Quantum Mechanical Anharmonic Oscillator Research Articles

Scaling limit of the ground state Bethe roots for the inhomogeneous XXZ spin - [formula omitted] chain

It is known that for the Heisenberg XXZ spin - 12 chain in the critical regime, the scaling limit of the vacuum Bethe roots yields an infinite set of numbers that coincide with the energy spectrum of the quantum mechanical 3D anharmonic oscillator. The discovery of this curious relation, among others, gave rise to the approach referred to as the ODE/IQFT (or ODE/IM) correspondence. Here we consider a multiparametric generalization of the Heisenberg spin chain, which is associated with the inhomogeneous six-vertex model. When quasi-periodic boundary conditions are imposed the lattice system may be explored within the Bethe Ansatz technique. We argue that for the critical spin chain, with a properly formulated scaling limit, the scaled Bethe roots for the ground state are described by second order differential equations, which are multi-parametric generalizations of the Schrödinger equation for the anharmonic oscillator.

Derivation of Miller’s rule for the nonlinear optical susceptibility of a quantum anharmonic oscillator

Miller’s rule is an empirical relation between the nonlinear and linear optical coefficients that applies to a large class of materials but has only been rigorously derived for the classical Lorentz model with a weak anharmonic perturbation. In this work, we extend the proof and present a detailed derivation of Miller’s rule for an equivalent quantum-mechanical anharmonic oscillator. For this purpose, the classical concept of velocity-dependent damping inherent to the Lorentz model is replaced by an adiabatic switch-on of the external electric field, which allows a unified treatment of the classical and quantum-mechanical systems using identical potentials and fields. Although the dynamics of the resulting charge oscillations, and hence the induced polarizations, deviate due to the finite zero-point motion in the quantum-mechanical framework, we find that Miller’s rule is nevertheless identical in both cases up to terms of first order in the anharmonicity. With a view to practical applications, especially in the context of ab initio calculations for the optical response where adiabatically switched-on fields are widely assumed, we demonstrate that a correct treatment of finite broadening parameters is essential to avoid spurious errors that may falsely suggest a violation of Miller’s rule, and we illustrate this point by means of a numerical example.

Potential energy curves, partition functions and line intensities of HF and HCl molecules under the improved Hulburt‐Hirschfelder quantum mechanical anharmonic oscillator model

AbstractAn improved Hulburt‐Hirschfelder (IHH) quantum mechanical anharmonic oscillator model which describes the exact analytical potential energy curves (PECs) of diatomic molecules is presented. The IHH analytical potential function that involves a variable parameter, four empirically determined low‐order molecular vibrational and rotational spectral constants and the dissociation energy was applied to the ground electronic states of HF and HCl, yielding a good agreement between the Rydberg‐Klein‐Rees (RKR) potential energy points and the calculations. Correspondingly, with the determined IHH PECs, the partition functions at temperatures 300–6000 Kelvin (K) and rovibrational line intensities are evaluated, and coincide well with those quoted in the HITRAN2020 database compilation. The methodology could be valuable for the detection of molecules in interstellar clouds using infrared spectroscopy, as well as thermochemical reactions.

Quantum mechanical out-of-time-ordered-correlators for the anharmonic (quartic) oscillator

Out-of-time-ordered correlators (OTOCs) have been suggested as a means to study quantum chaotic behavior in various systems. In this work, I calculate OTOCs for the quantum mechanical anharmonic oscillator with quartic potential, which is classically integrable and has a Poisson-like energy-level distribution. For low temperature, OTOCs are periodic in time, similar to results for the harmonic oscillator and the particle in a box. For high temperature, OTOCs exhibit a rapid (but power-like) rise at early times, followed by saturation consistent with 2〈x2〉T〈p2〉T at late times. At high temperature, the spectral form factor decreases at early times, bounces back and then reaches a plateau with strong fluctuations.

A quantum-mechanical anharmonic oscillator with a most interesting spectrum

We revisit the problem posed by an anharmonic oscillator with a potential given by a polynomial function of the coordinate of degree six that depends on a parameter λ. The ground state can be obtained exactly and its energy E0=1 is independent of λ. This solution is valid only for λ>0 because the eigenfunction is not square integrable otherwise. Here we show that the perturbation series for the expectation values are Padé and Borel–Padé summable for λ>0. When λ<0 the spectrum exhibits an infinite number of avoided crossings at each of which the eigenfunctions undergo dramatic changes in their spatial distribution that we analyse by means of the expectation values 〈x2〉.

MonteCarlo Study of Real Time Dynamics on the Lattice.

