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

Physical Review A

Equivalent multipole operators for degenerate Rydberg states

Valentin N. Ostrovsky, D. Vrinceanu, M. Raymond Flannery

Published Aug 23, 2006• 6 citations
Publisher

Abstract

As shown by Pauli, [Z. Phys. 36, 336 (1926)], the electric dipole operator $\mathbf{r}$ can be replaced by the Runge-Lenz vector $\mathbf{A}$ when operating within the ${n}^{2}$ degenerate manifold of hydrogenic states of principal quantum number $n$. We seek to develop similar rules for higher multipole operators by expressing equivalent operators in terms only of the two vector constants of motion---the orbital angular momentum $\mathbf{L}$ and the Runge-Lenz vector $\mathbf{A}$---appropriate to the degenerate hydrogenic shell. Equivalence of two operators means here that they yield identical matrix elements within a subspace of Hilbert space that corresponds to fixed $n$. Such equivalent-operator techniques permit direct algebraic calculation of perturbations of Rydberg atoms by external fields and often exact analytical results for transition probabilities. Explicit expressions for equivalent quadrupole and octupole operators are derived, examples are provided, and general aspects of the problem are discussed.

DOI
10.1103/physreva.74.022720

Related papers

Hydrogen Atom: Its Spectrum and Degeneracy Importance of the Laplace-Runge-Lenz Vector

Pal, Akshay, Sen, Siddhartha

Consider the problem: why does the bound state spectrum $E(n) < 0$, of hydrogen atom Hamiltonian $H$ have more degenerate eigenstates than those required by rotational symmetry? The answer is well known and was demonstrated by Pauli. It is due to an additional conserved vector, $\vec A$, of $H$, called the Laplace-Runge-Lenz vector, that was first discovered for planetary orbits. However, surprisingly, a direct link between degenerate eigenstates of $H$ and the physical labels that describe them is missing. To provide such a link requires, as we show, solving a subtle problem of self adjoint operators. In our discussions we address a number of conceptual historical aspects regarding hydrogen atom that also include a careful discussion of both the classical as well as the quantum vector $\vec A$.

arXiv:2104.04577•Apr 9, 2021•0 citations

Calculation of Rydberg interaction potentials

Sebastian Weber, Christoph Tresp, Henri Menke +4 more

Abstract The strong interaction between individual Rydberg atoms provides a powerful tool exploited in an ever-growing range of applications in quantum information science, quantum simulation and ultracold chemistry. One hallmark of the Rydberg interaction is that both its strength and angular dependence can be fine-tuned with great flexibility by choosing appropriate Rydberg states and applying external electric and magnetic fields. More and more experiments are probing this interaction at short atomic distances or with such high precision that perturbative calculations as well as restrictions to the leading dipole–dipole interaction term are no longer sufficient. In this tutorial, we review all relevant aspects of the full calculation of Rydberg interaction potentials. We discuss the derivation of the interaction Hamiltonian from the electrostatic multipole expansion, numerical and analytical methods for calculating the required electric multipole moments and the inclusion of electromagnetic fields with arbitrary direction. We focus specifically on symmetry arguments and selection rules, which greatly reduce the size of the Hamiltonian matrix, enabling the direct diagonalization of the Hamiltonian up to higher multipole orders on a desktop computer. Finally, we present example calculations showing the relevance of the full interaction calculation to current experiments. Our software for calculating Rydberg potentials including all features discussed in this tutorial is available as open source.

arXiv:1612.08053•Journal of Physics B Atomic Molecular and Optical Physics•May 19, 2017•174 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

Vector operators and spherical harmonics in quantum mechanics

M. Andrews

The usual explicit expressions for spherical harmonics are derived by a method which does not involve the solution of differential equations or the use of angular momentum operators in spherical polar form. Instead a raising operator constructed from an arbitrary vector is used together with the usual angular momentum raising and lowering operators. The method is first presented as simply as possible in Schrödinger representation using position as the vector. Then it is generalized to an arbitrary vector and arbitrary integral angular momentum without using a particular representation. Finally it is shown how a conserved vector (other than angular momentum) of a central system can be used to reveal properties of the energy eigenstates. As an example, eigenstates of the hydrogen atom are found using the Runge-Lenz vector.

American Journal of Physics•Mar 1, 1979•5 citations

A systematic analysis of cancellations in atomic-multipole transitions

Reinhard K. Kremer, Constantine E. Theodosiou

The authors have carried out exhaustive calculations of monopole, dipole, and quadrupole transition radial matrix elements as functions of the effective principal quantum numbers of the initial and final electronic state. The general features of the results of these calculations have been displayed in three-dimensional graphs for the general cases. The locations in (n*,n{prime}*)-space of the zeroes of these matrix elements have been found and appear to follow a generic behavior. This behavior has been fitted to a simple analytic form and the parameters of the fits are given for monopole, dipole and quadrupole transitions involving l = 0, 1, 2, and 3 states. These results are expected to be of great value to experimentalists working with atomic Rydberg states, especially of alkali-like systems.

Bulletin of the American Physical Society•Dec 1, 1992•0 citations

Rydberg-atom - ion collision: classical theory of intrashell transitions

Andrey K. Kazansky, Valentin N. Ostrovsky

Transitions between states with the same principal quantum number n in the collision of a Rydberg atom with an ion are described within the classical approximation for the Rydberg electron. The quantum electron state is represented by an ensemble of classical trajectories. Within the dipole approximation for the interaction the final value of the orbital momentum l is calculated analytically for each initial trajectory in the ensemble. Averaging over the ensemble gives the distribution function over l. For the initial s state of the Rydberg atom the distribution functions are found also for the pairs of variables, namely: {the angular momentum; its projection on the collision velocity}; {projections of the angular momentum and the Runge - Lenz vector on the collision velocity} and {projections of the angular momentum and the Runge - Lenz vector on the plane perpendicular to the collision velocity}. The l-distributions for the orbitally polarized initial p and d Rydberg states with definite projections of the initial angular momentum on the velocity axis are also obtained. The influence of the non-Coulomb core is taken into account by an approximation. The final l-distributions (calculated for the initial s-state) are shifted towards the large-l values as the relative collision velocity vn decreases. Good agreement is achieved with the recent experimental data for Na(28d) and Na(26d) Rydberg atoms.

