Abstract
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.
Highlights
Stationary states of a quantum-mechanical system are described by the time-independent Schrödinger equation and there are many systems whose exact analytic solutions are known as can be seen in textbooks such as Refs. [1, 2]
The standard time-independent perturbation theory [3] that usually appears in most textbooks like Refs. [1, 2] of quantum mechanics is the Rayleigh–Schrödinger perturbation theory that was first introduced by Schrödinger
We present a new alternative method to find the solution of the perturbative system by employing the Lagrange-multiplier regularization formalism for the eigenproblem, which was suggested in Ref. [7] and successfully applied to the several eigenvalue problems in classical and quantum mechanics
Summary
Stationary states of a quantum-mechanical system are described by the time-independent Schrödinger equation and there are many systems whose exact analytic solutions are known as can be seen in textbooks such as Refs. [1, 2]. We present a new alternative method to find the solution of the perturbative system by employing the Lagrange-multiplier regularization formalism for the eigenproblem, which was suggested in Ref. The dependence on the regularization parameter can be removed by requiring a constraint equation This procedure is similar to adding a gauge-fixing term to the gauge-field Lagrangian density in gauge-field theories and turned out to be very powerful to solve several eigenvalue problems in physics [8, 9]. We provide a new approach to find the energy and wavefunction of the time-independent Schödinger equation with perturbation by regularizing the eigenvalue equation of the Hamiltonian with matrix-valued Lagrange undetermined multipliers. We provide a new Lagrange-multiplier-regularization formalism of determining the energy and wavefunction of the time-independent Schrödinger equation with perturbation in Sect. A rigorous proof of the property of the adjugate matrix which is a crucial part of the formulation is presented in appendix
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