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

The Journal of Chemical Physics

Constrained energy functionals obtained as solutions to a partial differential equation

Paul A. Westhaus

Published Oct 15, 1984• 2 citations
Publisher

Abstract

The constrained energy functional, defined by E0(a) =minSa{a〈Ψ‖Ĥ0‖Ψ〉a}, ai=a〈Ψ‖Âi‖Ψ〉a, i=1, ..., M, ‖Ψ〉a∈Sa, with a={ai} a preassigned set of expectation values generated by any state vector ‖Ψ〉a in the class Sa, is shown to satisfy a first order, but nonlinear, partial differential equation. The equation results by replacing the Lagrange multipliers introduced in an earlier paper by the corresponding partial derivatives of the to-be-found energy functional with respect to the constraints. Two alternative expressions for E0(a), one focusing upon the ground state eigenvalue of a modified Hamiltonian operator and the other upon the corresponding eigenvector, lead via standard Rayleigh–Schrödinger perturbation theory to two approximate partial differential equations for the energy functional. Both equations together with the imposed subsidiary conditions lead in turn to the same approximate E0(a). An example based on the linear harmonic oscillator illustrates the concepts presented in the formalism.

DOI
10.1063/1.448104

Related papers

Derivation of energy expressions in the Rayleigh–Schrödinger perturbation theory

Kalbe Razi Naqvi

It is shown that repeated differentiation, with respect to the perturbation parameter λ, of the equation 〈ψm (λ)‖H0+λV‖ψn(λ)〉 =En(λ)δmn, and methodical application of the closure relation ∑‖ψn〉〈ψn‖=1, provide a painless technique for deriving the energy corrections in the Rayleigh–Schrödinger perturbation theory. The procedure is compared with one based on solving the secular equation D(E)=0, where D is the determinant whose elements are 〈ψi‖H0+V‖ψj〉−Eδij.

American Journal of Physics•Mar 1, 1987•1 citation

Time-independent perturbation theory with Lagrange multipliers

Chaehyun Yu, Dong‐Won Jung, U-Rae Kim +1 more

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.

Journal of the Korean Physical Society•Dec 1, 2021•1 citation

Studies in Perturbation Theory. IX. Connection Between Various Approaches in the Recent Development—Evaluation of Upper Bounds to Energy Eigenvalues in Schrödinger's Perturbation Theory

Per‐Olov Löwdin

The treatment of Schrödinger's perturbation theory based on the use of a series of inhomogeneous differential equations of iterative character is briefly surveyed. As an illustration, the method is used to derive the general expression for the expectation value of the Hamiltonian to any order which provides an upper bound for the ground-state energy. It is indicated how the well-known theory for inhomogeneous equations may be utilized also in this special case. The solution of the Schrödinger equation by means of the partitioning technique and the concept of reduced resolvents is then treated. It is shown that the expressions obtained are most conveniently interpreted in terms of inhomogeneous differential equations. A study of the connection with the first approach reveals that the two methods are essentially equivalent, but also that the use of reduced resolvents and inverse operators may give an alternative insight in the mathematical structure of perturbation theory, particularly with respect to the ``bracketing theorem'' and the use of power series expansions with a remainder. In conclusion, it is emphasized that the combined use of the two methods provides a simpler and more powerful tool than any one of them taken separately.

Journal of Mathematical Physics•Aug 1, 1965•178 citations

Quantization of constrained systems using the WKB approximation

Eqab M. Rabei, Khaled I. Nawafleh, H. B. Ghassib

A general theory for solving the Hamilton-Jacobi partial differential equations for constrained Hamiltonian systems is proposed. The quantization of constrained systems is then applied using the WKB approximation. The constraints become conditions on the wave function to be satisfied in the semiclassical limit.

Physical Review A•Aug 5, 2002•38 citations

Energy Ambiguity in Nonlinear Quantum Mechanics

Waldemar Puszkarz

We observe that in nonlinear quantum mechanics, unlike in the linear theory, there exists, in general, a difference between the energy functional defined within the Lagrangian formulation as an appropriate conserved component of the canonical energy-momentum tensor and the energy functional defined as the expectation value of the corresponding nonlinear Hamiltonian operator. Some examples of such ambiguity are presented for a particularly simple model and some known modifications. However, we point out that there exist a class of nonlinear modifications of the Schrödinger equation where this difference does not occur, which makes them more consistent in a manner similar to that of the linear Schrödinger equation. It is found that necessary but not sufficient a condition for such modifications is the homogeneity of the modified Schrödinger equation or its underlying Lagrangian density which is assumed to be ``bilinear'' in the wave function in some rather general sense. Yet, it is only for a particular form of this density that the ambiguity in question does not arise. A salient feature of this form is the presence of phase functionals. The present paper thus introduces a new class of modifications characterized by this desirable and rare property.

