Supersymmetry and duality in relativistic oscillator systems

A simple mapping is defined which takes the relativistic oscillator on a (1+1)-dimensional spacetime into a dual system consisting of a nonrelativistic oscillator confined to a circular space. The radius of the circle in the dual system measures the size of relativistic effects in the original system. It is shown that both of the dual oscillator systems have supersymmetric partners, and the exact wavefunctions and energy levels of all four of the intertwined systems are found.     

Wronskian differential formula for confluent supersymmetric quantum mechanics

A Wronskian differential formula, useful for applying the confluent second-order SUSY transformations to arbitrary potentials, will be obtained. This expression involves a parametric derivative with respect to the factorization energy which, in many cases, is simpler for calculations than the previously found integral equation. This alternative mechanism will be applied to the free particle and the single-gap Lamé potential.     

Complex oscillator and Painlevé IV equation

Supersymmetric quantum mechanics is a powerful tool for generating exactly solvable potentials departing from a given initial one. In this article the first- and second-order supersymmetric transformations will be used to obtain new exactly solvable potentials departing from the complex oscillator. The corresponding Hamiltonians turn out to be ruled by polynomial Heisenberg algebras. By applying a mechanism to reduce to second the order of these algebras, the connection with the Painlevé IV equation is achieved, thus giving place to new solutions for the Painlevé IV equation.     

Time-dependent harmonic oscillator in a magnetic field and an electric field on the non-commutative plane

In the non-commutative space, wave functions and geometric phases are derived for the time-dependent harmonic oscillator in external time-dependent magnetic and electric field. Explicit forms of the coherent states are also given, which are not the minimum uncertainty states for the coordinates and momenta.     

Solvable rational extensions of the isotonic oscillator

Combining recent results on rational solutions of the Riccati–Schrödinger equations for shape invariant potentials to the finite difference Bäcklund algorithm and specific symmetries of the isotonic potential, we show that it is possible to generate the three infinite sets (L1, L2 and L3 families) of regular rational solvable extensions of this potential in a very direct and transparent way.     

Painlevé IV coherent states

A simple way to find solutions of the Painlevé IV equation is by identifying Hamiltonian systems with third-order differential ladder operators. Some of these systems can be obtained by applying supersymmetric quantum mechanics (SUSY QM) to the harmonic oscillator. In this work, we will construct families of coherent states for such subset of SUSY partner Hamiltonians which are connected with the Painlevé IV equation. First, these coherent states are built up as eigenstates of the annihilation operator, then as displaced versions of the extremal states, both involving the related third-order ladder operators, and finally as extremal states which are also displaced but now using the so called linearized ladder operators. To each SUSY partner Hamiltonian corresponds two families of coherent states: one inside the infinite subspace associated with the isospectral part of the spectrum and another one in the finite subspace generated by the states created through the SUSY technique.     

The multidimensional Darboux transformation

A generalization of the classical one-dimensional Darboux transformation to arbitrary n-dimensional oriented Riemannian manifolds is constructed using an intrinsic formulation based on the properties of twisted Hodge Laplacians. The classical two-dimensional Moutard transformation is also generalized to non-compact oriented Riemannian manifolds of dimension n ≥ 2. New examples of quasi-exactly solvable multidimensional matrix Schrödinger operators on curved manifolds are obtained by applying the above results.     

非线性振子的矩阵元及其经典近似

对于势能为V(x)=1/2 mω2x2+λx4的非线性谐振子,不能用微扰论对经典方程进行求解.这里利用海森伯对应原理,由量子力学的矩阵元得到了非线性振子的经典解,从而对于非线性振子的性质有了进一步的理解.     

Rational solutions for the Riccati-Schrödinger equations associated to translationally shape invariant potentials

We develop a new approach to build the eigenfunctions of a translationally shape invariant potential. For this we show that their logarithmic derivatives can be expressed as terminating continued fractions in an appropriate variable. We give explicit formulas for all the eigenstates, their specific form depending on the Barclay-Maxwell class to which the considered potential belongs.     

The effect of singular potentials on the harmonic oscillator

We address the problem of a quantum particle moving under interactions presenting singularities. The self-adjoint extension approach is used to guarantee that the Hamiltonian is self-adjoint and to fix the choice of boundary conditions. We specifically look at the harmonic oscillator added of either a δ-function potential or a Coulomb potential (which is singular at the origin). The results are applied to Landau levels in the presence of a topological defect, the Calogero model and to the quantum motion on the noncommutative plane.     

