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

Ferroelectrics

Quantum mechanics of a proton or deuteron on a single h-bond: Fluctuating potential and friction term

Andreas Becker, R. Siems

Published Dec 1, 1991• 0 citations
Publisher

Abstract

The dynamics of protons and deuterons in double minimum hydrogen bond potentials is investigated. Results for fluctuating potentials are derived and compared with time dependent solutions for constant potentials. The influence of different friction operators and of stochastically varying terms in the Hamiltonian is studied both separately and for the case of combined action. Quantities studied are ψ, <x>, and <H> as functions of t.

DOI
10.1080/00150199108209408

Related papers

Lingering Time of the Proton in the Wells of the Double-Minimum Potential of Hydrogen Bonds

Jürgen Brickmann, Herbert W. Zimmermann

The movement of a proton which moves in a one-dimensional effective double-minimum potential V(x) and can pass by means of the tunnel effect from one well into the other was investigated. The lingering times of the proton in the potential wells (a) and (b) were calculated for more than 150 potential models as functions of the height Vmax of the potential barrier separating the two wells, the energy difference ΔV, and separation d of the minima. In the determination of the lingering times, it was assumed that the proton at time t = 0 is to be found with certainty in one of the wells and can be represented by means of a statistical operator related to the eigenstates of a one-well Hamiltonian H(a)orH(b), respectively. The times determined on this basis were compared with those which result with a quasiclassical method from the classical frequency of oscillation of a particle in a well and from the penetration coefficient of the potential barrier. The deviation between the values determined with both methods is largest if an eigenvalue of H(a) becomes approximately equal to an eigenvalue of H(b). For the potential model V(x) a polynomial of fourth degree in the position coordinate x was chosen. The eigenfunctions of H(a)andH(b) as well as those of the effective Hamiltonian H corresponding to V(x) were represented as linear combinations of eigenfunctions of the harmonic oscillator with angular frequency ω. ω was chosen so that the mean square deviations of the approximated eigenvalues of H became as small as possible for the states of lowest energy.

The Journal of Chemical Physics•Feb 15, 1969•90 citations

Quantum mechanics of two coupled H-bond protons: eigenstates, time evolution, spectral fluctuations, and chaos

Thomas Pattard, R. Siems

A system of two bilinearly coupled particles moving in two equal symmetric one dimensional double well potentials is considered. Varying the coupling strength changes the classical behaviour drastically. From a quantum mechanical treatment statistical properties of the spectrum (term spacing distributions, spectral rigidity) are derived and discussed in connection with the occurence of classical chaos. Furthermore, the time evolutions of wavepackets are discussed.

Ferroelectrics•Mar 1, 1994•2 citations

Interacting protons in two double minimum potentials: Quantum mechanical calculations

H. Blaurock, R. Siems

A system consisting of two protons in two double minimum potentials coupled by a bilinear interaction was studied. Explicit calculations were performed for local potentials described by a combination of harmonic oscillator and Gaussian terms. To determine eigenfunctions and -values variational procedures were employed which made use of numerically determined 1-particle solutions. Eigenvalues and (graphical representations of) 2d eigenfunctions are presented, classified according to their symmetry, and used to determine the time evolution of the system for given initial conditions.

Ferroelectrics•Dec 1, 1991•1 citation

The use of quantum damping self-frictional theory in a study of hydrogen-like atomic energies and forces

Nurşen Seçkin Görgün, I. I. Guseinov, Ramazan Aydın

By the use of complete orthogonal sets of ψ ( δ ∗ ) -damping self-frictional (DSF) exponential type orbitals ( ψ ( δ ∗ ) -DSFETO) introduced by one of the authors, the quantum DSF study of hydrogen-like atomic energies and forces are suggested. Here, δ ∗ = α ∗ or δ ∗ = p l ∗ , p l ∗ = 2 l + 2 - α ∗ and α ∗ = α - nl ∗ ( z ) are the integer and non-integer DSF quantum numbers which depend on quantities n , l and z , where z = H, He + , Li ++ , Be +++ , B ++++ , C +++++ , N ++++++ , O +++++++ , F ++++++++ , Ne +++++++++ . The average values of potentials, kinetic and total energies and forces of the hydrogen-like atomic systems are calculated. The given values are compared with the results obtained in the non-DSF (NDSF) theory.