arXiv:quant-ph/9802001•Feb 3, 1998•7 citations

A moment equation reformulation of Rayleigh - Ritz theory

Carlos R. Handy

We present a moment equation Rayleigh - Ritz (RR) variational theory for the ground-state energy. An extended Fourier transform space can be defined through the moment equation corresponding to a given Schrödinger Hamiltonian. Within this extended space, we can implement a variational ansatz with respects to configurations of the form in which the energy-dependent functions : (i) satisfy the momentum space Schrödinger equation; (ii) are uniquely prescribed; (iii) are (most likely) non-integrable; and (iv) yield the physical solution for physical E and values. On the basis of this representation, one can minimize the energy expectation value with respects to the E and missing moment variables, . The proposed approach is in sharp contrast to traditional configuration-space RR implementations in which the selection of a variational basis is not manifest a priori, particularly for problems of spatial dimension greater than one. The analysis of one- and two-dimension problems is presented.

Journal of Physics A Mathematical and General•Jul 21, 1996•4 citations
The constrained energy functional, defined by E0(a) =minSa{a〈Ψ‖Ĥ0‖Ψ〉a}, ai=a〈Ψ‖Âi‖Ψ〉a, i=1, ..., M, ‖Ψ〉a∈Sa, with a={ai} a preassigned set of expectation values generated by any state vector ‖Ψ〉a in the class Sa, is shown to satisfy a first order, but nonlinear, partial differential equation. The equation results by replacing the Lagrange multipliers introduced in an earlier paper by the corresponding partial derivatives of the to-be-found energy functional with respect to the constraints. Two alternative expressions for E0(a), one focusing upon the ground state eigenvalue of a modified Hamiltonian operator and the other upon the corresponding eigenvector, lead via standard Rayleigh–Schrödinger perturbation theory to two approximate partial differential equations for the energy functional. Both equations together with the imposed subsidiary conditions lead in turn to the same approximate E0(a). An example based on the linear harmonic oscillator illustrates the concepts presented in the formalism.
It is shown that repeated differentiation, with respect to the perturbation parameter λ, of the equation 〈ψm (λ)‖H0+λV‖ψn(λ)〉 =En(λ)δmn, and methodical application of the closure relation ∑‖ψn〉〈ψn‖=1, provide a painless technique for deriving the energy corrections in the Rayleigh–Schrödinger perturbation theory. The procedure is compared with one based on solving the secular equation D(E)=0, where D is the determinant whose elements are 〈ψi‖H0+V‖ψj〉−Eδij.
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.
The treatment of Schrödinger's perturbation theory based on the use of a series of inhomogeneous differential equations of iterative character is briefly surveyed. As an illustration, the method is used to derive the general expression for the expectation value of the Hamiltonian to any order which provides an upper bound for the ground-state energy. It is indicated how the well-known theory for inhomogeneous equations may be utilized also in this special case. The solution of the Schrödinger equation by means of the partitioning technique and the concept of reduced resolvents is then treated. It is shown that the expressions obtained are most conveniently interpreted in terms of inhomogeneous differential equations. A study of the connection with the first approach reveals that the two methods are essentially equivalent, but also that the use of reduced resolvents and inverse operators may give an alternative insight in the mathematical structure of perturbation theory, particularly with respect to the ``bracketing theorem'' and the use of power series expansions with a remainder. In conclusion, it is emphasized that the combined use of the two methods provides a simpler and more powerful tool than any one of them taken separately.
We observe that in nonlinear quantum mechanics, unlike in the linear theory, there exists, in general, a difference between the energy functional defined within the Lagrangian formulation as an appropriate conserved component of the canonical energy-momentum tensor and the energy functional defined as the expectation value of the corresponding nonlinear Hamiltonian operator. Some examples of such ambiguity are presented for a particularly simple model and some known modifications. However, we point out that there exist a class of nonlinear modifications of the Schrödinger equation where this difference does not occur, which makes them more consistent in a manner similar to that of the linear Schrödinger equation. It is found that necessary but not sufficient a condition for such modifications is the homogeneity of the modified Schrödinger equation or its underlying Lagrangian density which is assumed to be ``bilinear'' in the wave function in some rather general sense. Yet, it is only for a particular form of this density that the ambiguity in question does not arise. A salient feature of this form is the presence of phase functionals. The present paper thus introduces a new class of modifications characterized by this desirable and rare property.
We present a moment equation Rayleigh - Ritz (RR) variational theory for the ground-state energy. An extended Fourier transform space can be defined through the moment equation corresponding to a given Schrödinger Hamiltonian. Within this extended space, we can implement a variational ansatz with respects to configurations of the form in which the energy-dependent functions : (i) satisfy the momentum space Schrödinger equation; (ii) are uniquely prescribed; (iii) are (most likely) non-integrable; and (iv) yield the physical solution for physical E and values. On the basis of this representation, one can minimize the energy expectation value with respects to the E and missing moment variables, . The proposed approach is in sharp contrast to traditional configuration-space RR implementations in which the selection of a variational basis is not manifest a priori, particularly for problems of spatial dimension greater than one. The analysis of one- and two-dimension problems is presented.