Quantum Mechanical Eigenvalues Research Articles

The ground electronic state of CS: A global multi-isotopologue direct potential fit analysis

A collection of 6759 pure-rotational and vibrational–rotational spectroscopic line positions available for seven isotopologues of the ground X1Σ+ electronic state of carbon monosulfide has been employed in a weighted nonlinear least-squares direct fit to the potential energy function of 12C32S and supplementary functions describing breakdown of the Born-Oppenheimer approximation. All radial functions are represented by compact analytical models having proper theoretical asymptotic behavior. The spectroscopic line positions are reproduced on average to within the associated experimental uncertainties by the quantum–mechanical eigenvalues of the derived Hamiltonians for individual isotopologues. The potential energy function is constrained to obey the theoretical radial behavior both at short-range and at long-range. Accurate quantum–mechanical vibrational term values and rotational and centrifugal distortion constants are provided for all stable isotopologues of CS included in the least-squares fits.

Some quantum mechanical solutions in nonrelativistic anti-Snyder framework

In this paper, we investigate nonrelativistic anti-Snyder model in momentum representation and obtain quantum mechanical eigenvalues and eigenfunctions. Using this framework, first, in one dimension, we study a particle in a box and the harmonic oscillator problems. Then, for more investigations, in three dimensions, the quantum mechanical eigenvalues and eigenfunctions of a free particle problem and the radius of the neutron star are obtained.

Accurate internuclear potential energy functions for the ground electronic states of NeH+ and ArH+

All pure rotational and vibrational-rotational spectroscopic line positions available on the ground X1Σ+ electronic states of the rare gas hydride cations NeH+ and ArH+ have been employed in weighted least-squares direct fits to the potential energy functions, together with auxiliary functions describing breakdown of the Born-Oppenheimer approximation. All radial functions are represented by compact analytical models, and the spectroscopic line positions are reproduced to within the associated experimental uncertainties by the quantum-mechanical eigenvalues of the derived Hamiltonians. The potential energy functions are constrained to approach the theoretical radial behavior at long-range. Accurate vibrational term values and rotational and centrifugal distortion constants are provided for all stable isotopologues of NeH+ and ArH+ included in the least-squares fits.

Algebraic Treatment of 𝓟𝓣-Symmetric Coupled Oscillators

The purpose of this paper is the discussion of a pair of coupled linear oscillators that has recently been proposed as a model of a system of two optical resonators. By means of an algebraic approach we show that the frequencies of the classical and quantum-mechanical interpretations of the optical phenomenon are exactly the same. Consequently, if the classical frequencies are real, then the quantum-mechanical eigenvalues are also real.

Nonlinear eigenvalue problems

This paper presents an asymptotic study of the differential equation y′(x) = cos [πxy(x)] subject to the initial condition y(0) = a. While this differential equation is nonlinear, the solutions to the initial-value problem bear a striking resemblance to solutions to the linear time-independent Schrödinger eigenvalue problem. As x increases from 0, y(x) oscillates and thus resembles a quantum wave function in a classically allowed region. At a critical value x = xcrit, where xcrit depends on a, the solution y(x) undergoes a transition; the oscillations abruptly cease and y(x) decays to 0 monotonically as x → ∞. This transition resembles the transition in a wave function at a turning point as one enters the classically forbidden region. Furthermore, the initial condition a falls into discrete classes; in the nth class of initial conditions an − 1 < a < an (n = 1, 2, 3, …), y(x) exhibits exactly n maxima in the oscillatory region. The boundaries an of these classes are the analogues of quantum-mechanical eigenvalues. An asymptotic calculation of an for large n is analogous to a high-energy semiclassical (WKB) calculation of eigenvalues in quantum mechanics. The principal result of this paper is that as n → ∞, , where A = 25/6. Numerical analysis reveals that the first Painlevé transcendent has an eigenvalue structure that is quite similar to that of the equation y′(x) = cos [πxy(x)] and that the nth eigenvalue grows with n like a constant times n3/5 as n → ∞. Finally, it is noted that the constant A is numerically very close to the lower bound on the power-series constant P in the theory of complex variables, which is associated with the asymptotic behavior of zeros of partial sums of Taylor series.

