We give an algebraic derivation of the eigenvalues of energy of a quantum harmonic oscillator on the surface of constant curvature, i.e. on the sphere or on the hyperbolic plane. We use the method proposed by Daskaloyannis for fixing the energy eigenvalues of two-dimensional (2D) quadratically superintegrable systems by assuming that they are determined by the existence of finite-dimensional representation of the polynomial algebra of the motion integral operators. The tool for realizing representations is the deformed parafermionic oscillator. The eigenvalues of energy are calculated and the result derived by us algebraically agrees with the known energy eigenvalues calculated by classical analytical methods. This assertion which is the main result of this article is demonstrated by a detailed presentation. We also discuss the qualitative difference of the energy spectra on the sphere and on the hyperbolic plane.
Starting from a total Lagrangian describing an oscillator-bath system, a novel derivation of exact quantum propagator is presented. Having the quantum propagator, the exact density matrix, reduced density matrix of the main oscillator and thermal equilibrium fixed point are obtained. The problem is generalised to the cases where the main oscillator is under the influence of a classical external force. By introducing generalised auxiliary classical fields, the generalised quantum propagator or generating functional of position correlation functions is obtained.
In this note we show that the standard \mbox{Rayleigh-Schrödinger} (RS) perturbation method gives the same result as the hypervirial pertubative method (HPM), for an approximate analytic expression for the energy eigenvalues of the bounded quartic oscillator. This connection between the HPM and the RS method went unnoticed for a long time, apparently because it was not obvious that the resulting polygamma sums to be evaluated in the RS method could, in fact, be expressed in closed form.
This paper presents an application of white noise functional approach to derive the quantum propagator and the evolution of the reduced density matrix for an open quantum system consisting of coupled harmonic oscillators which are coupled to a bath consisting of a multimode harmonic oscillator. It is shown that the full quantum propagator is a product of three individual propagators. These propagators were obtained after two successive transformations of the coordinates for the system, and the coordinates for the coupling between the bath and one of the two harmonic oscillators in the system. …
We present a full algebraic derivation of the wavefunctions of the simple harmonic oscillator in coordinate and momentum space. This derivation illustrates the abstract approach to the simple harmonic oscillator by completing the derivation of the representation-dependent wavefunctions from the representation-independent energy eigenfunctions. It is simple to incorporate into the undergraduate and graduate curricula. This new derivation begins with the standard approach that was first presented by Dirac in 1947 (and is modified slightly here in the spirit of the Schroedinger factorization meth…
A systematic procedure for deriving the master equation of a dissipative system is reported in the framework of stochastic description. For the Caldeira-Leggett model of the harmonic-oscillator bath, a detailed and elementary derivation of the bath-induced stochastic field is presented. The dynamics of the system is thereby fully described by a stochastic differential equation and the desired master equation would be acquired with statistical averaging. It is shown that the existence of a closed-form master equation depends on the specificity of the system as well as the feature of the dissipa…
In this paper we analyze the double Caldeira-Leggett model: the path integral approach to two interacting dissipative harmonic oscillators. Assuming a general form of the interaction between the oscillators, we consider two different situations: i) when each oscillator is coupled to its own reservoir, and ii) when both oscillators are coupled to a common reservoir. After deriving and solving the master equation for each case, we analyze the decoherence process of particular entanglements in the positional space of both oscillators. To analyze the decoherence mechanism we have derived a general…
The Light Front Holographic (LFH) wave equation, which is the conformal scalar equation on the plane, is revisited from the perspective of the supersymmetric quantum mechanics, and attention is drawn to the fact that it naturally emerges in the small hyperbolic angle approximation to the "curved" Higgs oscillator on the hyperbolic plane, i.e. on the upper part of the two-dimensional hyperboloid of two sheets, a space of constant negative curvature. Such occurs because the particle dynamics under consideration reduces to the one dimensional Schrödinger equation with the second hyperbolic Pöschl…
The classical and quantum solutions of a nonlinear model describing harmonic oscillators on the sphere and the hyperbolic plane, derived in polar coordinates in a recent paper [Phys.\ Lett.\ A 379 (2015) 1589], are extended by the inclusion of an isotonic term.
A novel derivation of quantum propagator of a system described by a general quadratic Lagrangian is presented in the framework of Heisenberg equations of motion. The general corresponding density matrix is obtained for a derived quantum harmonic oscillator and a particle confined in a one dimensional Paul trap. Total mean energy, work and absorbed heat, Wigner function and excitation probabilities are found explicitly. The method presented here is based on the Heisenberg representation of position and momentum operators and can be generalized to a system consisting of a set of linearly interac…
A nonlinear model representing the quantum harmonic oscillator on the three-dimensional spherical and hyperbolic spaces, $S_\k^3$ ($κ>0$) and $H_k^3$ ($κ<0$), is studied. The curvature $\k$ is considered as a parameter and then the radial Schrödinger equation becomes a $\k$-dependent Gauss hypergeometric equation that can be considered as a $\k$-deformation of the confluent hypergeometric equation that appears in the Euclidean case. The energy spectrum and the wavefunctions are exactly obtained in both the three-dimensional sphere $S_\k^3$ ($κ>0$) and the hyperbolic space $H_k^3$ ($κ<0$). A co…
A simple derivation of the classical solutions of a nonlinear model describing a harmonic oscillator on the sphere and the hyperbolic plane is presented in polar coordinates. These solutions are then related to those in cartesian coordinates, whose form was previously guessed. In addition, the nature of the classical orthogonal polynomials entering the bound-state radial wavefunctions of the corresponding quantum model is identified.
