We analyze the behavior of a quantum system described by a one-dimensional asymmetric potential consisting of a step plus a harmonic barrier. We solve the eigenvalue equation by the integral representation method, which allows us to classify the independent solutions as equivalence classes of homotopic paths in the complex plane. We then consider the propagation of a wave packet reflected by the harmonic barrier and obtain an expression for the interaction time as a function of the peak energy. For high energies we recover the classical half-period limit.
We study the time evolution for the quantum harmonic oscillator subjected to a sudden change of frequency. It is based on an approximate analytic solution to the time dependent Ermakov equation for a step function. This approach allows for a continuous treatment that differs from former studies that involve the matching of two time independent solutions at the time when the step occurs.
The type III Hermite $X_m$ exceptional orthogonal polynomial family is generalized to a double-indexed one $X_{m_1,m_2}$ (with $m_1$ even and $m_2$ odd such that $m_2 > m_1$) and the corresponding rational extensions of the harmonic oscillator are constructed by using second-order supersymmetric quantum mechanics. The new polynomials are proved to be expressible in terms of mixed products of Hermite and pseudo-Hermite ones, while some of the associated potentials are linked with rational solutions of the Painlevé IV equation. A novel set of ladder operators for the extended oscillators is also…
An exactly-solvable model of the non-relativistic harmonic oscillator with a position-dependent effective mass is constructed. The model behaves itself as a semi-infinite quantum well of the non-rectangular profile. Such a form of the profile looks like a step-harmonic potential as a consequence of the certain analytical dependence of the effective mass from the position and semiconfinement parameter $a$. Both states of the discrete and continuous spectrum are studied. In the case of the discrete spectrum, wavefunctions of the oscillator model are expressed through the Bessel polynomials. The …
Bound-state solutions of the singular harmonic oscillator and singular Coulomb potentials in arbitrary dimensions are generated in a simple way from the solutions of the one-dimensional generalized Morse potential. The nonsingular harmonic oscillator and nonsingular Coulomb potentials in arbitrary dimensions with their additional accidental degeneracies are obtained as particular cases. Added bonuses from these mappings are the straightforward determination of the critical attractive singular potential, the proper boundary condition on the radial eigenfunction at the origin and the inexistence…
We present a new trajectory formulation of high harmonic generation that treats classically allowed and classically forbidden processes within a single dynamical framework. Complex trajectories orbit the nucleus, producing the stationary Coulomb ground state. When the field is turned on, these complex trajectories continue their motion in the field-dressed Coulomb potential and therefore tunnel ionization, unbound evolution and recollision are described within a single, seamless framework. The new formulation can bring mechanistic understanding to a broad range of strong field physics effects.
In the Madelung-Bohm approach to quantum mechanics, we consider a (time dependent) phase that depends quadratically on position and show that it leads to a Bohm potential that corresponds to a time dependent harmonic oscillator, provided the time dependent term in the phase obeys an Ermakov equation.
The paper presents a new method of geometric solution of a Schrodinger equation by a construction of an equivalent first-order partial differential equation with a bigger number of variables. The equivalent equation shall be restricted to a specific subspace with auxiliary conditions which are obtained from a coherent state transform. The method is applied to the fundamental case of the harmonic oscillator and coherent state transform generated by the minimal nilpotent step three Lie group---the shear group (also known as quartic group in literature). We obtain a geometric solution for an arbi…
In this work, we present the analytical approach to the evaluation of the conditional measure Wiener path integral. We consider the time-dependent model parameters. We find the differential equation for the variable, determining the behavior of the harmonic as well the an-harmonic parts of the oscillator. We present the an-harmonic part of the result in the form of the operator function.
Eigenstates of a particle in a localized and unconfined harmonic potential well are investigated. Effects due to the variation of the potential parameters as well as certain results from asymptotic expansions are discussed.
