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The Runge-Lenz Vector

Carl W David

Published Jan 1, 2006• 0 citations
OpenAlex

Abstract

The Runge-Lenz vector is a constant of the motion in the Kepler problem. As a precursor to developing a ladder operator formalism for the H-atom's quantum mechanics, various relations concerning the Runge-Lenz vector are obtained.

Related papers

The Commutator of the Runge Lenz Vector with other operators of the H-atoms physics using Maxima

Carl W David

Commutators of components of the Runge-Lenz vector and constants of the motion and ladder operators for the H-atom problem are evaluated using brute force calculus and Maxima.

•Jan 1, 2018•0 citations

On the connections between the classical and quantum-mechanical Kepler problems

Joachim Dahl, Thomas Godsk Jørgensen

The Runge-Lenz vector, which accounts for the accidental degeneracy of the non-relativistic Kepler problem, has been the subject matter of many studies, both in quantum mechanics and in classical mechanics. Much less attention has been paid to the Johnson-Lippmann operator which accounts for the accidental degeneracy of the relativistic Kepler problem in Dirac`s quantum-mechanical description. In the present communication we discuss the properties of the Johnson-Lippmann operator. We show its relation to the non-relativistic Runge-Lenz vector and draw a connection to Sommerfield`s early discussion of the relativistic Kepler problem. This enables us, inter alia, to give an explanation of the apparent coincidence of the energy expressions of the two theories.

Bulletin of the American Physical Society•May 1, 1993•0 citations

The Runge0Lenz Vector (continued)

Carl W David

We continue our discussion of the Runge-Lenz vector in a quantum mechanical context. The traditional form of the Runge-Lenz vector is obtained, and the commutation relations between the Runge-Lenz vector, the Hamiltonian, and the Angular Momentum are obtained using Maple.

•Jan 1, 2009•0 citations

Obtaining Hydrogen energy wave functions using the Runge-Lenz vector

Chun-Khiang Chua

The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. It is well known that the energy spectrum is identical with the one obtained from the Schrödinger equation and the consistency contributed significantly to the development of Quantum Mechanics in the early days. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate $|n,l,m\rangle$ with other degenerate states $|n, l\pm 1,m'\rangle$. Recursive relations can be obtained and the wave functions of the whole spectrum can be obtained easily. Note that the recursive relations are consistent with those used in factorizing the Schrödinger equation. Nevertheless, the present analysis provide a better reasoning originated from the conserved vector, the Runge-Lenz vector. As in the Pauli analysis, group theory or symmetry plays a prominent role in the present analysis, while the rest of the derivations are mostly elementary.

arXiv:1805.01610•May 4, 2018•0 citations

Obtaining hydrogen energy eigenstate wave functions using the Runge-Lenz vector

Chun-Khiang Chua

The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. It is well known that the energy spectrum is identical with the one obtained from the Schrödinger equation and the consistency contributed significantly to the development of Quantum Mechanics in the early days. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate with other degenerate states . Recursive relations can be obtained and the energy eigenstate wave functions of the whole spectrum can be obtained easily. Note that the recursive relations are consistent with those used in factorizing the Schrödinger equation. Nevertheless, this analysis provides a better reasoning originated from the conserved vector, the Runge-Lenz vector. As in the Pauli analysis, group theory or symmetry plays a prominent role in this analysis, while the rest of the derivations are elementary.

Journal of Physics Communications•Aug 7, 2018•3 citations

Runge–Lenz Operator in the Momentum Space

Sergei P. Efimov

The fundamental quantum Coulomb problem in the momentum space is considered. A differential equation with SO(4) symmetry has been obtained in the momentum space instead of the integral Fock equation. The corresponding equation in the coordinate space is the sum of the squares of the angular momentum and Runge–Lenz operators. This approach is unknown in the momentum space where the Runge–Lenz operator is not applied. The Runge–Lenz operator obtained in the momentum space is simpler than that in the coordinate space and allows one to effectively consider the Coulomb problem in the momentum space. A relation of new operator to the infinitesimal rotation operator of the three-dimensional Fock sphere has been determined.

Письма в Журнал экспериментальной и теоретической физики•May 15, 2023•0 citations
The Runge-Lenz vector, which accounts for the accidental degeneracy of the non-relativistic Kepler problem, has been the subject matter of many studies, both in quantum mechanics and in classical mechanics. Much less attention has been paid to the Johnson-Lippmann operator which accounts for the accidental degeneracy of the relativistic Kepler problem in Dirac`s quantum-mechanical description. In the present communication we discuss the properties of the Johnson-Lippmann operator. We show its relation to the non-relativistic Runge-Lenz vector and draw a connection to Sommerfield`s early discussion of the relativistic Kepler problem. This enables us, inter alia, to give an explanation of the apparent coincidence of the energy expressions of the two theories.
The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. It is well known that the energy spectrum is identical with the one obtained from the Schrödinger equation and the consistency contributed significantly to the development of Quantum Mechanics in the early days. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate ∣n,l,m⟩|n,l,m\rangle∣n,l,m⟩ with other degenerate states ∣n,l±1,m′⟩|n, l\pm 1,m'\rangle∣n,l±1,m′⟩. Recursive relations can be obtained and the wave functions of the whole spectrum can be obtained easily. Note that the recursive relations are consistent with those used in factorizing the Schrödinger equation. Nevertheless, the present analysis provide a better reasoning originated from the conserved vector, the Runge-Lenz vector. As in the Pauli analysis, group theory or symmetry plays a prominent role in the present analysis, while the rest of the derivations are mostly elementary.
The Pauli method of quantizing the Hydrogen system using the Runge-Lenz vector is ingenious. It is well known that the energy spectrum is identical with the one obtained from the Schrödinger equation and the consistency contributed significantly to the development of Quantum Mechanics in the early days. Since the Runge-Lenz vector is a vector and it commutes with the Hamiltonian, it is natural to use it to connect energy eigenstate with other degenerate states . Recursive relations can be obtained and the energy eigenstate wave functions of the whole spectrum can be obtained easily. Note that the recursive relations are consistent with those used in factorizing the Schrödinger equation. Nevertheless, this analysis provides a better reasoning originated from the conserved vector, the Runge-Lenz vector. As in the Pauli analysis, group theory or symmetry plays a prominent role in this analysis, while the rest of the derivations are elementary.
The fundamental quantum Coulomb problem in the momentum space is considered. A differential equation with SO(4) symmetry has been obtained in the momentum space instead of the integral Fock equation. The corresponding equation in the coordinate space is the sum of the squares of the angular momentum and Runge–Lenz operators. This approach is unknown in the momentum space where the Runge–Lenz operator is not applied. The Runge–Lenz operator obtained in the momentum space is simpler than that in the coordinate space and allows one to effectively consider the Coulomb problem in the momentum space. A relation of new operator to the infinitesimal rotation operator of the three-dimensional Fock sphere has been determined.