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.     

Reconsidering harmonic and anharmonic coherent states: Partial differential equations approach

This article presents a new approach to dealing with time dependent quantities such as autocorrelation function of harmonic and anharmonic systems using coherent states and partial differential equations. The approach that is normally used to evaluate dynamical quantities involves formidable operator algebra. That operator algebra becomes insurmountable when employing Morse oscillator coherent states. This problem becomes even more complicated in case of Morse oscillator as it tends to exhibit divergent dynamics. This approach employs linear partial differential equations, some of which may be solved exactly and analytically, thereby avoiding the cumbersome noncommutative algebra required to manipulate coherent states of Morse oscillator. Additionally, the arising integrals while using the herein presented method feature stability and high numerical efficiency. The correctness, applicability, and utility of the above approach are tested by reproducing the partition and optical autocorrelation function of the harmonic oscillator. A closed-form expression for the equilibrium canonical partition function of the Morse oscillator is derived using its coherent states and partial differential equations. Also, a nonequilibrium autocorrelation function expression for weak electron–phonon coupling in condensed systems is derived for displaced Morse oscillator in electronic state. Finally, the utility of the method is demonstrated through further simplifying the Morse oscillator partition function or autocorrelation function expressions reported by other researchers in unevaluated form of second-order derivative exponential. Comparison with exact dynamics shows identical results.     

SU(1,1) Lie Algebra Applied to the General Time-dependent Quadratic Hamiltonian System

Exact quantum states of the time-dependent quadratic Hamiltonian system are investigated using SU(1,1) Lie algebra. We realized SU(1,1) Lie algebra by defining appropriate SU(1,1) generators and derived exact wave functions using this algebra for the system. Raising and lowering operators of SU(1,1) Lie algebra expressed by multiplying a time-constant magnitude and a time-dependent phase factor. Two kinds of the SU(1,1) coherent states, i.e., even and odd coherent states and Perelomov coherent states are studied. We applied our result to the Caldirola–Kanai oscillator. The probability density of these coherent states for the Caldirola–Kanai oscillator converged to the center as time goes by, due to the damping constant γ. All the coherent state probability densities for the driven system are somewhat deformed. PACS Numbers: 02.20.Sv, 03.65.-w, 03.65.Fd     

高激发振动态氰化氢分子的解离研究

运用李代数陪集空间方法，结合莫尔斯（Morse）振子的解离特点研究了具有多重共振的高 激发振动态氰化氢（HCN）分子的解离.结果表明，分子振动模之间的共振（主要是体系的高 次共振）作用，有助于化学键的解离，共振的作用导致H—C键解离能量的降低. 关键词： 高激发振动态 共振 解离     

Construction of coherent states for Morse potential: A su(2)-like approach

We propose a generalized su(2) algebra that perfectly describes the discrete energy part of the Morse potential. Then, we examine particular examples and the approach can be applied to any Morse oscillator and to practically any physical system whose spectrum is finite. Further, we construct the Klauder coherent state for Morse potential satisfying the resolution of identity with a positive measure, obtained through the solution of truncated Stieltjes moment problem. The time evolution of the uncertainty relation of the constructed coherent states is analyzed. The uncertainty relation is more localized for small values of radius of convergence.     

SU(1, 1) and SU(2) Perelomov number coherent states: algebraic approach for general systems

We study some properties of the SU(1, 1) Perelomov number coherent states. The Schrödinger's uncertainty relationship is evaluated for a position and momentum-like operators (constructed from the Lie algebra generators) in these number coherent states. It is shown that this relationship is minimized for the standard coherent states. We obtain the time evolution of the number coherent states by supposing that the Hamiltonian is proportional to the third generator K₀ of the su(1, 1) Lie algebra. Analogous results for the SU(2) Perelomov number coherent states are found. As examples, we compute the Perelomov coherent states for the pseudoharmonic oscillator and the two-dimensional isotropic harmonic oscillator.     

The rotating Morse potential energy eigenvalues solved by using the analytical transfer matrix method

We study the eigenvalues of the rotating Morse potential by using the quantization condition from the analytical transfer matrix(ATM) method.A hierarchy of supersymmetric partner potentials is obtained with Pekeris approximation,which can be used to calculate the energies of higher rotational states from the energies of lower states.The energies of rotational states of the hydrogen molecule are calculated by the ATM condition,and comparison of the results with those from the hypervirial perturbation method reveals that the accuracy of the approximate expression of Pekeris for the eigenvalues of the rotating Morse potential can be improved substantially in the framework of supersymmetric quantum mechanics.     