Results in Physics•Aug 29, 2018•1 citation

Quantum Langevin Dynamics

Mohammad Attrash, Roi Baer

Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of $E_{gs}+k_BT$. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.

arXiv:2509.09743•Sep 11, 2025•0 citations

Hydrogen Spectrum for the Proton Hopping Randomly Between Two Sites

José Angel Lesteiro Tejeda

The hydrogen spectrum is explored as a sensitive probe of a massive object being in a superposition of two distinct locations. Here I solve the simpler but instructive semiclassical problem of the proton hopping stochastically between two sites but forbidden to be anywhere else, analogous to the superposition over two locations. Although the model is semiclassical, it captures symmetry properties and the structure of the energy spectrum that one would expect in a fully quantum treatment. The hydrogen atom is a well-understood system whose spectroscopy reaches exceptional precision, and its semiclassical description displays symmetry with respect to two fixed locations. I develop a classical model for an electron and a proton, with the proton stochastically hopping between two fixed sites; and describe its classical dynamics mathematically via a one-particle effective Hamiltonian. For this Hamiltonian I identify a set of invariance properties that are characteristic of heteronuclear diatomic molecules and are not as extensive as the SO(4) symmetries present in the traditional models of the hydrogen atom. I connect the classical description of the proton-hopping model with the two-fixed-centre problem with time-dependent charges. I find that there is a lower limit to the hopping distance of the proton below which state-of-the-art measurements cannot resolve the spectral signature predicted; and also an upper limit above which approximation methods are not applicable. I gain meaningful insight into the mechanisms that shape the quantum dynamics of proton tunnelling in hydrogen by analyzing the classical counterpart, leveraging the shared symmetries between quantum and classical analogous systems. My results predict an observable spectral shift in the ground-state energy when the proton is delocalized over two sites separated by a distance large enough to be considered macroscopically distinct. With this research, I support the idea that proton tunnelling inside the hydrogen atom is worth exploring as a probe for a delocalized massive object.

