

Dive into the research topics of 'Physics analysis with Leibniz’s differential operators d^{n}'. Together they form a unique fingerprint.