This paper proposes a new mixed quantum mechanics (QM)-molecular mechanics (MM) approach, where MM is replaced by quantum Hamilton mechanics (QHM), which inherits the modeling capability of MM, while preserving the state-dependent nature of QM. QHM, a single mechanics playing the roles of QM and MM simultaneously, will be employed here to derive the three-dimensional quantum dynamics of diatomic molecules. The resulting state-dependent molecular dynamics including vibration, rotation and spin are shown to completely agree with the QM description and well match the experimental vibration-rotation spectrum. QHM can be incorporated into the framework of a mixed quantum-classical Bohmian method to enable a trajectory interpretation of orbital-spin interaction and spin entanglement in molecular dynamics.This paper proposes a new mixed quantum mechanics (QM)-molecular mechanics (MM) approach, where MM is replaced by quantum Hamilton mechanics (QHM), which inherits the modeling capability of MM, while preserving the state-dependent nature of QM. QHM, a single mechanics playing the roles of QM and MM simultaneously, will be employed here to derive the three-dimensional quantum dynamics of diatomic molecules. The resulting state-dependent molecular dynamics including vibration, rotation and spin are shown to completely agree with the QM description and well match the experimental vibration-rotation spectrum. QHM can be incorporated into the framework of a mixed quantum-classical Bohmian method to enable a trajectory interpretation of orbital-spin interaction and spin entanglement in molecular dynamics.

MoleculesOct 9, 20141 citation

Hamiltonian dynamics of a quantum of space: hidden symmetries and spectrum of the volume operator, and discrete orthogonal polynomials

Vincenz̊o Aquilanti, Dimitri Marinelli, Annalisa Marzuoli

The action of the quantum mechanical volume operator, introduced in connection with a symmetric representation of the three-body problem and recently recognized to play a fundamental role in discretized quantum gravity models, can be given as a second order difference equation which, by a complex phase change, we turn into a discrete Schrödinger-like equation. The introduction of discrete potential-like functions reveals the surprising crucial role here of hidden symmetries, first discovered by Regge for the quantum mechanical 6j symbols; insight is provided into the underlying geometric features. The spectrum and wavefunctions of the volume operator are discussed from the viewpoint of the Hamiltonian evolution of an elementary "quantum of space", and a transparent asymptotic picture emerges of the semiclassical and classical regimes. The definition of coordinates adapted to Regge symmetry is exploited for the construction of a novel set of discrete orthogonal polynomials, characterizing the oscillatory components of torsion-like modes.The action of the quantum mechanical volume operator, introduced in connection with a symmetric representation of the three-body problem and recently recognized to play a fundamental role in discretized quantum gravity models, can be given as a second order difference equation which, by a complex phase change, we turn into a discrete Schrödinger-like equation. The introduction of discrete potential-like functions reveals the surprising crucial role here of hidden symmetries, first discovered by Regge for the quantum mechanical 6j symbols; insight is provided into the underlying geometric features. The spectrum and wavefunctions of the volume operator are discussed from the viewpoint of the Hamiltonian evolution of an elementary "quantum of space", and a transparent asymptotic picture emerges of the semiclassical and classical regimes. The definition of coordinates adapted to Regge symmetry is exploited for the construction of a novel set of discrete orthogonal polynomials, characterizing the oscillatory components of torsion-like modes.

arXiv:1301.1949Journal of Physics A Mathematical and TheoreticalApr 15, 201322 citations

What Kind of Insight Provide Analytical Solutions of Quantum Models?