Extending the supersymmetric method proposed by Tkachuk to the complex domain, we obtain general expressions for superpotentials allowing generation of quasi-exactly solvable PT-symmetric potentials with two known real eigenvalues (the ground state and first-excited state energies). We construct examples, namely those of complexified non-polynomial oscillators and of a complexified hyperbolic potential, to demonstrate how our scheme works in practice. For the former we provide a connection with the sl(2) method, illustrating the comparative advantages of the supersymmetric one.
We use the Fourier operator to transform a time dependent mass quantum harmonic oscillator into a frequency dependent one. Then we use Lewis-Ermakov invariants to solve the Schrödinger equation by using squeeze operators. Finally we give two examples of time dependencies: quadratically and hyperbolically growing masses.
The Schwinger quantum correction to the classic Thomas-Fermi atom is directly derived by solving for the latter without recourse to a modeling after the harmonic oscillator potential and without solving for the particle density.
The correspondence between classical and quantum invariants is established. The Ermakov Lewis quantum invariant of the time dependent harmonic oscillator is translated from the coordinate and momentum operators into amplitude and phase operators. In doing so, Turski's phase operator as well as Susskind-Glogower operators are generalized to the time dependent harmonic oscillator case. A quantum derivation of the Manley-Rowe relations is shown as an example.
We show that quantum Bateman's system which arises in the quantization of a damped harmonic oscillator is equivalent to a quantum problem with 2D parabolic potential barrier known also as 2D inverted isotropic oscillator. It turns out that this system displays the family of complex eigenvalues corresponding to the poles of analytical continuation of the resolvent operator to the complex energy plane. It is shown that this representation is more suitable than the hyperbolic one used recently by Blasone and Jizba.
We re-derive the Schrödinger-Robertson uncertainty principle for the position and momentum of a quantum particle. Our derivation does not directly employ commutation relations, but works by reduction to an eigenvalue problem related to the harmonic oscillator, which can then be further exploited to find a larger class of constrained uncertainty relations. We derive an uncertainty relation under the constraint of a fixed degree of Gaussianity and prove that, remarkably, it is saturated by all eigenstates of the harmonic oscillator. This goes beyond the common knowledge that the (Gaussian) groun…
In this paper I discuss by means of path integrals the quantum dynamics of a charged particle on the hyperbolic plane under the influence of an Aharonov-Bohm gauge field. The path integral can be solved in terms of an expansion of the homotopy classes of paths. I discuss the interference pattern of scattering by an Aharonov-Bohm gauge field in the flat space limit, yielding a characteristic oscillating behavior in terms of the field strength. In addition, the cases of the isotropic Higgs-oscillator and the Kepler-Coulomb potential on the hyperbolic plane are shortly sketched.
In this paper we establish a relation between two exactly-solvable problems on one-dimensional hyperbolics space, namely singular Coulomb and singular oscillator systems.
A many particle Hamiltonian, where the interaction term conserves the number of particles, is considered. A master equation for the populations of the different levels is derived in an exact way. It results in a local equation with time-dependent coefficients, which can be identified with the transition probabilities in the golden rule approximation. A reinterpretation of the model as a set of coupled harmonic oscillators enables one to obtain for one of them an exact local Langevin equation, with time-dependent coefficients.
We present a unified treatment of exact solutions for a class of four quantum mechanical models, namely the singular anharmonic potential, the generalized quantum isotonic oscillator, the soft-core Coulomb potential, and the non-polynomially modified oscillator. We show that all four cases are reducible to the same basic ordinary differential equation, which is quasi-exactly solvable. A systematic and closed form solution to the basic equation is obtained via the Bethe ansatz method. Using the result, general exact expressions for the energies and the allowed potential parameters are given exp…
In this paper we derive an exact master equation for two coupled quantum harmonic oscillators interacting via bilinear coupling with a common environment at arbitrary temperature made up of many harmonic oscillators with a general spectral density function. We first show a simple derivation based on the observation that the two-harmonic oscillator model can be effectively mapped into that of a single harmonic oscillator in a general environment plus a free harmonic oscillator. Since the exact one harmonic oscillator master equation is available [Hu, Paz and Zhang, Phys. Rev. D \textbf{45}, 284…
The problem of the quantum harmonic oscillator is investigated in the framework of bicomplex numbers, which are pairs of complex numbers making up a commutative ring with zero divisors. Starting with the commutator of the bicomplex position and momentum operators, and adapting the algebraic treatment of the standard quantum harmonic oscillator, we find eigenvalues and eigenkets of the bicomplex harmonic oscillator Hamiltonian. We construct an infinite-dimensional bicomplex module from these eigenkets. Turning next to the differential equation approach, we obtain coordinate-basis eigenfunctions…
We revise the construction of creation/annihilation operators in quantum mechanics based on the representation theory of the Heisenberg and symplectic groups. Besides the standard harmonic oscillator (the elliptic case) we similarly treat the repulsive oscillator (hyperbolic case) and the free particle (the parabolic case). The respective hypercomplex numbers turn to be handy on this occasion, this provides a further illustration to Similarity and Correspondence Principle. Keywords: Heisenberg group, Kirillov's method of orbits, geometric quantisation, quantum mechanics, classical mechanics, P…