用自洽平均值法计算非线性谐振子本征值

提出了一种自洽平均值法计算非线性谐振子本征值的方法.研究结果表明,等效势能曲线、坐标算符的一次项和二次项的平均值及能量本征值均在主量子数n较小时与微扰量子理论相吻合,在n=0时两者完全一致.     

Equity prices as a simple harmonic oscillator with noise

The centred return on the London Stock Exchange's FTSE All Share Index is modelled as a simple harmonic oscillator with noise over the period from 1 January, 1994 until 30 June 2006. Our empirical results are compatible with the hypothesis that there is a period in the FTSE All Share Index of between two and two and one half years. This means the centred return will on average continue to increase for about a year after reaching the minimum in its oscillatory cycle; alternatively, it will continue on average to decline for about a year after reaching a maximum. Our analysis also shows that there is potential to exploit the harmonic nature of the returns process to earn abnormal profits. Extending our analysis to the low energy states of a quantum harmonic oscillator is also suggested.     

Influence of experimental resolution on the quantum-to-classical transition in the quartic oscillator

The quantum-classical transition problem is investigated for the quartic oscillator coupled to a thermal reservoir. We show for this model that the combination of relevant diffusion, classical action (represented by the amplitude of an initial coherent state) and the experimental uncertainties is necessary to achieve the classical regime. In order to study the role of limited resolutions of measurement apparatuses on the correspondence between the quantum and classical dynamics, we consider experimental errors due the preparation of the initial state of the quartic oscillator and the inaccuracies in the time measurements. A quantum break time depending on the diffusion constant, the amplitude of the initial coherent state and the inaccuracy of measurements is defined. We found, for this model, a regime where the increasing of diffusion does not anticipate classicality. In such regime, there is a minimum value for the classical action associated to classical behavior of the system.     

Higher dimensional supersymmetric quantum mechanics and Dirac equation

We exhibit the supersymmetric quantum mechanical structure of the full 3+1 dimensional Dirac equation considering ‘mass’ as a function of coordinates. Its usefulness in solving potential problems is discussed with specific examples. We also discuss the ‘physical’ significance of the supersymmetric states in this formalism.     

Exactness of SWKB for shape invariant potentials

The supersymmetry-based semiclassical method (SWKB) is known to produce exact spectra for conventional shape invariant potentials. In this paper we prove that this exactness follows from their additive shape invariance.     

量子谐振子与经典谐振子的比较

运用广义线性量子变换的普遍理论求解了量子谐振子,同经典谐振子比较给出了量子谐振子趋近于经典极限的条件,并得出相干态是最理想的经典极限态．     

Kink mass quantum shifts from SUSY quantum mechanics

In this paper a new version of the DHN (Dashen–Hasslacher–Neveu) formula, which is used to compute the one-loop order kink mass correction in (1+1)-dimensional scalar field theory models, is constructed. The new expression is written in terms of the spectral data coming from the supersymmetric partner operator of the second-order small kink fluctuation operator and allows us to compute the kink mass quantum shift in new models for which this calculation was out of reach by means of the old formula.     

The kinetic energy partition method applied to a confined quantum harmonic oscillator in a one-dimensional box

A new, physically motivated, basis set expansion method for solving quantum eigenvalue problems with competing interaction potentials is presented. In contrast to the usual dissection of the potential energy into unperturbed and perturbing terms, we divide the kinetic energy into partial terms by modifying the mass factor. The partition scheme results in partial kinetic energies with their effective mass factors. By distributing each partial kinetic energy to a respective potential energy to form a subsystem, the total Hamiltonian is written as the sum of subsystem Hamiltonians. Using a linear combination of the subsystem wave-functions to represent the system wave-function we obtain a set of coupled equations for the expansion coefficients, by solving these energies and wave-functions can be obtained. We demonstrate the solution scheme with a standard model system: a confined harmonic oscillator in a one-dimensional box. With only a few (less than ten) basis functions from each subsystem, we can reproduce the exact solutions very accurately, thus showing the applicability of this method.     

Overlaps of deformed and non-deformed harmonic oscillator basis states

A systematic approach for expanding non-deformed harmonic oscillator basis states in terms of deformed ones, and vice versa, is presented. The objective is to provide analytical results for calculating these overlaps (transformation brackets) between deformed and non-deformed basis states in spherical, cylindrical, and Cartesian coordinates. These overlaps can be used for reducing the complexity of different research problems that employ three-dimensional harmonic oscillator basis states, for example as used in coherent state theory and the nuclear shell-model, especially within the context of ab initio symmetry-adapted no-core shell model.