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

Progress of Theoretical Physics

Q-Number Variational Method for Non-Linear Lagrangian in Quantum Mechanics

T. Ohtani, Reiji Sugano

Published Nov 1, 1973• 15 citations
Publisher

Abstract

A previous q-number variational method is extended to be applicable to any quantum system for which the classical Lagrangian is given by Lc=½gij\dotqi\dotqj+ui\dotqi-v. The q-number variation is necessary for the formulation to be form-invariant under a general space-time transformation. From the action principle, not only the Euler-Lagrange equation but also the commutation relations up to a constant factor are obtained. Futhermore the form of the quantum Lagrangian is, to some extent, decided in order for solutions to exist for the action principle. It is shown that the first Noether theorem holds and the quantization is consistent with the canonical formalism.

DOI
10.1143/ptp.50.1715

Related papers

Variation Principle for Non-Linear Lagrangian in Quantum Mechanics

Teruya Ohtani, Reiji Sugano

The variation principle has not been well defined in quantum mechanics for a Lagrangian containing a velocity dependent potential. We investigate the method of variation to derive an Euler-Lagrange equation in connection with Noether's theorem when the Lagrangian is invariant under some point transformation. Obtained are the Euler-Lagrange equation retaining the above invariance by means of the variation method and the Hamiltonian with which the canonical equation of motion leads to the Euler-Lagrange equation.

Progress of Theoretical Physics•May 1, 1972•8 citations

Schwinger's Variation Principle by Means ofQ-Number Variation for Non-Linear Lagrangian

Reiji Sugano

Since the ordinary c-number variation principle does not lead, for a non-linear Lagrangian, to the result consistent between the Lagrangian and Hamiltonian formalisms, Schwinger's variation principle is reformulated for a type of Lagrangian L=½\dotqigij(q)\dotqj-u(q) by means of a q-number variation. The canonical momentum, the Hamiltonian, the canonical commutation relations and equation of motion are derived. Also the Euler-Lagrange equation is obtained, which is consistent with the canonical equation of motion. These consequences are exactly the same as those of previous papers, but different from the ordinary ones in the Euler-Lagrange equation and the Hamiltonian.

Progress of Theoretical Physics•Apr 1, 1973•1 citation

Schwinger's Variational Principle in Quantum Mechanics with Velocity Dependent Potential. III

Hideki Kamo, Toshiharu Kawai

Schwinger’s variational principle is formulated for the multi-dimensional quantum system which corresponds to the classical system described by the Lagrangian Lc(\dotx,x)=(M/2)gij(x)\dotxi\dotxj-v(x). The c-number variations of coordinates and time are sufficient to give the laws of quantum mechanics. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relations are derived from this principle and there is no inconsistency. An appropriate choice of the Lagrangian operator is essential in our formulation. It is shown that an arbitrary point transformation is entitled to be called a canonical transformation. Ambiguities in the quantal Lagrangian are also discussed.

Progress of Theoretical Physics•Aug 1, 1973•12 citations

Schwinger's Variational Principle in Quantum Mechanics with Velocity Dependent Potential. I

Toshiharu Kawai

Schwinger's variational principle is formulated for the quantum system which corresponds to the one-dimensional classical system described by the Lagrangian Lc(\dotx, x) = (M/2) f-1(x)\dotx2 - v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider only c-number variations of coordinate and time. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relation are derived from the principle. All resulting relations are consistent with one another. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant and is suitable to be called a canonical transformation. The appropriate choice of the Lagrangian operator is essential in our formulation.

Progress of Theoretical Physics•Dec 1, 1972•6 citations

A NEW VARIATIONAL PRINCIPLE IN QUANTUM MECHANICS

Theodore Joseph Newman

Quantum theory is developed from a q-number (operator) action principle with a representation-invariant technique for limiting the number of independent system variables. It is shown that in a q-number theory such a limitation on the number of variations is necessary, since a completely arbitrary q-number variation implies an infinite number of conditions to be satisfied. This implication arises because, in a q-number theory, the system variables are matrices which, in most cases, have an infinite number of elements, each of which is independently varied in a completely arbitrary q-number variation. A review of Schwinger's quantum field theory formalism shows that the limitation to c- number variations, while accomplishing the desired limitation on the number of independent variations, is not representation-invariant, i.e., under a change of representation, c-number variations applied to the original system variables transform to q-number variations of the new system variables. In the Hamiltonian formulation of classical mechanics, invariant infinitesimal canonical transformations are generated by constants of the motion. It is also possible to generate infinitesimal invariant transformations from constants of the motion in the classical Lagrangian formulation. This latter possibility is exploited to obtain a q-number theory. It is demanded that the variations of the system variablesmore » in the action principle be the invariant infinitesimal transformations generated by constants of the motion. This requirement accomplishes the desired limitation of the number of independent variations in a representation-invariant manner. The resulting theory is developed in detail for a certain class of Lagrangian functions. A set of commutation relations is derived and a dynamical principle for determining the temporal development of the system is postulated. A discussion of the results, including consideration of the self-consistency of the formalism developed, is presented. Finally, a specific problem is worked through and it is shown that the results obtained correspond with those of the canonical method of quantization.« less

Dissertation Abstr.•Oct 1, 1961•0 citations

On Consistency between Lagrange and Hamilton Formalisms in Quantum Mechanics

Reiji Sugano

Consistency between the Lagrangean and the Hamiltonian formalisms in the quantum mechanics is investigatedc for the type of Lagrangean L=½\dotqigij(q)\dotqj-v(q) as an extension of a previous paper. The variations δqi and δ\dotqi should be considered as q-numbers. When the Lagrangean can be transformed into the standard form L=½\dotQα2-V(Q), the commutation relations of δqi and δ\dotqi with qj and \dotqj are found with the help of the Q-coordinate system. It is shown that using the commutation relations, the variation principle leads to the same equation of motion as the canonical equation of motion which is obtained in the previous paper.