•Dec 16, 2025•0 citations
The dynamics of protons and deuterons in double minimum hydrogen bond potentials is investigated. Results for fluctuating potentials are derived and compared with time dependent solutions for constant potentials. The influence of different friction operators and of stochastically varying terms in the Hamiltonian is studied both separately and for the case of combined action. Quantities studied are ψ, <x>, and <H> as functions of t.
The movement of a proton which moves in a one-dimensional effective double-minimum potential V(x) and can pass by means of the tunnel effect from one well into the other was investigated. The lingering times of the proton in the potential wells (a) and (b) were calculated for more than 150 potential models as functions of the height Vmax of the potential barrier separating the two wells, the energy difference ΔV, and separation d of the minima. In the determination of the lingering times, it was assumed that the proton at time t = 0 is to be found with certainty in one of the wells and can be represented by means of a statistical operator related to the eigenstates of a one-well Hamiltonian H(a)orH(b), respectively. The times determined on this basis were compared with those which result with a quasiclassical method from the classical frequency of oscillation of a particle in a well and from the penetration coefficient of the potential barrier. The deviation between the values determined with both methods is largest if an eigenvalue of H(a) becomes approximately equal to an eigenvalue of H(b). For the potential model V(x) a polynomial of fourth degree in the position coordinate x was chosen. The eigenfunctions of H(a)andH(b) as well as those of the effective Hamiltonian H corresponding to V(x) were represented as linear combinations of eigenfunctions of the harmonic oscillator with angular frequency ω. ω was chosen so that the mean square deviations of the approximated eigenvalues of H became as small as possible for the states of lowest energy.
A system consisting of two protons in two double minimum potentials coupled by a bilinear interaction was studied. Explicit calculations were performed for local potentials described by a combination of harmonic oscillator and Gaussian terms. To determine eigenfunctions and -values variational procedures were employed which made use of numerically determined 1-particle solutions. Eigenvalues and (graphical representations of) 2d eigenfunctions are presented, classified according to their symmetry, and used to determine the time evolution of the system for given initial conditions.
By the use of complete orthogonal sets of ψ ( δ ∗ ) -damping self-frictional (DSF) exponential type orbitals ( ψ ( δ ∗ ) -DSFETO) introduced by one of the authors, the quantum DSF study of hydrogen-like atomic energies and forces are suggested. Here, δ ∗ = α ∗ or δ ∗ = p l ∗ , p l ∗ = 2 l + 2 - α ∗ and α ∗ = α - nl ∗ ( z ) are the integer and non-integer DSF quantum numbers which depend on quantities n , l and z , where z = H, He + , Li ++ , Be +++ , B ++++ , C +++++ , N ++++++ , O +++++++ , F ++++++++ , Ne +++++++++ . The average values of potentials, kinetic and total energies and forces of the hydrogen-like atomic systems are calculated. The given values are compared with the results obtained in the non-DSF (NDSF) theory.
Previous years researchers began to simulate open quantum system, taking into account the interaction between system and the environment. One approach to deal with this problem is to use the density matrix within the Liouville-von-Neumann formalism or the Markovian variant the Lindblad equations. Another way is to use a stochastic approach where a random force is added to the system. The benefit of the stochastic approach is to solve the dynamics of the system with less time and memory than the density matrix approaches. In this project we want to develop a stochastic approach that can deal with the stochastic wave functions approach. We did this on a 2-level system and found that it works well when comparing to a density matrix approach. Next, we tested a quantum particle connect to a bath of harmonic oscillators using the stochastic approach. We found that a friction term is necessary and applied it. Like in the classical Langevin equations the friction constant and the random force fluctuations are related by the fluctuation-dissipation constant. We showed that with friction the dynamics decays to an ensemble with energy of Egs+kBTE_{gs}+k_BTEgs​+kB​T. However, we also found here are problems. The system seems to absorb energy indefinitely if the temperature is higher than the zero point energy or if the system is a Morse oscillator. Thus more research is required to make this method work.
The hydrogen spectrum is explored as a sensitive probe of a massive object being in a superposition of two distinct locations. Here I solve the simpler but instructive semiclassical problem of the proton hopping stochastically between two sites but forbidden to be anywhere else, analogous to the superposition over two locations. Although the model is semiclassical, it captures symmetry properties and the structure of the energy spectrum that one would expect in a fully quantum treatment. The hydrogen atom is a well-understood system whose spectroscopy reaches exceptional precision, and its semiclassical description displays symmetry with respect to two fixed locations. I develop a classical model for an electron and a proton, with the proton stochastically hopping between two fixed sites; and describe its classical dynamics mathematically via a one-particle effective Hamiltonian. For this Hamiltonian I identify a set of invariance properties that are characteristic of heteronuclear diatomic molecules and are not as extensive as the SO(4) symmetries present in the traditional models of the hydrogen atom. I connect the classical description of the proton-hopping model with the two-fixed-centre problem with time-dependent charges. I find that there is a lower limit to the hopping distance of the proton below which state-of-the-art measurements cannot resolve the spectral signature predicted; and also an upper limit above which approximation methods are not applicable. I gain meaningful insight into the mechanisms that shape the quantum dynamics of proton tunnelling in hydrogen by analyzing the classical counterpart, leveraging the shared symmetries between quantum and classical analogous systems. My results predict an observable spectral shift in the ground-state energy when the proton is delocalized over two sites separated by a distance large enough to be considered macroscopically distinct. With this research, I support the idea that proton tunnelling inside the hydrogen atom is worth exploring as a probe for a delocalized massive object.