The potential energy function of the ground electronic state of 16O2

The potential energy function of the ground X (3)Σ(g)(-) electronic state of (16)O(2) has been obtained in analytical form by employing a fully quantum-mechanical direct potential fitting method. Term values generated from highly precise Dunham coefficients were employed in the direct fit, yielding a potential energy function that is accurate up to υ = 31, or up to an energy that is 90% of the dissociation limit. The term values are represented with a standard deviation of 0.0031 cm(-1) by the quantum-mechanical eigenvalues of the derived potential energy function. There is considerable evidence for N-dependent nonadiabatic coupling between the ground X (3)Σ(g)(-) electronic state and the b (1)Σ(g)(+) excited electronic state.

An “adiabatic-hindered-rotor” treatment allows para-H2 to be treated as if it were spherical

In para-H(2)-{molecule} interactions, the common assumption that para-H(2) may be treated as a spherical particle is often substantially in error. For example, quantum mechanical eigenvalues on a full four-dimensional (4D) potential energy surface for para H(2)-{linear molecule} species often differ substantially from those calculated from the corresponding two-dimensional (2D) surface obtained by performing a simple spherical average over the relative orientations of the H(2) moiety. However, use of an "adiabatic-hindered-rotor" approximation can yield an effective 2D surface whose spectroscopic properties are an order of magnitude closer to those yielded by a full 4D treatment.

Application of direct potential fitting to line position data for the [formula omitted] and [formula omitted] states of Li 2

A collection of 15 222 spectroscopic Li 2 line positions, all with measurement precision in the range 0.005–0.02 cm −1, and which sample 99.97 and 99.98% of the A and X state well depths, respectively, have been employed in a direct least squares fit of the effective potential energy and Born–Oppenheimer breakdown functions for the two states. The fully analytical potential function for the A state of 6Li 2 is determined out to R ≈ 55 Å, well into the long-range region. For the three isotopologues, 6Li 2, 6Li 7Li, and 7Li 2, most of the data are comprised of rotationally resolved transitions in the A– X system recorded by Fourier transform spectroscopies. For 6Li 2 and 7Li 2, complementary fluorescence data are also included for high vibrational levels of both the A and X states. The reduced standard deviation of the fit was close to unity, indicating that the quantum mechanical eigenvalues calculated from the analytical functions of the Hamiltonians of the two states, which are described by a total of only 45 fitted parameters, represent the line positions, on average, to within the estimated uncertainties. The dissociation energies for 6Li 2 are estimated precisely as D e( X) = 8516.74(1) cm −1 and D e( A) = 9351.96(1) cm −1.

A large number of higher-energy eigenvalues of a huge dimensional matrix for a quantum chaotic study of a quartic potential

A system of two quartic oscillators coupled by a quartic perturbation is numerically studied for quantum mechanical eigenvalues and classical periodic orbits. The coupling strength serves as a control parameter to simulate the transition from integrable to chaotic regimes. In order to obtain higher-energy eigenvalues of a huge dimensional matrix, the Lanczos method and the equi-energy method are investigated for a practical use.

Quantum level structures at a Fermi resonance with angular momentum: classical periodic orbits, catastrophe maps and quantum monodromy

The Xiao-Kellman catastrophe map, for the classification of classical periodic orbits of the standard 2:1 Fermi resonance Hamiltonian is extended to species with finite vibrational angular momentum. The influence of the classical periodic orbit structure on different organizations of the quantum mechanical eigenvalues, in the four regions of the map, is strikingly demonstrated. The quantum eigenvalue lattices in angular momentum and energy space show dislocations attributable to a topological effect, termed quantum monodromy. Analogues with quantum monodromy in quasi-linear molecules and LiCN/LiNC isomerisation are demonstrated.

Application of direct potential fitting to line position data for the X 1Σ+ and A 1Σ+ states of LiH

A collection of 9089 spectroscopic LiH line positions, of widely varying precision, which sample 84.9% and 98.6% of the A and X state well depths, respectively, have been employed in a direct least-squares fit of the effective potential energy and Born-Oppenheimer breakdown functions for the two states. For the four isotopomers (6)LiH, (7)LiH, (6)LiD, and (7)LiD, the data comprise both pure rotational and vibration-rotational transitions within the ground state, as well as rotationally resolved transitions in the A-X system. Despite the unusual shape and associated anomalous properties of the A state potential, no special features or considerations were required in the direct potential fitting approach. The reduced standard deviation of the fit was close to unity, indicating that the quantum mechanical eigenvalues calculated from the fully analytical functions of the Hamiltonians of the two states, which are characterized by a total of only 53 fitted parameters, represent the line positions, on average, to within the estimated uncertainties. A quantum mechanical calculation of the molecular constants G(nu), B(nu), D(nu), H(nu), L(nu), M(nu), N(nu), and O(nu) from the fitted potential for the A state of (7)LiH confirms that the usual polynomial expansion in J(J+1) is an unsatisfactory representation for the rotational terms of the lowest vibrational levels.

