Motivated by studies of ultimate speed of computers, we examine the question of minimum time of orthogonalization in a simple anharmonic oscillator and find an upper bound on the rate of computations. Furthermore, we investigate the growth rate of complexity of operation when the system undergoes a definite perturbation. At the phase space of the parameters, by numerical analysis, we find the critical point where beyond that the rate of complexity changes its behavior.
We use a power-series expansion to calculate the eigenvalues of anharmonic oscillators bounded by two infinite walls. We show that for large finite values of the separation of the walls, the calculated eigenvalues are of the same high accuracy as the values recently obtained for the unbounded case by the inner-product quantization method. We also apply our method to the Morse potential. The eigenvalues obtained in this case are in excellent agreement with the exact values for the unbounded Morse potential.
We outline a remarkably efficient method for generating solutions to quantum anharmonic oscillators with an x^{2M} potential. We solve the Schroedinger equation in terms of a free parameter which is then tuned to give the correct boundary condition by generating a power series expansion of the wavefunction in x and applying a modified Borel resummation technique to obtain the large x behaviour. The process allows us to calculate energy eigenvalues to an arbitrary level of accuracy. High degrees of precision are achieved even with modest computing power. Our technique extends to all levels of e…
Classical oscillators of sextic and octic anharmonicities are solved analytically up to the linear power of λ(Anharmonic Constant) by using Taylor series method. These solutions exhibit the presence of secular terms which are summed up for all orders. The frequency shifts of the oscillators for small anharmonic constants are obtained. It is found that the calculated shifts agree nicely with the available results to-date. The solutions for classical anharmonic oscillators are used to obtain the solutions corresponding to quantum anharmonic oscillators by imposing fundamental commutation relatio…
In this work, we obtained energy levels of one dimensional quartic anharmonic oscillator by using neural network system. Quartic anharmonic oscillator is a very important tool in quantum mechanics and also in quantum field theory. Our results are in good agreement in high accuracy with the reference studies.
A simple method for the calculation of higher orders of the logarithmic perturbation theory for bound states of the spherical anharmonic oscillator is developed. The structure of the perturbation series for energy eigenvalues of the sextic doubly anharmonic oscillator is investigated. The recursion technique for deriving renormalized perturbation expansions is offered.
The treatment of anharmonic oscillators (including double-wells) by instanton methods is wellknown. The alternative differential equation method is not so wellknown. Here we reformulate the latter completely parallel to the strong coupling case of the cosine potential and Mathieu equation for which extensive literature and monographs exist.
The energy gap between the first excited state and the ground state is calculated for the quantized anharmonic oscillator in the framework of the functional renormalization group method. The compactly supported smooth regulator is used which includes various types of regulators as limiting cases. It was found that the value of the energy gap depends on the regulator parameters. We argue that the optimization based on the disappearance of the false, broken symmetric phase of the model leads to the Litim's regulator. The least sensitivity on the regulator parameters leads however to an IR regula…
The machinery of computing vacuum expectation values of a time-ordered sequence of position operators of the simple harmonic oscillator is already well established. It rests on a Wick theorem, which enables one to decompose such a quantity in terms of products of \emph{pairwise contractions}, which are vacuum expectation values of a time-ordered sequence of position operators taken two at a time. This result naturally leads to a diagrammatic approach of computing such correlators, and is already well known in the form of Feynman diagrams. We generalise this setup to encompass expectation value…
Functional renormalization group methods formulated in the real-time formalism are applied to the $O(N)$ symmetric quantum anharmonic oscillator, considered as a $0+1$ dimensional quantum field-theoric model, in the next-to-leading order of the gradient expansion of the one- and two-particle irreducible effective action. The infrared scaling laws and the sensitivity-matrix analysis show the existence of only a single, symmetric phase. The field-independent term of the wavefunction renormalization turned out to be negligible, but its field-dependent piece is noticeable. It is shown that the inf…
We introduce a functional perturbative method for treating weakly nonlinear systems coupled with a quantum field bath. We demonstrate using this method to obtain the covariance matrix elements and the correlation functions of a quantum anharmonic oscillator interacting with a heat bath. We identify a fluctuation-dissipation relation based on the nonequilibrium dynamics of this nonlinear open quantum system. To establish its connection with dynamical equilibration, we further examine the energy flows between the anharmonic oscillator and the bath field. The vanishing of the net flow {is an indi…
We will describe how a new, quite simple, but highly effective algorithm, together with the asymptotically fast FFT-based high-precision number multiplication of Mathematica 4 can calculate the ground state of the x^4 anharmonic oscillator to the new record of more than 1000 digits.