We study the energy properties of a particle in one dimensional semi-harmonic rectangular wells and barriers. The integration of energies is obtained by solving a simple transcendental equation. Scattering states are shown to include cases in which the impinging particle is 'captured' by the semi-harmonic rectangular potentials. The 'time of capture' is connected with the dwell time of the scattered wave. Using the particle absorption method, it is shown that the dwell time $τ^a_D$ coincides with the phase time $τ_W$ of Eisenbud and Wigner, calculated as the energy derivative of the reflected …
We derive a one-step extension of the well known Swanson oscillator that describes a specific type of pseudo-Hermitian quadratic Hamiltonian connected to an extended harmonic oscillator model. Our analysis is based on the use of the techniques of supersymmetric quantum mechanics and address various representations of the ladder operators starting from a seed solution of the harmonic oscillator given in terms of a pseudo-Hermite polynomial. The role of the resulting chain of Hamiltonians related via similarity transformation is then exploited. In the second part we write down a two dimensional …
Dynamical phases are obtained for a quantum thermal engine, whose working medium is a single harmonic oscillator. The dynamics of this engine is obtained by using four steps where in two steps the time dependent frequency is changing. In the other two steps, the thermal engine is coupled alternatively to hot and cold heat baths. Similar dynamical phases are obtained in a quantum thermal engine whose working medium is spin 1/2 system. The role of times durations of such steps in the quantum engines for getting maximal efficiency is analyzed. The dynamic of charge pumping in a quantum dot couple…
We show how the recently discovered solvable rational extensions of Harmonic Oscillator and Morse potentials can be constructed in a direct and systematic way, without the need of supersymmetry, shape invariance, Darboux-Crum and Darboux-Bäcklund transformations
In this article, the problem of the charged harmonic plus an inverse harmonic oscillator with time-dependent mass and frequency in a time-dependent electromagnetic field is investigated. It is reduced to the problem of the inverse harmonic oscillator with time-independent parameters and the exact wave function is obtained.
The equilibrium properties of an open harmonic oscillator are considered in three steps: First the creation and destruction operators are generalized for open dynamics and the creation operator is used to construct coherent states. The second step consists of the introduction of the Heisenberg representation where the dynamical decoherence can be identified without making reference to the state of the system. Finally it is pointed out that the quantum fluctuations generate non-continuous limit for infinitesimal system-environment interactions and at the border of the under- and over-damped osc…
We apply the Schrödinger factorization to construct the ladder operators for hydrogen atom, Mie-type potential, harmonic oscillator and pseudo-harmonic oscillator in arbitrary dimensions. By generalizing these operators we show that the dynamical algebra for these problems is the $su(1,1)$ Lie algebra.
Supersymmetry transformations of first and second order are used to generate Hamiltonians with known spectra departing from the harmonic oscillator with an infinite potential barrier. It is studied also the way in which the eigenfunctions of the initial Hamiltonian are transformed. The first and certain second order supersymmetric partners of the initial Hamiltonian possess third-order differential ladder operators. Since systems with this kind of operators are linked with the Painlevé IV equation, several solutions of this non-linear second-order differential equation will be simply found.
Applying the general framework for local zeta regularization proposed in Part I of this series of papers, we renormalize the vacuum expectation value of the stress-energy tensor (and of the total energy) for a scalar field in presence of an external harmonic potential.
The unitary operator which transforms a harmonic oscillator system of time-dependent frequency into that of a simple harmonic oscillator of different time-scale is found, with and without an inverse-square potential. It is shown that for both cases, this operator can be used in finding complete sets of wave functions of a generalized harmonic oscillator system from the well-known sets of the simple harmonic oscillator. Exact invariants of the time-dependent systems can also be obtained from the constant Hamiltonians of unit mass and frequency by making use of this unitary transformation. The g…
We analyze the existences, properties, and stabilities of the self-localized solutions of the nonlinear quantum harmonic oscillator (NQHO) using spectral renormalization method (SRM). We show that self-localized single, dual and triple soliton solutions of the NQHO do exists, however, only single and dual soliton solutions satisfy the necessary Vakhitov and Kolokolov slope condition, therefore triple soliton solution is found to be unstable, at least for the parameter ranges considered. Additionally, we investigate the stability characteristics of the single and dual soliton solutions using a …
The Levi-Civita transformation is applied in the two-dimensional (2D) Dirac and Klein-Gordon (KG) equations with equal external scalar and vector potentials. The Coulomb and harmonic oscillator problems are connected via the Levi-Civita transformation. These connections lead to an approach to solve the Coulomb problems using the results of the harmonic oscillator potential in the above-mentioned relativistic systems.
