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

ASME International Design Engineering Technical Conferences and Computers and Information in Engineering Conference (IDETC/CIE)

The Hydrogen Molecular Ion With Time-Dependent Magnetic Field Strength as Control

Katherine A. Kime

Published Aug 17, 2014• 0 citations
Publisher

Abstract

We consider the hydrogen molecular ion with time-dependent magnetic field strength. We discretize the corresponding Schroedinger equation with the Hamiltonian written in prolate spheroidal coordinates. We formulate a control problem and give an example of steering a restricted initial state to a restricted terminal state.
DOI
10.1115/detc2014-35345

Related papers

Solving for Quantum Controls

Katherine A. Kime

We study the use of a time-dependent potential barrier to control quantum wave packets, in a discretization of the Schroedinger equation. We consider computational issues in solving for a control which steers an initial single peak wave packet to a terminal double peak wave packet.

IEEE/ASME International Conference on Mechatronic and Embedded Systems and Applications•Jan 1, 2011•0 citations

Hydrogen Atom and Hydrogen Molecule Ion in Homogeneous Magnetic Fields of Arbitrary Strength

Ranjan Bhaduri, Yozo Nogami, Chindhu S. Warke

A simple and accurate variational method is formulated and applied to the ground and excited states of the hydrogen atom as well as the hydrogen molecule ion in a homogeneous magnetic field of arbitrary strength. For the ground state, the electronic wave function o is assumed to be a function of a single, appropriately chosen variable R which contains a variational parameter. The functional form of o, and the corresponding energy, are determined by solving a Schr6dinger- like equation, in which the variational parameter appears explicitly in the kinetic and potential energy terms. The variational parameter is optimized by minimizing the lowest eigenvalue of the equation. Excited states may also be obtained by a slight modification of the method, and explicit calculations are carried out for the m1 = - 1 first excited state of hydrogen and the ( )-state of H2 +. Numerical comparisons are made with some recent papers in the literature. Subject headings: atomic processes - magnetic fields - molecular processes - quantum mechanics

The Astrophysical Journal•Oct 1, 1977•36 citations

The Berry phase and the hydrogen molecular ion in an external magnetic field

W. N. Cottingham, N Hassan

The authors consider the rotational spectrum of the hydrogen molecular ion in a magnetic field. It is shown that the Berry phase factor for this system acts like an additional magnetic field which effectively reduces the external field by about 5%.

Journal of Physics B Atomic Molecular and Optical Physics•Jan 28, 1990•9 citations

Electronic quantum motions in hydrogen molecule ion

Ciann‐Dong Yang, Hung‐Jen Weng

Abstract The hydrogen molecule ion is a two‐center force system expressed under the prolate spheroidal coordinates, whose quantum motions and quantum trajectories have never been addressed in the literature before. The momentum operators in this coordinate system are derived for the first time from the Hamilton equations of motion and used to construct the Hamiltonian operator. The resulting Hamiltonian comprises a kinetic energy T and a total potential VTotal consisting of the Coulomb potential and a quantum potential. It is shown that the participation of the quantum potential and the accompanied quantum forces in the force interaction within H2+ is essential to develop an electronic motion consistent with the prediction of the probability density function |Ψ|2. The motion of the electron in H2+ can be either described by the Hamilton equations derived from the Hamiltonian H = TK + VTotal or by the Lagrange equations derived from the Lagrangian H = TK − VTotal. Solving the equations of motion with different initial positions, we show that the solutions yield an assembly of electronic quantum trajectories whose distribution and concentration reconstruct the σ and π molecular orbitals in H2+. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011

International Journal of Quantum Chemistry•Jun 10, 2010•8 citations

H+2in a strong magnetic field described via a solvable model

Rafael D. Benguria, Raymond Brummelhuis, Pierre Duclos +1 more

We consider the hydrogen molecular ion $H^+_2$ in the presence of a strong homogeneous magnetic field. In this regime, the effective Hamiltonian is almost one dimensional with a potential energy which looks like a sum of two Dirac delta functions. This model is solvable, but not close enough to our exact Hamiltonian for relevant strenght of the magnnetic field. However we show that the correct values of the equilibrium distance as well as the binding energy of the ground state of the ion, can be obtained when incorporating perturbative corrections up to second order. Finally, we show that $ He_2^{3+}$ exists for sufficiently large magnetic fields.

arXiv:math-ph/0403034•Journal of Physics B Atomic Molecular and Optical Physics•May 18, 2004•17 citations