Using an ansatz motivated by the classical form of $e^{iϕ}$, where $ϕ$ is the angle variable, we construct operators which satisfy the commutation relations of the creation-annihilation operators for the anharmonic oscillator. The matrix elements of these operators can be expressed in terms of {\it entire} functions in the position complex plane. These functions provide solutions of the Ricatti equation associated with the time-independent Schrödinger equation. We relate the normalizability of the eigenstates to the global properties of the flows of this equation. These exact results yield app…
The two-dimensional hydrogen with a linear potential in a magnetic field is solved by two different methods. Furthermore the connection between the model and an anharmonic oscillator had been investigated by methods of KS transformation.
The energy eigenvalues of the anharmonic oscillator characterized by the cubic potential for various eigenstates are determined within the framework of the hypervirial-Padé summation method. For this purpose the E[3,3] and E[3,4] Padé approximants are formed to the energy perturbation series and given the energy eigenvalues up to fourth order in terms of the anharmonicity parameter $λ$.
The paper introduces a simple quantum model to calculate in a general way allowed frequencies and energy levels of the anharmonic oscillator. The theoretical basis of the approach has been introduced in two early papers aimed to infer the properties of quantum systems exploiting the uncertainty principle only. For clarity the anharmonic oscillator is described having in mind the lattice oscillations of atoms/ions, yet quantum formalism of the model and approach have general character and can be extended to any oscillating system. The results show that the harmonic energy levels split into a co…
A new recursion procedure for deriving renormalized perturbation expansions for the one-dimensional anharmonic oscillator is offered. Based upon the $\hbar$-expansions and suitable quantization conditions, the recursion formulae obtained have the same simple form both for ground and excited states and can be easily applied to any renormalization scheme. As an example, the renormalized expansions for the sextic anharmonic oscillator are considered.
We examine a generalized script{PT}-Symmetric quartic anharmonic oscillator model to determine the various physical variables perturbatively in powers of a small quantity ε. We make use of the Bender-Dunne operator basis elements and exploit the properties of the totally symmetric operator T_{m,n}.
Anharmonic oscillators with the sextic and decatic potentials are studied employing the refinable interpolating scale functions. This method yields highly accurate values of both energy eigenvalues and eigenfunctions for the sextic and decatic oscillator without constraining the potential parameters. Convergence of the solutions in the present method is noticed to be very fast.
The concept of a generalized nonanalytic expansion which involves nonanalytic combinations of exponentials, logarithms and powers of a coupling is introduced and its use illustrated in various areas of physics. Dispersion relations for the resonance energies of odd anharmonic oscillators are discussed, and higher-order formulas are presented for cubic and quartic potentials.
The explicit semiclassical treatment of the logarithmic perturbation theory for the bound-state problem for the spherical anharmonic oscillator is developed. Based upon the $\hbar$-expansions and suitable quantization conditions a new procedure for deriving perturbation expansions is offered. Avoiding disadvantages of the standard approach, new handy recursion formulae with the same simple form both for ground and excited states have been obtained. As an example, the perturbation expansions for the energy eigenvalues of the quartic anharmonic oscillator are considered.
We use the Liouville-von Neumann (LvN) approach to study the dynamics and the adiabaticity of a time-dependent driven anharmonic oscillator as an eample of non-equilibrium quantum dynamics. We show that the adiabaticity is minimally broken in the sense that a gaussian wave-packet at the past infinity evolves to coherent states, however slowly the potential changes, its coherence factor is order of the coupling. We also show that the dynamics are governed by an equation of motion similar to the Kepler motion which satisfies the angular momentum conservation.