Schrödinger equation for two center Coulomb plus harmonic oscillator potential is solved by the method of ethalon equation at large intercenter separations. Asymptotical expansions for energy term and wave function are obtained in the analytical form.
We investigate a single laser driven atom trapped in a non-harmonic potential. We present the performance of ground-state laser cooling and Doppler cooling and the signatures of the center-of-mass motion in the power spectrum of the scattered light. In order to illustrate the results we provide two explicit examples for the confining potential: the infinite square well and the Morse potential.
We derive the energy levels associated with the even-parity wave functions of the harmonic oscillator with an additional delta-function potential at the origin. Our results bring to the attention of students a non-trivial and analytical example of a modification of the usual harmonic oscillator potential, with emphasis on the modification of the boundary conditions at the origin. This problem calls the attention of the students to an inaccurate statement in quantum mechanics textbooks often found in the context of solution of the harmonic oscillator problem.
This work sets the exact equations for the quasiclassical response function and susceptibility of a Brownian particle immersed in a bath of quantum harmonic oscillators driving by nonlinear harmonic potentials. A delta force perturbation gives rise to a response whose susceptibility is the combination of a linear term, own of the harmonic oscillator, plus a nonlinear one involving an integral \textcolor{black}{equation. It is provided a recursion method to find its solutions based on functional equations in the Banach space.} The ODE for the response function is a highly nonlinear damped non-a…
Assuming the validity of grand canonical statistics, we study the properties of a spin-polarized Fermi gas in harmonic traps. Universal forms of Fermi temperature $T_F$, internal energy $U$ and the specific heat per particle of the trapped Fermi gas are calculated as a {\it function} of particle number, and the results compared with those of infinite number particles.
We describe a procedure to systematically improve direct diagonalization results for few-particle systems trapped in one-dimensional harmonic potentials interacting by contact interactions. We start from the two-body problem to define a renormalization method for the interparticle interactions. The procedure is benchmarked with state-of-the-art numerical results for three and four symmetric fermions.
In this study, we have derived close form of the Casimir force for the non-interacting ideal Bose gas between two slabs in harmonic-optical lattice potential by using Ketterle and van Druten approximation. We find that Bose-Einstein condensation temperature $T_{c}$ is a critical point for different physical behavior of the Casimir force. We have shown that Casimir force of confined Bose gas in the presence of the harmonic-optical potential decays with inversely proportional to $d^{5}$ when $T\leq T_{c}$. However, in the case of $T>T_{c}$, it decays exponentially depends on separation $d$ of th…
We discuss the solutions of the Schroedinger equation for piecewise potentials, given by the harmonic oscillator potential for $\vert x\vert >a$ and an arbitrary function for $\vert x\vert <a$, using elementary methods. The study of this problem sheds light on usual errors when discussing the asymptotic behavior of the eigenfunctions of the quantum harmonic oscillator and can also be used for the analysis of the eigenfunctions of the hydrogen atom. We present explicit results for the energy levels of a potential of this class, used to model the confinement of electrons in nanostructures.
Using the technique of tridiagonal representation approach; for the first time, we extend this method to study quantum systems with literally perturbed Hamiltonians. Specifically, we consider a quantum system in a 3D spherical oscillator symmetric potential function coupled with an electric and a magnetic field. We obtain the energy spectrum and wavefunction in the two cases.