Journal of Physics B Atomic Molecular and Optical Physics•Aug 28, 1996•18 citations
As shown by Pauli, [Z. Phys. 36, 336 (1926)], the electric dipole operator r\mathbf{r}r can be replaced by the Runge-Lenz vector A\mathbf{A}A when operating within the n2{n}^{2}n2 degenerate manifold of hydrogenic states of principal quantum number nnn. We seek to develop similar rules for higher multipole operators by expressing equivalent operators in terms only of the two vector constants of motion---the orbital angular momentum L\mathbf{L}L and the Runge-Lenz vector A\mathbf{A}A---appropriate to the degenerate hydrogenic shell. Equivalence of two operators means here that they yield identical matrix elements within a subspace of Hilbert space that corresponds to fixed nnn. Such equivalent-operator techniques permit direct algebraic calculation of perturbations of Rydberg atoms by external fields and often exact analytical results for transition probabilities. Explicit expressions for equivalent quadrupole and octupole operators are derived, examples are provided, and general aspects of the problem are discussed.
Consider the problem: why does the bound state spectrum E(n)<0E(n) < 0E(n)<0, of hydrogen atom Hamiltonian HHH have more degenerate eigenstates than those required by rotational symmetry? The answer is well known and was demonstrated by Pauli. It is due to an additional conserved vector, A⃗\vec AA, of HHH, called the Laplace-Runge-Lenz vector, that was first discovered for planetary orbits. However, surprisingly, a direct link between degenerate eigenstates of HHH and the physical labels that describe them is missing. To provide such a link requires, as we show, solving a subtle problem of self adjoint operators. In our discussions we address a number of conceptual historical aspects regarding hydrogen atom that also include a careful discussion of both the classical as well as the quantum vector A⃗\vec AA.
Abstract The strong interaction between individual Rydberg atoms provides a powerful tool exploited in an ever-growing range of applications in quantum information science, quantum simulation and ultracold chemistry. One hallmark of the Rydberg interaction is that both its strength and angular dependence can be fine-tuned with great flexibility by choosing appropriate Rydberg states and applying external electric and magnetic fields. More and more experiments are probing this interaction at short atomic distances or with such high precision that perturbative calculations as well as restrictions to the leading dipole–dipole interaction term are no longer sufficient. In this tutorial, we review all relevant aspects of the full calculation of Rydberg interaction potentials. We discuss the derivation of the interaction Hamiltonian from the electrostatic multipole expansion, numerical and analytical methods for calculating the required electric multipole moments and the inclusion of electromagnetic fields with arbitrary direction. We focus specifically on symmetry arguments and selection rules, which greatly reduce the size of the Hamiltonian matrix, enabling the direct diagonalization of the Hamiltonian up to higher multipole orders on a desktop computer. Finally, we present example calculations showing the relevance of the full interaction calculation to current experiments. Our software for calculating Rydberg potentials including all features discussed in this tutorial is available as open source.
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 usual explicit expressions for spherical harmonics are derived by a method which does not involve the solution of differential equations or the use of angular momentum operators in spherical polar form. Instead a raising operator constructed from an arbitrary vector is used together with the usual angular momentum raising and lowering operators. The method is first presented as simply as possible in Schrödinger representation using position as the vector. Then it is generalized to an arbitrary vector and arbitrary integral angular momentum without using a particular representation. Finally it is shown how a conserved vector (other than angular momentum) of a central system can be used to reveal properties of the energy eigenstates. As an example, eigenstates of the hydrogen atom are found using the Runge-Lenz vector.
The authors have carried out exhaustive calculations of monopole, dipole, and quadrupole transition radial matrix elements as functions of the effective principal quantum numbers of the initial and final electronic state. The general features of the results of these calculations have been displayed in three-dimensional graphs for the general cases. The locations in (n*,n{prime}*)-space of the zeroes of these matrix elements have been found and appear to follow a generic behavior. This behavior has been fitted to a simple analytic form and the parameters of the fits are given for monopole, dipole and quadrupole transitions involving l = 0, 1, 2, and 3 states. These results are expected to be of great value to experimentalists working with atomic Rydberg states, especially of alkali-like systems.
Transitions between states with the same principal quantum number n in the collision of a Rydberg atom with an ion are described within the classical approximation for the Rydberg electron. The quantum electron state is represented by an ensemble of classical trajectories. Within the dipole approximation for the interaction the final value of the orbital momentum l is calculated analytically for each initial trajectory in the ensemble. Averaging over the ensemble gives the distribution function over l. For the initial s state of the Rydberg atom the distribution functions are found also for the pairs of variables, namely: {the angular momentum; its projection on the collision velocity}; {projections of the angular momentum and the Runge - Lenz vector on the collision velocity} and {projections of the angular momentum and the Runge - Lenz vector on the plane perpendicular to the collision velocity}. The l-distributions for the orbitally polarized initial p and d Rydberg states with definite projections of the initial angular momentum on the velocity axis are also obtained. The influence of the non-Coulomb core is taken into account by an approximation. The final l-distributions (calculated for the initial s-state) are shifted towards the large-l values as the relative collision velocity vn decreases. Good agreement is achieved with the recent experimental data for Na(28d) and Na(26d) Rydberg atoms.