Conventional weak-coupling perturbation theory suffers from problems that arise from resonant coupling of successive orders in the perturbation series. Multiple-scale perturbation theory avoids such problems by implicitly performing an infinite reordering and resummation of the conventional perturbation series. Multiple-scale analysis provides a good description of the classical anharmonic oscillator. Here, it is extended to study the Heisenberg operator equations of motion for the quantum anharmonic oscillator. The analysis yields a system of nonlinear operator differential equations, which i…
A mixed supersymmetric-algebraic approach to construction of the minimum uncertainty coherent states of anharmonic oscillators is presented. It permits generating not only the well-known coherent states of the harmonic and Morse oscillators but also the so far unknown coherent states of the Wei Hua, Kratzer-Fues and generalized Morse and Kratzer-Fues oscillators. The method can be applied also to generate superpotentials indispensable for deriving the Schrödinger equation in the supersymmetric form amenable to direct solution in the SUSYQM scheme.
For a system of coupled anharmonic oscillators we compare the convergence rate of the variational collocation approach presented recently by Amore and Fernandez (2010 Phys.Scr.81 045011) with the one obtained using the optimized Rayleigh-Ritz (RR) method. The monotonic convergence of the RR method allows us to obtain more accurate results at a lower computational cost.
The determination of the eigenenergies of a quantum anharmonic oscillator consists merely in finding the zeros of a function of the energy, namely the Wronskian of two solutions of the Schroedinger equation which are regular respectively at the origin and at infinity. We show in this paper how to evaluate that Wronskian exactly, except for numerical rounding errors. The procedure is illustrated by application to the gx2+x2N (N a positive integer) oscillator.
As an application of a recently developed variational perturbation theory we find the first 22 terms of the convergent strong-coupling series expansion for the ground state energy of the quartic anharmonic oscillator.
It is proved that the divergent Rayleigh-Schrodinger perturbation expansions for the eigenvalues of any odd anharmonic oscillator are Borel summable in the distributional sense to the resonances naturally associated with the system.
This paper illustrates the application of group theory to a quantum-mechanical three-dimensional quartic anharmonic oscillator with $O_{h}$ symmetry. It is shown that group theory predicts the degeneracy of the energy levels and facilitates the application of perturbation theory and the Rayleigh-Ritz variational method as well as the interpretation of the results in terms of the symmetry of the solutions . We show how to obtain suitable symmetry-adapted basis sets.
The renormalization group method is applied for obtaining the asymptotic form of the wave function of the quantum anharmonic oscillator by resumming the perturbation series. It is shown that the resumed series is the cumulant of the naive perturbation series. Working out up to the sixth order and performing a further resummation proposed by Bender and Bettencourt, we find that the agreement with the WKB result becomes worse in the higher orders than the fourth at which the agreement is the best.
The aim of this paper is twofold. First of all, we study the behaviour of the lowest eigenvalues of the quantum anharmonic oscillator under influence of an external field. We try to understand this behaviour using perturbation theory and compare the results with numerical calculations. This brings us to the second aim of selecting the best method to carry out the numerical calculations accurately.
We consider an elastic medium with the distortion of a circular curve into a vertical spiral, and investigate the influence of this topological defect on the doubly anharmonic oscillator. We show that the Schrödinger equation for the doubly anharmonic oscillator in the presence of this linear topological defect can be solved analytically. We also obtain the exact expressions for the permitted energies of the ground state of the doubly anharmonic oscillator, and show that the topology of the screw dislocation modifies the spectrum of energy of the doubly anharmonic oscillator.
Utilizing an appropriate ansatz to the wave function, we reproduce the exact bound-state solutions of the radial Schrodinger equation to various exactly solvable sextic anharmonic oscillator and confining perturbed Coulomb models in $D$-dimensions. We show that the perturbed Coulomb problem with eigenvalue $E$ can be transformed to a sextic anharmonic oscillator problem with eigenvalue E. We also check the explicit relevance of these two related problems in higher-space dimensions. It is shown that exact solutions of these potentials exist when their coupling parameters with $k=D+2\ell $ appea…
A new pseudoperturbative (artificial in nature) methodical proposal [15] is used to solve for Schrodinger equation with a class of phenomenologically useful and methodically challenging anharmonice oscillator potentials V(q)=α_o q^2 + αq^4. The effect of the [4,5] Pade' approximant on the leading eigenenergy term is studied. Comparison with results from numerical (exact) and several eligible (approximation) methods is made.