Daniel Braak

Abstract There are several concepts of what constitutes the analytical solution of a quantum model, as opposed to the mere “numerically exact” one. This applies even if one considers only the determination of the discrete spectrum of the corresponding Hamiltonian, setting aside such important questions as the asymptotic dynamics for long times. In the simplest case, the spectrum can be given in closed form, the eigenvalues $E_{j}, j=0,\ldots ,N\le \infty$ E j , j = 0 , … , N ≤ ∞ read $E_{j} =f(j,\{p_{k}\})$ E j = f ( j , { p k } ) , where f is a known function of the label $j\in \mathbb {N}_{0}$ j ∈ N 0 and the $\{p_k\}$ { p k } are a set of numbers parameterizing the Hamilton operator. This kind of solution exists only in cases where the classical limit of the model is Liouville-integrable. Some quantum-mechanical many-body systems allow the determination of the spectrum in terms of auxiliary parameters $[\{k_j\},\{n_l\}]$ [ { k j } , { n l } ] as $E(\{n_l\}) = f(\{k_{j}(\{n_{l}\})\})$ E ( { n l } ) = f ( { k j ( { n l } ) } ) where the $\{k_{j}(\{n_{l}\})\}$ { k j ( { n l } ) } satisfy a coupled set of transcendental equations, following from a certain ansatz for the eigenfunctions. These systems (integrable in the sense of Yang-Baxter (Eckle 2019)) may have a Hilbert space dimension growing exponentially with the system size L, i.e., $N\sim e^{L}$ N ∼ e L . The simple enumeration of the energies with the label j is replaced by the multi-index $\{n_{l}\}$ { n l } . Although no priori knowledge about the spectrum is available, its statistical properties can be computed exactly (Berry and Tabor 1977). Other integrable and also non-integrable models exist where N depends polynomially on L and the energies $E_j$ E j are the zeroes of an analytically computable transcendental function, the so-called G-function $G(E,\{p_k\})$ G ( E ,Abstract There are several concepts of what constitutes the analytical solution of a quantum model, as opposed to the mere “numerically exact” one. This applies even if one considers only the determination of the discrete spectrum of the corresponding Hamiltonian, setting aside such important questions as the asymptotic dynamics for long times. In the simplest case, the spectrum can be given in closed form, the eigenvalues $$E_{j}, j=0,\ldots ,N\le \infty $$ E j , j = 0 , … , N ≤ ∞ read $$E_{j} =f(j,\{p_{k}\})$$ E j = f ( j , { p k } ) , where f is a known function of the label $$j\in \mathbb {N}_{0}$$ j ∈ N 0 and the $$\{p_k\}$$ { p k } are a set of numbers parameterizing the Hamilton operator. This kind of solution exists only in cases where the classical limit of the model is Liouville-integrable. Some quantum-mechanical many-body systems allow the determination of the spectrum in terms of auxiliary parameters $$[\{k_j\},\{n_l\}]$$ [ { k j } , { n l } ] as $$E(\{n_l\}) = f(\{k_{j}(\{n_{l}\})\})$$ E ( { n l } ) = f ( { k j ( { n l } ) } ) where the $$\{k_{j}(\{n_{l}\})\}$$ { k j ( { n l } ) } satisfy a coupled set of transcendental equations, following from a certain ansatz for the eigenfunctions. These systems (integrable in the sense of Yang-Baxter (Eckle 2019)) may have a Hilbert space dimension growing exponentially with the system size L, i.e., $$N\sim e^{L}$$ N ∼ e L . The simple enumeration of the energies with the label j is replaced by the multi-index $$\{n_{l}\}$$ { n l } . Although no priori knowledge about the spectrum is available, its statistical properties can be computed exactly (Berry and Tabor 1977). Other integrable and also non-integrable models exist where N depends polynomially on L and the energies $$E_j$$ E j are the zeroes of an analytically computable transcendental function, the so-called G-function $$G(E,\{p_k\})$$ G ( E ,

Mathematics for industryOct 22, 20200 citations

E_{j}, j=0,\ldots ,N\le \infty

E j , j = 0 , … , N ≤ ∞ read

E_{j} =f(j,\{p_{k}\})

E j = f ( j , { p k } ) , where f is a known function of the label

j\in \mathbb {N}_{0}

j ∈ N 0 and the

\{p_k\}

{ p k } are a set of numbers parameterizing the Hamilton operator. This kind of solution exists only in cases where the classical limit of the model is Liouville-integrable. Some quantum-mechanical many-body systems allow the determination of the spectrum in terms of auxiliary parameters

[\{k_j\},\{n_l\}]

[ { k j } , { n l } ] as

E(\{n_l\}) = f(\{k_{j}(\{n_{l}\})\})

E ( { n l } ) = f ( { k j ( { n l } ) } ) where the

\{k_{j}(\{n_{l}\})\}

{ k j ( { n l } ) } satisfy a coupled set of transcendental equations, following from a certain ansatz for the eigenfunctions. These systems (integrable in the sense of Yang-Baxter (Eckle 2019)) may have a Hilbert space dimension growing exponentially with the system size L, i.e.,

N\sim e^{L}

N ∼ e L . The simple enumeration of the energies with the label j is replaced by the multi-index

\{n_{l}\}

{ n l } . Although no priori knowledge about the spectrum is available, its statistical properties can be computed exactly (Berry and Tabor 1977). Other integrable and also non-integrable models exist where N depends polynomially on L and the energies

E_j

E j are the zeroes of an analytically computable transcendental function, the so-called G-function

G(E,\{p_k\})

G ( E ,