Direct potential fit analysis of the X1sigma+ ground state of CO.

A collection of 21,559 highly precise spectroscopic line positions from pure rotational and vibration-rotational spectra for seven isotopomers of carbon monoxide in the X1sigma+ ground electronic state has been employed in direct least-squares fits of the rovibrational Hamiltonian operator obtained from Watson's work [J. Mol. Spectrosc. 80, 411 (1980)] and that obtained by Herman and Ogilvie [Adv. Chem. Phys 103, 187 (1998)]. Fully analytical models are used for the various functions, including the Born-Oppenheimer internuclear potential function, and an account is taken of breakdown of the Born-Oppenheimer approximation. The resulting representations are more compact than currently available traditional Ukl/deltakl extended Dunham descriptions, and they generate quantum-mechanical eigenvalues that reproduce reliably the spectroscopic line positions to within the associated measurement uncertainties. Rayleigh-Schrodinger perturbation theory has been used to calculate highly accurate rotational and centrifugal distortion constants Bupsilon-Oupsilon for nine isotopomers of carbon monoxide. These constants are just as successful at reconstructing the observed spectroscopic information as the quantum-mechanical eigenvalues of the fitted Hamiltonian operators.

EIGENVALUE SPECTRUM FOR A SINGLE PARTICLE IN A SPHEROIDAL CAVITY: A SEMICLASSICAL APPROACH

Following the semiclassical formalism of Strutinsky et al.,11 we have obtained the complete eigenvalue spectrum for a particle enclosed in an infinitely high spheroidal cavity. Our spheroidal trace formula also reproduces the results of a spherical billiard in the limit η → 1.0. Inclusion of repetition of each family of the orbits with reference to the largest one significantly improves the eigenvalues of the sphere and an exact comparison with the quantum mechanical results is observed up to the second decimal place for kR0 ≥ 7. The contributions of the equatorial, planar (in the axis of symmetry plane) and non-planar (three-dimensional) orbits are obtained from the same trace formula by using the appropriate conditions. The resulting eigenvalues compare very well with the quantum mechanical eigenvalues at normal deformation. It is interesting that the partial sum of equatorial orbits leads to eigenvalues with maximum angular momentum projection, while the summing of planar orbits leads to eigenvalues with Lz = 0 except for L = 1. The remaining quantum mechanical eigenvalues are observed to arise from the three-dimensional (3D) orbits. Very few spurious eigenvalues arise in these partial sums. This result establishes the important role of 3D orbits even at normal deformations.

Resonance Zones in Action Space

Abstract The classical and quantum mechanics of isolated, nonlinear resonances in integrable systems with N ≥ 2 degrees of freedom is discussed in terms of geometry in the space of action vari­ ables. Energy surfaces and frequencies are calculated and graphically presented for invariant tori inside and outside the resonance zone. The quantum mechanical eigenvalues, computed in the sem iclassical WKB approximation, show a regular pattern when transformed into the action space of the associated symmetry reduced system: eigenvalues inside the resonance zone are arranged on iV-dimensional cubic lattices, whereas those outside are, in general, non-periodically distributed. However, TV-dimensional triclinic (skewed) lattices exist locally. Both kinds of lattices are joined smoothly across the classical separatrix surface. The statements are illustrated with the help of two and three coupled rotors. The energy-level statistics of this system are found numerically to be in very good agreement with the Poisson distribution, despite of the complex lattice structure. PACS: 03.65.Sq, 05.45.-a

A new technique for the construction of effective Hamiltonians

A new technique, based on the self-consistent projection-operator method due to Seke, is used for the derivation of a sequence of diagonalized effective Hamiltonians. As demonstrated in an example, the new technique is a powerful method for the calculation of quantum-mechanical eigenvalues and eigenstates. Unlike other methods, not only the problem of degenerate eigenvalues is treated adequately, but the constructed effective Hamiltonians open up the way for investigations of the dynamical behaviour of quantum systems as well.

The Seke self-consistent projection-operator approach for the calculation of quantum-mechanical eigenvalues and eigenstates

Originally, the Seke self-consistent projection-operator method has been developed for treating non-Markovian time evolution of probability amplitudes of a relevant set of state vectors. In the so-called Born approximation the method leads automatically to an Hamiltonian restricted to a subspace and thus enables the construction of effective Hamiltonians. In the present paper, in order to explain the efficiency of Seke's method its algebraic operator structure is analyzed and a new successive approximation technique for the calculation of eigenstates and eigenvalues of an arbitrary quantum-mechanical system is developed. Since, unlike most perturbative techniques, in the present approach a well-defined effective (restricted) Hamiltonian in each order of the applied approximation exists, the self-consistency of all obtained results is self-evident. Finally, to illustrate the efficiency of the developed formalism, the optical polaron model is investigated.