In our previous paper I (del Valle--Turbiner, Int. J. Mod. Phys. A34, 1950143, 2019) it was developed the formalism to study the general $D$-dimensional radial anharmonic oscillator with potential $V(r)= \frac{1}{g^2}\,\hat{V}(gr)$. It was based on the Perturbation Theory (PT) in powers of $g$ (weak coupling regime) and in inverse, fractional powers of $g$ (strong coupling regime) in both $r$-space and in $(gr)$-space, respectively. As the result it was introduced - the Approximant - a locally-accurate uniform compact approximation of a wave function. If taken as a trial function in variationa…
Two quantum quartic anharmonic many-body oscillators are introduced. One of them is the celebrated Calogero model (rational $A_n$ model) modified by quartic anharmonic two-body interactions which support the same symmetry as the Calogero model. Another model is the three-body Wolfes model (rational $G_2$ model) with quartic anharmonic interaction added which has the same symmetry as the Wolfes model. Both models are studied in the framework of algebraic perturbation theory and by the variational method.
We have applied a collocation approach to obtain the numerical solution to the stationary Schrödinger equation for systems of coupled oscillators. The dependence of the discretized Hamiltonian on scale and angle parameters is exploited to obtain optimal convergence to the exact results. A careful comparison with results taken from the literature is performed, showing the advantages of the present approach.
A mapping is obtained relating radial screened Coulomb systems with low screening parameters to radial anharmonic oscillators in N-dimensional space. Using the formalism of supersymmetric quantum mechanics, it is shown that exact solutions of these potentials exist when the parameters satisfy certain constraints.
We find a one-parameter Gaussian state for an anharmonic oscillator with quadratic and quartic terms, which depends on the energy expectation value. For the weak coupling constant, the Gaussian state is a squeezed state of the vacuum state. However, for the strong coupling constant, the Gaussian state represents a different kind of condensation of bosonic particles through a nonlinear Bogoliubov transformation of the vacuum state.
To lowest order of perturbation theory we show that an equivalence can be established between a $\cal PT$-symmetric generalized quartic anharmonic oscillator model and a Hermitian position-dependent mass Hamiltonian $h$. An important feature of $h$ is that it reveals a domain of couplings where the quartic potential could be attractive, vanishing or repulsive. We also determine the associated physical quantities.
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 $2\langle x^2\rangle_T \langle p^2\rangle_T$…
We study the synchronization of a van der Pol self-oscillator with Kerr anharmonicity to an external drive. We demonstrate that the anharmonic, discrete energy spectrum of the quantum oscillator leads to multiple resonances in both phase locking and frequency entrainment not present in the corresponding classical system. Strong driving close to these resonances leads to nonclassical steady-state Wigner distributions. Experimental realizations of these genuine quantum signatures can be implemented with current technology.
The application of the optimized expansion for the quantum-mechanical propagation in the anharmonic potential $λx^4$ is discussed for real and imaginary time. The first order results in the imaginary time formalism provide approximations to the free energy and particle density which agree well with the exact results in the whole range of temperatures.
We have developed a simple method to solve anharmonic oscillators equations. The idea of our method is mainly based on the partitioning of the potential curve into (n+1) small intervals, solving the Schrödinger equation in each subintervals, and writing the continuity conditions of each solutions at each subintervals boundary, which leads to the energy quantification condition, so to the energy levels, and finally, one can calculate the exact wave function associated to each energy level. Our method has been applied to three examples: the well-known square well, the bounded and unbounded harmo…
We prove a powerful scaling property for the extremality condition in the recently developed variational perturbation theory which converts divergent perturbation expansions into exponentially fast convergent ones. The proof is given for the energy eigenvalues of an anharmonic oscillator with an arbitrary $x^p$-potential. The scaling property greatly increases the accuracy of the results.