Progress of Theoretical Physics•Jul 1, 1971•19 citations
A previous q-number variational method is extended to be applicable to any quantum system for which the classical Lagrangian is given by Lc=½gij\dotqi\dotqj+ui\dotqi-v. The q-number variation is necessary for the formulation to be form-invariant under a general space-time transformation. From the action principle, not only the Euler-Lagrange equation but also the commutation relations up to a constant factor are obtained. Futhermore the form of the quantum Lagrangian is, to some extent, decided in order for solutions to exist for the action principle. It is shown that the first Noether theorem holds and the quantization is consistent with the canonical formalism.
The variation principle has not been well defined in quantum mechanics for a Lagrangian containing a velocity dependent potential. We investigate the method of variation to derive an Euler-Lagrange equation in connection with Noether's theorem when the Lagrangian is invariant under some point transformation. Obtained are the Euler-Lagrange equation retaining the above invariance by means of the variation method and the Hamiltonian with which the canonical equation of motion leads to the Euler-Lagrange equation.
Since the ordinary c-number variation principle does not lead, for a non-linear Lagrangian, to the result consistent between the Lagrangian and Hamiltonian formalisms, Schwinger's variation principle is reformulated for a type of Lagrangian L=½\dotqigij(q)\dotqj-u(q) by means of a q-number variation. The canonical momentum, the Hamiltonian, the canonical commutation relations and equation of motion are derived. Also the Euler-Lagrange equation is obtained, which is consistent with the canonical equation of motion. These consequences are exactly the same as those of previous papers, but different from the ordinary ones in the Euler-Lagrange equation and the Hamiltonian.
Schwinger’s variational principle is formulated for the multi-dimensional quantum system which corresponds to the classical system described by the Lagrangian Lc(\dotx,x)=(M/2)gij(x)\dotxi\dotxj-v(x). The c-number variations of coordinates and time are sufficient to give the laws of quantum mechanics. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relations are derived from this principle and there is no inconsistency. An appropriate choice of the Lagrangian operator is essential in our formulation. It is shown that an arbitrary point transformation is entitled to be called a canonical transformation. Ambiguities in the quantal Lagrangian are also discussed.
Schwinger's variational principle is formulated for the quantum system which corresponds to the one-dimensional classical system described by the Lagrangian Lc(\dotx, x) = (M/2) f-1(x)\dotx2 - v(x). It is sufficient for the purpose of deriving the laws of quantum mechanics to consider only c-number variations of coordinate and time. The Euler-Lagrange equation, the canonical equations of motion and the canonical commutation relation are derived from the principle. All resulting relations are consistent with one another. Further, it is shown that an arbitrary point transformation leaves the forms of the fundamental equations invariant and is suitable to be called a canonical transformation. The appropriate choice of the Lagrangian operator is essential in our formulation.
Quantum theory is developed from a q-number (operator) action principle with a representation-invariant technique for limiting the number of independent system variables. It is shown that in a q-number theory such a limitation on the number of variations is necessary, since a completely arbitrary q-number variation implies an infinite number of conditions to be satisfied. This implication arises because, in a q-number theory, the system variables are matrices which, in most cases, have an infinite number of elements, each of which is independently varied in a completely arbitrary q-number variation. A review of Schwinger's quantum field theory formalism shows that the limitation to c- number variations, while accomplishing the desired limitation on the number of independent variations, is not representation-invariant, i.e., under a change of representation, c-number variations applied to the original system variables transform to q-number variations of the new system variables. In the Hamiltonian formulation of classical mechanics, invariant infinitesimal canonical transformations are generated by constants of the motion. It is also possible to generate infinitesimal invariant transformations from constants of the motion in the classical Lagrangian formulation. This latter possibility is exploited to obtain a q-number theory. It is demanded that the variations of the system variablesmore » in the action principle be the invariant infinitesimal transformations generated by constants of the motion. This requirement accomplishes the desired limitation of the number of independent variations in a representation-invariant manner. The resulting theory is developed in detail for a certain class of Lagrangian functions. A set of commutation relations is derived and a dynamical principle for determining the temporal development of the system is postulated. A discussion of the results, including consideration of the self-consistency of the formalism developed, is presented. Finally, a specific problem is worked through and it is shown that the results obtained correspond with those of the canonical method of quantization.« less
Consistency between the Lagrangean and the Hamiltonian formalisms in the quantum mechanics is investigatedc for the type of Lagrangean L=½\dotqigij(q)\dotqj-v(q) as an extension of a previous paper. The variations δqi and δ\dotqi should be considered as q-numbers. When the Lagrangean can be transformed into the standard form L=½\dotQα2-V(Q), the commutation relations of δqi and δ\dotqi with qj and \dotqj are found with the help of the Q-coordinate system. It is shown that using the commutation relations, the variation principle leads to the same equation of motion as the canonical equation of motion which is obtained in the previous paper.