Hidden symmetry and explicit spheroidal eigenfunctions of the hydrogen atom

Stella M. Sung, Dudley R. Herschbach

The Schrödinger equation for a hydrogenic atom is separable in prolate spheroidal coordinates, as a consequence of the ‘‘hidden symmetry’’ stemming from the fixed spatial orientation of the classical Kepler orbits. One focus is at the nucleus and the other a distance R away along the major axis of the elliptic orbit. The separation constant α is not an elementary function of Z or R or quantum numbers. However, for given principal quantum number n and angular momentum projection m, the allowed values of α and corresponding eigenfunctions in spheroidal coordinates are readily obtained from a secular equation of order n−m. We evaluate α(n,m;ZR) and the coefficients gl(α) that specify the spheroidal eigenfunctions as hybrids of the familiar ‖nlm〉 hydrogen-atom states with fixed n and m but different l values. Explicit formulas and plots are given for α and gl and for the probability distributions derived from the hybrid wave functions, ∑lgl(α)‖nlm〉, for all states up through n=4. In the limit R→∞ these hybrids become the solutions in parabolic coordinates, determined simply by geometrical Clebsch–Gordan coefficients that account for conservation of angular momentum and the hidden symmetry. We also briefly discuss some applications of the spheroidal eigenfunctions, particularly to exact analytic solutions of two-center molecular orbitals for special values of R and the nuclear charge ratio Za/Zb.

The Journal of Chemical Physics•Nov 15, 1991•21 citations
A simple and accurate variational method is formulated and applied to the ground and excited states of the hydrogen atom as well as the hydrogen molecule ion in a homogeneous magnetic field of arbitrary strength. For the ground state, the electronic wave function o is assumed to be a function of a single, appropriately chosen variable R which contains a variational parameter. The functional form of o, and the corresponding energy, are determined by solving a Schr6dinger- like equation, in which the variational parameter appears explicitly in the kinetic and potential energy terms. The variational parameter is optimized by minimizing the lowest eigenvalue of the equation. Excited states may also be obtained by a slight modification of the method, and explicit calculations are carried out for the m1 = - 1 first excited state of hydrogen and the ( )-state of H2 +. Numerical comparisons are made with some recent papers in the literature. Subject headings: atomic processes - magnetic fields - molecular processes - quantum mechanics
Abstract The hydrogen molecule ion is a two‐center force system expressed under the prolate spheroidal coordinates, whose quantum motions and quantum trajectories have never been addressed in the literature before. The momentum operators in this coordinate system are derived for the first time from the Hamilton equations of motion and used to construct the Hamiltonian operator. The resulting Hamiltonian comprises a kinetic energy T and a total potential VTotal consisting of the Coulomb potential and a quantum potential. It is shown that the participation of the quantum potential and the accompanied quantum forces in the force interaction within H2+ is essential to develop an electronic motion consistent with the prediction of the probability density function |Ψ|2. The motion of the electron in H2+ can be either described by the Hamilton equations derived from the Hamiltonian H = TK + VTotal or by the Lagrange equations derived from the Lagrangian H = TK − VTotal. Solving the equations of motion with different initial positions, we show that the solutions yield an assembly of electronic quantum trajectories whose distribution and concentration reconstruct the σ and π molecular orbitals in H2+. © 2010 Wiley Periodicals, Inc. Int J Quantum Chem, 2011
We consider the hydrogen molecular ion H2+H^+_2H2+​ in the presence of a strong homogeneous magnetic field. In this regime, the effective Hamiltonian is almost one dimensional with a potential energy which looks like a sum of two Dirac delta functions. This model is solvable, but not close enough to our exact Hamiltonian for relevant strenght of the magnnetic field. However we show that the correct values of the equilibrium distance as well as the binding energy of the ground state of the ion, can be obtained when incorporating perturbative corrections up to second order. Finally, we show that $ He_2^{3+}$ exists for sufficiently large magnetic fields.
The Schrödinger equation for a hydrogenic atom is separable in prolate spheroidal coordinates, as a consequence of the ‘‘hidden symmetry’’ stemming from the fixed spatial orientation of the classical Kepler orbits. One focus is at the nucleus and the other a distance R away along the major axis of the elliptic orbit. The separation constant α is not an elementary function of Z or R or quantum numbers. However, for given principal quantum number n and angular momentum projection m, the allowed values of α and corresponding eigenfunctions in spheroidal coordinates are readily obtained from a secular equation of order n−m. We evaluate α(n,m;ZR) and the coefficients gl(α) that specify the spheroidal eigenfunctions as hybrids of the familiar ‖nlm〉 hydrogen-atom states with fixed n and m but different l values. Explicit formulas and plots are given for α and gl and for the probability distributions derived from the hybrid wave functions, ∑lgl(α)‖nlm〉, for all states up through n=4. In the limit R→∞ these hybrids become the solutions in parabolic coordinates, determined simply by geometrical Clebsch–Gordan coefficients that account for conservation of angular momentum and the hidden symmetry. We also briefly discuss some applications of the spheroidal eigenfunctions, particularly to exact analytic solutions of two-center molecular orbitals for special values of R and the nuclear charge ratio Za/Zb.