We analyse the one dimensional quartic oscillator using the effective operator methodology of Lee and Suzuki. We reproduce known results for low lying energy eigenvalues.
Unique set of coherent states for the anharmonic oscillator is obtained by requiring i. under the quantum mechanical time evolution a coherent state evolves into another, governed by trajectory in the classical phase space (of a related hamiltonian); ii. the resolution of identity involves exactly the classical phase space measure.The rules are invariant under unitary transformations of the quantum theory and canonical transformations of the classical theory. The states are almost, but not quite, minimal uncertainty wave packets. The construction can be generalized to quantum versions of integ…
In this contribution to Peter Suranyi Festschrift, we study the Halliday-Suranyi perturbation method for calculating the energy eigenvalues of the quartic anharmonic oscillator.
In this paper we analyze a recent application of perturbation theory by the moment method to a family of two-dimensional anharmonic oscillators. By means of straightforward unitary transformations we show that two of the models studied by the authors are separable. Other is unbounded from below and therefore cannot be successfully treated by perturbation theory unless a complex harmonic frequency is introduced in the renormalization process. We calculate the lowest resonance by means of complex-coordinate rotation and compare its real part with the eigenvalue estimated by the authors. A pair o…
We obtain sufficiently accurate eigenvalues and eigenfunctions for the anharmonic oscillator with potential $V(x,y)=x^{2}y^{2}$ by means of three different methods. Our results strongly suggest that the spectrum of this oscillator is discrete in agreement with early rigorous mathematical proofs and against a recent statement that cast doubts about it.
We construct a W^*-dynamical system describing the dynamics of a class of anharmonic quantum oscillator lattice systems in the thermodynamic limit. Our approach is based on recently proved Lieb-Robinson bounds for such systems on finite lattices.
The dynamics of a quantum anharmonic oscillator whose position is monitored by a Bose-Einstein condensate (BEC) trapped in a symmetric double well potential is studied. The (non-exponential) decoherence induced on the oscillator by the measuring device is analysed. A detailed quasiclassical and quantum analysis is presented. In the first case, for an arbitrary initial coherent state, two different decoherence regimes are observed: An initial Gaussian decay followed by a power law decay for longer times. The characteristic time scales of both regimes are reported. Analytical approximated expres…
For the model of a linearly driven quantum anharmonic oscillator, the role of damping is investigated. We compare the position of the stable points in phase space obtained from a classical analysis to the result of a quantum mechanical analysis. The solution of the full master equation shows that the stable points behave qualitatively similar to the classical solution but with small modifications. Both the quantum effects and additional effects of temperature can be described by renormalizing the damping.
We point out a misleading treatment in a recent paper published in this Journal [Eur. Phys. J. C (2018)78:106] regarding solutions for the Schrödinger equation with a anharmonic oscillator potential embedded in the background of a cosmic string mapped into biconfluent Heun equation. This fact jeopardizes the thermodynamical properties calculated in this system.
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.
Dunkl derivative enriches solutions by discussing parity due to its reflection operator. Very recently, one of the authors of this manuscript presented one of the most general forms of Dunkl derivative that depends on three Wigner parameters to have a better tuning. In this manuscript, we employ the latter generalized Dunkl derivative in a relativistic equation to examine two-dimensional harmonic and anharmonic oscillators solutions. We obtain the solutions by Nikiforov-Uvarov and QES methods, respectively. We show that degenerate states can occur according to the Wigner parameter values.
We measure the state dynamics of a tunable anharmonic quantum system, the Josephson phase circuit, under the excitation of a frequency-chirped drive. At small anharmonicity, the state evolves like a wavepacket - a characteristic response in classical oscillators; in this regime we report exponentially enhanced lifetimes of highly excited states, held by the drive. At large anharmonicity, we observe sharp steps, corresponding to the excitation of discrete energy levels. The continuous transition between the two regimes is mapped by measuring the threshold of these two effects.
We solve analytically master equations that describe a cavity filled with a kerr medium, taking into account the dissipation induced by the environment, and parametric down conversion processes. We use superoperator techniques.
