Time-independent perturbation theory with Lagrange multipliers

Chaehyun Yu, Dong Won Jung, U. Rae Kim, Jungil Lee

Research output: Contribution to journalArticlepeer-review


We derive the formulas for the energy and wavefunction of the time-independent Schrödinger equation with perturbation in a compact form. Unlike the conventional approaches based on Rayleigh–Schrödinger or Brillouin–Wigner perturbation theories, we employ a recently developed approach of matrix-valued Lagrange multipliers that regularizes an eigenproblem. The Lagrange-multiplier regularization makes the characteristic matrix for an eigenproblem invertible. After applying the constraint equation to recover the original equation, we find the solutions of the energy and wavefunction consistent with the conventional approaches. This formalism does not rely on an iterative way and the order-by-order corrections are easily obtained by taking the Taylor expansion. The Lagrange-multiplier regularization formalism for perturbation theory presented in this paper is completely new and can be extended to the degenerate perturbation theory in a straightforward manner. We expect that this new formalism is also pedagogically useful to give insights on the perturbation theory in quantum mechanics.

Original languageEnglish
Pages (from-to)1104-1113
Number of pages10
JournalJournal of the Korean Physical Society
Issue number12
Publication statusPublished - 2021 Dec

Bibliographical note

Publisher Copyright:
© 2021, The Author(s).


  • Eigenvalue problem
  • Lagrange multipliers
  • Regularization
  • Time-independent perturbation theory

ASJC Scopus subject areas

  • General Physics and Astronomy


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