MonteCarlo studies involving real time dynamics are severely restricted by the sign problem that emerges from a highly oscillatory phase of the path integral. In this Letter, we present a new method to compute real time quantities on the lattice using the Schwinger-Keldysh formalism via MonteCarlo simulations. The key idea is to deform the path integration domain to a complex manifold where the phase oscillations are mild and the sign problem is manageable. We use the previously introduced "contraction algorithm" to create a Markov chain on this alternative manifold. We substantiate our approach by analyzing the quantum mechanical anharmonic oscillator. Our results are in agreement with the exact ones obtained by diagonalization of the Hamiltonian. The method we introduce is generic and, in principle, applicable to quantum field theory albeit very slow. We discuss some possible improvements that should speed up the algorithm.

On the efficient numerical solution of lattice systems with low-order couplings

We apply the Quasi Monte Carlo (QMC) and recursive numerical integration methods to evaluate the Euclidean, discretized time path-integral for the quantum mechanical anharmonic oscillator and a topological quantum mechanical rotor model. For the anharmonic oscillator both methods outperform standard Markov Chain Monte Carlo methods and show a significantly improved error scaling. For the quantum mechanical rotor we could, however, not find a successful way employing QMC. On the other hand, the recursive numerical integration method works extremely well for this model and shows an at least exponentially fast error scaling.

A Brief Overview of Hard-Thermal-Loop Perturbation Theory

The poor convergence of quantum field theory at finite temperature has been one of the main obstacles in the practical applications of thermal QCD for decades. Here we briefly review the progress of hard-thermal-loop perturbation theory (HTLpt) in reorganizing the perturbative expansion in order to improve the convergence. The quantum mechanical anharmonic oscillator is used as a simple example to show the breakdown of weak-coupling expansion, and variational perturbation theory is introduced as an effective resummation scheme for divergent weak-coupling expansions. We discuss HTLpt thermodynamic calculations for QED, pure-glue QCD, and QCD with Nf = 3 up to three-loop order. The results suggest that HTLpt provides a systematic framework that can be used to calculate both static and dynamic quantities for temperatures relevant at LHC.

Anharmonic Oscillator and Double-Well Potential: Approximating Eigenfunctions

A simple uniform approximation of the logarithmic derivative of the ground state eigenfunction for both the quantum-mechanical anharmonic oscillator and the double-well potential given by $V= m^2 x^2+g x^4$ at arbitrary $g \geq 0$ for $m^2>0$ and $m^2<0$, respectively, is presented. It is shown that if this approximation is taken as unperturbed problem it leads to an extremely fast convergent perturbation theory.

Is there any unique frequency operator for quantum-mechanical anharmonic oscillators

We discuss some recently proposed methods that yield operator solutions free from secular terms for the quantum-mechanical anharmonic oscillators. The frequency operators obtained in those methods are not same. In the present work we show that the apparent differences between the frequency operators derived by different approaches are due to the different ordering of noncommuting observables. Here we derive some relations between the existing frequency operators. In some cases we generalize results from quartic to higher anharmonicities.

Physicochemical model of hypersonic rarefied gas flow past bodies

The effect of the internal molecular degrees of freedom on the flow field and heat transfer in hypersonic rarefied gas flow past a cylinder or sphere is investigated using the direct statistical simulation (Monte-Carlo) method. The variable-diameter rough spherical molecule model (VRS-model) is generalized to include the case of energy exchange between the translational and vibrational degrees of freedom. The interaction between diatomic molecules with allowance for vibrational degrees of freedom is simulated as the interaction of classical or quantum-mechanical harmonic and anharmonic oscillators in the external force approximation. A model of the dissociation of a diatomic gas is proposed.

Time-dependent quantum-mechanical calculation of ground and excited states of anharmonic and double-well oscillators

Ground- and excited-state energies as well as probability densities of quantum-mechanical anharmonic and double-well oscillators are obtained by numerically solving the time-dependent Schr\"odinger equation (TDSE). The methodology employed is exact in principle and is based on solving the TDSE in imaginary time, coupled with the minimization of energy expectation value. The present method works very well for deep wells and for double-well oscillators with pseudodegenerate energy levels where time-independent methods experience difficulties. The dependence of energy on anharmonicity parameter and the splitting of pseudodegeneracy by an asymmetry parameter as well as probability-density plots are discussed. Several potentials, especially those with pseudodegenerate energy levels, are reported here.

Perturbation theory free from secular terms for the equations of motion of anharmonic oscillators

We develop a time–independent perturbation theory in operator form for the equations of motion of classical and quantum-mechanical anharmonic oscillators. The method focuses on the frequency of the motion in the former case and in a frequency operator in the latter, producing a power series in a small coupling constant. The resulting expressions are free from secular instabilities that commonly appear in perturbation treatments of nonlinear problems.