Holomorphic quantization on the torus and finite quantum mechanics

We construct explicitly the quantization of classical linear maps of $SL(2, R)$ on toroidal phase space, of arbitrary modulus, using the holomorphic (chiral) version of the metaplectic representation. We show that Finite Quantum Mechanics (FQM) on tori of arbitrary integer discretization, is a consistent restriction of the holomorphic quantization of $SL(2, Z)$ to the subgroup $SL(2, Z)/\Gamma_l$, $\Gamma_l$ being the principal congruent subgroup mod l, on a finite dimensional Hilbert space. The generators of the ``rotation group'' mod l, $O_{l}(2)\subset SL(2,l)$, for arbitrary values of l are determined as well as their quantum mechanical eigenvalues and eigenstates.

Semiclassical quantization of KAM resonances in time-periodic systems

A semiclassical theory for the quasi-energy spectrum of time-periodic systems with accidental classical resonances is presented. The primitive EBK quantum conditions for integrable systems are extended to multiply periodic flux tubes occurring in resonant systems. Replacing classical actions by appropriate differential operators in a classical resonance Hamiltonian yields a uniform quantization of states related to a classical resonance region. The derivation being general for time-periodic systems unfolds the organization of the quasi-energy spectrum, reducing it to the spectrum of a single time-independent Hamiltonian of one degree of freedom with additional rational shifts of h(cross) omega . In a first-order approximation the resonance Hamiltonian is reduced to a pendulum leading to a differential equation of the Mathieu type for the quasi-energies. It is rigorously shown how parameters of the differential equation can be drawn from classical dynamics, using the data of the 'essential' orbits in the resonance zone, i.e. stability coefficients and actions of hyperbolic and elliptic orbits as well as actions of homoclinic orbits. The quasi-energy spectrum of a forced quartic oscillator is studied numerically and evaluated. Semiclassical quasi-energies related to a resonance of period three are computed and compared with exact quantum mechanical eigenvalues.

Comment on: Born–Oppenheimer breakdown in the ground state of carbon monoxide: A direct reduction of spectroscopic line positions to analytical radial Hamiltonian operators

A collection of 10 866 of the most precise ground-state (X1Σ+) vibration–rotational and pure rotational line positions of four carbon monoxide isptopomers (12C16O, 12C18O, 13C16O, and 13C18O) is employed simultaneously in a direct determination of the radial Hamiltonian operator in compact analytical form. The 22-parameter isotopically self-consistent operator takes full account of the Born–Oppenheimer breakdown and its quantum-mechanical eigenvalues represent all the available spectroscopic line positions of CO isotopomers to within the experimental uncertainties. Rayleigh–Schrodinger perturbation theory is employed to calculate quantum-mechanical molecular constants of rotation (Bν – Mν) for nine common isotopomeric forms of CO. Together with the quantum-mechanical vibrational eigenvalues these are fully consistent with the exact eigenvalues obtained by direct solution of the radial wave equation. The set of constants is expected to provide an accurate prediction of line positions of CO isotopomers that...

Born–Oppenheimer breakdown in the ground state of carbon monoxide: A direct reduction of spectroscopic line positions to analytical radial Hamiltonian operators

A collection of 10 866 of the most precise ground-state (X1Σ+) vibration–rotational and pure rotational line positions of four carbon monoxide isptopomers (12C16O, 12C18O, 13C16O, and 13C18O) is employed simultaneously in a direct determination of the radial Hamiltonian operator in compact analytical form. The 22-parameter isotopically self-consistent operator takes full account of the Born–Oppenheimer breakdown and its quantum-mechanical eigenvalues represent all the available spectroscopic line positions of CO isotopomers to within the experimental uncertainties. Rayleigh–Schrödinger perturbation theory is employed to calculate quantum-mechanical molecular constants of rotation (Bν – Mν) for nine common isotopomeric forms of CO. Together with the quantum-mechanical vibrational eigenvalues these are fully consistent with the exact eigenvalues obtained by direct solution of the radial wave equation. The set of constants is expected to provide an accurate prediction of line positions of CO isotopomers that have not yet been experimentally observed.