精确的量子化条件和不变量

提出并证明了一维量子系统和三维球对称量子系统的一个精确的量子化条件.在此精确量子化条件中, 除了通常的Nπ项外, 还有一积分项, 称为修正项. 发现该修正项正是在超对称量子力学中所谓的有形状不变势的量子系统的一个不变量，它不依赖于波函数的节点数.对这些系统, 可用基态能级和波函数确定此不变量的值, 从而由精确的量子化条件容易算出全部束缚态的能级. 计算得到能级的正确性又反过来验证了在有形状不变势的量子系统中此修正项确实是不变量.计算的有形状不变势的量子系统, 包括一维的有限方势阱、Morse势及其变形、R 关键词： 量子化条件 超对称量子力学 形状不变势 不变量     

NEW EXACTLY SOLVABLE SUPERSYMMETRIC PERIODIC POTENTIALS

Using the formalism of supersymmetric quantum mechanics, we give an exact solution for a family of one-dimensional periodic potentials, which are the supersymmetric partners of the potential proportional to the trigonometric function cos(2x) such that the Schr?dinger equation for this potential is named the Mathieu equation mathematically. We show that the new potentials are distinctly different from their original ones. However, both have the same energy band structure. All the potentials obtained in this paper are free of singularities.     

Spectral properties of supersymmetric shape invariant potentials

We present the spectral properties of supersymmetric shape invariant potentials (SIPs). Although the folded spectrum is completely random, unfolded spectrum shows that energy levels are highly correlated and absolutely rigid. All the SIPs exhibit harmonic oscillator-type spectral statistics in the unfolded spectrum. We conjecture that this is the reflection of shape invariant symmetry.      

On nature of supersymmetries

The problem of emergence of supersymmetry is considered. It is argued that putting of a set of harmonic oscillators into a thermal bath gives both quantum mechanics and supersymmetric structures.     

Supersymmetric Pair of q-Deformed Nonlocal Operators

A simple version of the q-deformed calculus is used to generate a pair ofq-nonlocal, second-order difference operators by means of deformed counterpartsof Darboux intertwining operators for the Schrödinger—Hermite oscillators atzero factorization energy. These deformed nonlocal operators may be consideredas supersymmetric partners and their structure contains contributions originatingin both the Hermite operator and the quantum harmonic oscillator operator. Thereare also extra ±x contributions. The undeformed limit, in which allq-nonlocalities wash out, corresponds to the usual supersymmetric pair of quantum mechanicalharmonic oscillator Hamiltonians. The more general case of negative factorizationenergy is briefly discussed as well.     

N = 4 Supersymmetric Morse Oscillator and Its Spectrum-Generating Algebra

In this paper N = 4 supersymmetry of generalized Morse oscillators in one dimension is studied. Both bound states and scattering states of its four superpartner Hamiltonians are analyzed by using unitary irreducible representations of the noncompact Lie algebra su(1,1). The spectrum-generating algebra governing the Hamiltonian of the N = 4 supersymmetric Morse oscillator is shown to be connected with the realization of Lie superalgebra osp(1,2)or B(0,1) in terms of the variables of a supersymmetric two-dimensional harmonic oscillator.     

The supersymmetric WKB approximation for Hartmann potential

Using the method of supersymmetric WKB approximation, the energy spectrum of some noncentral separable potentials can be exactly obtained in r and θ dimensions. We take the Hartmann potential as an important example for its validation, and the result is consistent with that obtained by using the supersymmetric quantum mechanics and shape invariance.     

Supersymmetric Isospectral Formalism for the Calculation of Near-Zero Energy States: Application to the Very Weakly Bound 4He Trimer Excited State

We propose a novel mathematical approach for the calculation of near-zero energy states by solving potentials which are isospectral with the original one. For any potential, families of strictly isospectral potentials (with very different shape) having desirable and adjustable features are generated by supersymmetric isospectral formalism. The near-zero energy Efimov state in the original potential is effectively trapped in the deep well of the isospectral family and facilitates more accurate calculation of the Efimov state. Application to the first excited state in ⁴He trimer is presented.     

Extension of PT-symmetric quantum mechanics to the Dirac theory with position-dependent mass

We present a new method to construct the exactly solvable PT-symmetric potentials within the framework of the position-dependent effective mass Dirac equation with the vector potential coupling scheme in 1 + 1 dimensions. In order to illustrate the procedure, we produce three PT-symmetric potentials as examples, which are PT-symmetric harmonic oscillator-like potential, PT-symmetric potential with the form of a linear potential plus an inversely linear potential, and PT-symmetric kink-like potential, respectively. The real relativistic energy levels and corresponding spinor components for the bound states are obtained by using the basic concepts of the supersymmetric quantum mechanics formalism and function analysis method.     

Supersymmetry and Shape Invariance of Hartmann Potential and Ring-Shaped Oscillator Potential in the γ and θ Dimensions of Spherical Polar Coordinates

This article shows that in spherical polar coordinates, some noncentral separable potentials have super-symmetry and shape invariance in the r and θ dimensions, we choose Hartmann potential and ring-shaped oscillator astwo important examples, thus in principle the energy eigenvalues and energy eigenfunctions of such the potentials in ther and θ dimensions can be obtained by the method of supersymmetric quantum mechanics. Here we use an alternativemethod to get the required results.