The finite temperature real-time ℏ 2 corrections in quantum mechanics

We study non-perturbative real-time correlation functions at finite temperature. In order to see whether the classical term gives a good approximation in the high temperature limit T ⪢ ℏω, we consider the first ℏ 2 quantum corrections. We find that for the simplest non-trivial case, the quantum mechanical anharmonic oscillator, the classical result is reliable only for moderately large times: after some time t ∗ the classical approximation breaks down even at high temperatures. Moreover, the result for the first quantum corrections cannot, in general, be reproduced by modifying the parameters of the classical theory.

Perturbative expansion around the Gaussian effective action: The background field method

We develop a systematic method of the perturbative expansion around the Gaussian effective action based on the background field method. We show, by applying the method to the quantum mechanical anharmonic oscillator problem, that even the first non-trivial correction terms greatly improve the Gaussian approximation.

The quartic anharmonic oscillator and its associated nonconstant magnetic field

Quantum mechanical anharmonic oscillators and Hamiltonians for particles in external magnetic fields are related to representations of nilpotent groups. Using this connection the eigenfunctions of the quartic anharmonic oscillator with potential Vα(x)=(α+(x2/2))2 can be used to determine the eigenfunctions of a charged particle in a nonconstant magnetic field, of the form Bz=β2+β3x. The quartic anharmonic oscillator eigenvalues for low-lying states are obtained numerically and a function which interpolates between α≪0 (a double harmonic oscillator) and α≫0 (a harmonic oscillator) is shown to give a good fit to the numerical data. Approximate expressions for the quartic anharmonic oscillator eigenfunctions are then used to get the eigenfunctions for the magnetic field Hamiltonian.

Unified description of the soft and the relaxational mode in the dielectric response of ferroelectrics.

We present a quantum-mechanical calculation of the temperature dependence of the dielectric susceptibility of displacive, order-disorder, and intermediate ferroelectrics. The ferroelectric is modeled as a lattice of coupled quantum-mechanical anharmonic oscillators. The coupling of the oscillators is described in a mean-field approximation. For all ferroelectrics the dielectric response consists of two contributions. The first contribution is formed by the response of the vibrational transitions. This response is calculated using a new equation for the dielectric susceptibility of low-frequency vibrational transitions. The second contribution to the response results from vibrational population transfer between levels with different mean displacements. The coupling of the two dielectric responses via the mean field leads to a rise of a strong low-frequency dielectric response close to the phase-transition temperature.

Convergence of Scaled Delta Expansion: Anharmonic Oscillator

We prove that the linear delta expansion for energy eigenvalues of the quantum mechanical anharmonic oscillator converges to the exact answer if the order dependent trial frequency Ω is chosen to scale with the order as Ω = CN γ; 1/3 < γ < 1/2, C > 0 as N → ∞. It converges also for γ = 1/3, if C ≥ α c g 1/3, α c ≃ 0.570875, where g is the coupling constant in front of the operator q 4/4. The extreme case with γ = 1/3, C = γ c g 1/3 corresponds to the choice discussed earlier by Seznec and Zinn-Justin and, more recently, by Duncan and Jones.

VARIATIONAL PERTURBATION THEORY

A nonperturbative method — variational perturbation theory (VPT) — is discussed. A quantity we are interested in is represented by a series, a finite number of terms of which not only describe the region of small coupling constant but reproduce well the strong coupling limit. The method is formulated only in terms of the Gaussian quadratures, and diagrams of the conventional perturbation theory are used. Its efficiency is demonstrated for the quantum-mechanical anharmonic oscillator. The properties of convergence are studied for series in VPT for the [Formula: see text] model. It is shown that it is possible to choose variational additions such that they lead to convergent series for any values of the coupling constant. Upper and lower estimates for the quantities under investigation are considered. The nonperturbative Gaussian effective potential is derived from a more general approach, VPT. Various versions of the variational procedure are explored and the preference for the anharmonic variational procedure in view of convergence of the obtained series is argued. We investigate the renormalization procedure in the φ4 model in VPT. The nonperturbative β function is derived in the framework of the proposed approach. The obtained result is in agreement with four-loop approximation and has the asymptotic behavior as g3/2 for a large coupling constant. We construct the VPT series for Yang-Mills theory and study its convergence properties. We introduce coupling to spinor fields and demonstrate that they do not influence the VPT series convergence properties.

Optimized perturbation methods for the free energy of the anharmonic oscillator

Two possible applications of the optimized expansion for the free energy of the quantum-mechanical anharmonic oscillator are discussed. The first method is for the finite temperature effective potential; the second for the classical effective potential. The results of both methods show a quick convergence and agree well with the exact free energy in the whole range of temperature.