The quantum dynamics of an atom with a magnetic quadrupole moment that interacts with an external field subject to a harmonic and a linear confining potentials is investigated. It is shown that the interaction between the magnetic quadrupole moment and an electric field gives rise to an analogue of the Coulomb potential and, by confining this atom to a harmonic and a linear confining potentials, a quantum effect characterized by the dependence of the angular frequency on the quantum numbers of the system is obtained. In particular, it is shown that the possible values of the angular frequency …
Experimental results from literature show equidistant energy levels in thin Bi films on surfaces, suggesting a harmonic oscillator description. Yet this conclusion is by no means imperative, especially considering that any measurement only yields energy levels in a finite range and with a nonzero uncertainty. Within this study we review isospectral potentials from the literature and investigate the applicability of the harmonic oscillator hypothesis to recent measurements. First, we describe experimental results from literature by a harmonic oscillator model, obtaining a realistic size and dep…
We have studied the quantum dissipative problem of a Gaussian wave packet under the influence of a harmonic potential. A phenomenological approach to dissipation is adopted in the light of the well-known model in which the environment is composed of a bath of non-interacting harmonic oscillators. As one of the effects of the coupling to the bath is the evolution of an initially pure wave packet into a statistical mixture, we estimate the characteristic time elapsed for this to occur for different regimes of temperature, damping, and also different initial states.
We here explore Majorana Fermion states in an s-wave superfluid of cold atoms in the presence of spin-orbital coupling and an additional harmonic potential. The superfluid boundary is induced by a harmonic trap. Two locally separated Majorana Fermion states are revealed numerically based on the self-consistent Bogoliubov-de Gennes equations. The local density of states are calculated, through which the signatures of Majorana excitations may be indicated experimentally.
We investigate Bose-Einstein condensation of noninteracting gases in a harmonic trap with an off-center dimple potential. We specifically consider the case of a tight and deep dimple potential which is modelled by a point interaction. This point interaction is represented by a Dirac delta function. The atomic density, chemical potential, critical temperature and condensate fraction, the role of the relative depth and the position of the dimple potential are analyzed by performing numerical calculations.
Quantum mechanical models and practical calculations often rely on some exactly solvable models like the Coulomb and the harmonic oscillator potentials. The $D$ dimensional generalized Coulomb potential contains these potentials as limiting cases, thus it establishes a continuous link between the Coulomb and harmonic oscillator potentials in various dimensions. We present results which are necessary for the utilization of this potential as a model and practical reference problem for quantum mechanical calculations. We define a Hilbert space basis, the generalized Coulomb-Sturmian basis, and ca…
This work is mainly based on some theoretical surveys on two dimensional quantum gravitational well, considering harmonic oscillator potential causes an effective plank constant. We find that there is a similarity between two different cases, potential free and oscillator potential in noncommutative phase space in commutative momentum, but in noncommutative, non zero space and momentum The Hamiltonian would a little bit different in energy levels.
The well known relation of Einstein relativistic energy for a free particle is extended to cover the total relativistic energy of a bound particle by calculating the relativistic potential energy. A non dissipative harmonic oscillator (NDHO) is a fundamental bound system. Therefore, the potential energy of an NDHO is analytically extended from the non relativistic to the relativistic regime for the first time. This study is essentially concerned with the relativistic mass, where the Lorentz factor transforms the state of the second-order differential equation of an NDHO from linear into nonlin…
In our previous paper, we have investigated the thermodynamics of the quantum ultra-cold neutron gas trapped in the two-dimensional square-well potential on the Earth's gravitational field. We have pointed out that they have strong dependence on the shape of the trapping potential. In the present paper, we estimate the Fermi energy and the density distribution of the Fermi gas with two-dimensional harmonic oscillator potential, instead of the square-well potential. Then we discuss them in terms of the Fermi energy and the density distribution as compared with both of cases.
We consider, in turn, three systems being acted upon by a regularly pulsed harmonic potential (PHP). These are i) a classical particle, ii) a quantum particle, and iii) a directed line. We contrast the mechanics of the first two systems by parameterizing their bands of stability and periodicity. Interesting differences due to quantum fluctuations are examined in detail. The fluctuations of the directed line are calculated in the two cases of a binding PHP, and an unbinding PHP. In the latter case there is a finite maximum line length for a given potential strength.
Precise control of quantum particles is required for many interesting or novel experiments. Here we consider the task of transporting an atom using an external harmonic potential from one well of an optical lattice to another without motional excitations. To achieve this we apply techniques from Shortcuts to Adiabaticity (STA) enabling fast and robust state manipulation. The process is split up into three independent building blocks; first the atom is loaded into an additional external harmonic trap; this trap is then transported from one lattice site to another and finally the atom is unloade…
