Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

European Journal of Physics

Derivation of the harmonic oscillator propagator using the Feynman path integral and recursive relations

Kiyoto Hira

Published Apr 8, 2013• 14 citations
Publisher

Abstract

We present the simplest and most straightforward derivation of the one-dimensional harmonic oscillator propagator, using the Feynman path integral and recursive relations. Our calculations have pedagogical benefits for those undergraduate students beginning to learn the path integral in quantum mechanics, in that they can follow its calculations very simply with only elementary mathematical manipulation. Further, our calculations do not require cumbersome matrix algebra.

DOI
10.1088/0143-0807/34/3/777

Related papers

Schwinger Method and Feynman Path Integral for a Harmonic Oscillator with Mass Growing with Time

Surarit Pepore, Bodinchat Sukbot

The exact propagator for a harmonic oscillator with mass growing like m(1+ατ )2 where m and α are constants and τ is time, is derived by the Schwinger method and the Feynman path integral. In the Schwinger formalism, the propagator can be calculated by basic operator algebra and elementary integrations. In the Feynman path integral method, it can be shown that such a propagator can be evaluated from that for a classical action with the application of a Pauli-Van Vleck formula.

Chinese Journal of Physics•Jun 1, 2015•4 citations

A simple evaluation of a determinant in a path integral calculation of a harmonic oscillator propagator

Kiyoto Hira

The calculation of a one-dimensional quantum harmonic oscillator propagator using the path integral has been reconsidered for more simplicity and more pedagogical significance. We focus on the evaluation of a determinant resulting from the action integral in the discretized form of the path integral. As a consequence we can make the commonly accepted calculations in the literature simpler and clearer.

European Journal of Physics•Sep 10, 2013•1 citation

One-dimensional Dunkl quantum mechanics: a path integral approach

Amar Benchikha, B Hamil, B. C. Lütfüoğlu +1 more

Abstract In the present manuscript, we employ the Feynman path integral method to derive the propagator in one-dimensional Wigner-Dunkl quantum mechanics. To verify our findings we calculate the propagator associated with the free particle and the harmonic oscillator in the presence of the Dunkl derivative. We also deduce the energy spectra and the corresponding bound-state wave functions from the spectral decomposition of the propagator.

arXiv:2407.12644•Physica Scripta•Sep 13, 2024•14 citations

Elementary derivation of the quantum propagator for the harmonic oscillator

Jiushu Shao

Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.

American Journal of Physics•Sep 21, 2016•9 citations

Hybrid derivation of the 1D harmonic oscillator propagator

Marco Moriconi

We provide an elementary derivation of the one-dimensional quantum harmonic oscillator propagator, using a mix of approaches, such as path integrals, canonical operators, and ladder operators. This way we take the best of each world, and find the propagator with as few tears as possible.

American Journal of Physics•Jun 18, 2020•3 citations

The Feynman Path Integral and Propagators

Horatiu Nastase

Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.

Cambridge University Press eBooks•Aug 4, 2022•0 citations
We present the simplest and most straightforward derivation of the one-dimensional harmonic oscillator propagator, using the Feynman path integral and recursive relations. Our calculations have pedagogical benefits for those undergraduate students beginning to learn the path integral in quantum mechanics, in that they can follow its calculations very simply with only elementary mathematical manipulation. Further, our calculations do not require cumbersome matrix algebra.
Operator algebra techniques are employed to derive the quantum evolution operator for the harmonic oscillator. The derivation begins with the construction of the annihilation and creation operators and the determination of the wave function for the coherent state as well as its time-dependent evolution, and ends with the transformation of the propagator in a mixed position-coherent-state representation to the desired one in configuration space. Throughout the entire procedure, besides elementary operator manipulations, it is only necessary to solve linear differential equations and to calculate Gaussian integrals.
Written for a two-semester graduate course in Quantum Mechanics, this comprehensive text helps develop the tools and formalism of Quantum Mechanics and its applications to physical systems. It suits students who have taken some introductory Quantum Mechanics and Modern Physics courses at undergraduate level, but it is self-contained and does not assume any specific background knowledge beyond appropriate fluency in mathematics. The text takes a modern logical approach rather than a historical one and it covers standard material, such as the hydrogen atom and the harmonic oscillator, the WKB approximations and Bohr-Sommerfeld quantization. Important modern topics and examples are also described, including Berry phase, quantum information, complexity and chaos, decoherence and thermalization, nonstandard statistics, as well as more advanced material such as path integrals, scattering theory, multiparticles and Fock space. Readers will gain a broad overview of Quantum Mechanics, as solid preparation for further study or research.