几类宏观量子态的内部结构及其压缩特性

谐振子,变形振子,非简谐振子以及变形非简谐振子湮没算符高次幂的正交归一本征态都具有奇偶结构形式.正是由于这种结构特点决定了它们振幅的高次幂压缩性质.     

On a supersymmetric nonlinear integrable equation in (2+1) dimensions

A supersymmetric integrable equation in (2+1) dimensions is constructed by means of the approach of the homogenous space of the super Lie algebra, where the super Lie algebra osp(3/2) is considered. For this (2+1) dimensional integrable equation, we also derive its Bäcklund transformation.     

Unified Method for Dynamical Groups of Some Anharmonic Potentials

Realizations of the creation and annihilation operators for some important anharmonic potentials, such as the Morse potential, the modified Pöschl–Teller potential (MPT), the pseudoharmonic oscillator, and infinitely deep square-well potential, are presented by a factorization method. It is shown that the operators for the Morse potential and the MPT potential satisfy the commutation relations of an SU(2) algebra, but those of the pseudoharmonic oscillator and the infinitely deep square-well potential constitute an SU(1, 1) algebra. The matrix elements of some related operators are analytically obtained. The harmonic limits of the SU(2) operators for the Morse and MPT potentials are studied as the Weyl algebra.     

Spin Number Coherent States and the Problem of Two Coupled Oscillators

From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2) generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.     

Spin Number Coherent States and the Problem of Two Coupled Oscillators

From the definition of the standard Perelomov coherent states we introduce the Perelomov number coherent states for any su(2) Lie algebra. With the displacement operator we apply a similarity transformation to the su(2)generators and construct a new set of operators which also close the su(2) Lie algebra, being the Perelomov number coherent states the new basis for its unitary irreducible representation. We apply our results to obtain the energy spectrum, the eigenstates and the partition function of two coupled oscillators. We show that the eigenstates of two coupled oscillators are the SU(2) Perelomov number coherent states of the two-dimensional harmonic oscillator with an appropriate choice of the coherent state parameters.     

sh(2/2) Superalgebra Eigenstates and Generalized Supercoherent and Supersqueezed States

The superalgebra eigenstates (SAES) concept is introduced and then applied to find SAES associated to the sh(2/2) superalgebra, also known as Heisenberg–Weyl Lie superalgebra. This implies to solve a Grassmannian eigenvalue superequation. Thus, the sh(2/2) SAES contain the class of supercoherent states associated to the supersymmetric harmonic oscillator and also a class of supersqueezed states associated to the osp(2/2)Ð sh(2/2) superalgebra, where osp(2/2) denotes the orthosymplectic Lie superalgebra generated by the set of operators formed from the quadratic products of the Heisenberg–Weyl Lie superalgebra generators. The properties of these states are investigated and compared with those of the states obtained by applying the group-theoretical technics. Moreover, new classes of generalized supercoherent and supersqueezed states are also obtained. As an application, the super-Hermitian and -pseudo-super-Hermitian Hamiltonians without a defined Grassmann parity and isospectral to the harmonic oscillator are constructed. Their eigenstates and associated supercoherent states are calculated.     

On the biHamiltonian structure of the supersymmetric KdV hierarchies. A Lie superalgebraic approach

We give a Lie superalgebraic interpretation of the biHamiltonian structure of known supersymmetric KdV equations. We show that the loop algebra of a Lie superalgebra carries a natural Poisson pencil, and we subsequently deduce the biHamiltonian structure of the supersymmetric KdV hierarchies by applying to loop superalgebras an appropriate reduction technique. This construction can be regarded as a superextension of the Drinfeld-Sokolov method for building a KdV-type hierarchy from a simple Lie algebra.     

Statistical Properties of the nonlinear SU（1,1） Coherent States

Using non-Hermitian realizations of SU(1,1) Lie algebra in terms of an f-oscillator,we generalize the notion of nonlinear coherent states to the single-mode and two-mode nonlinear SU(1,1) coherent states.Taking the nonlinearity function f(k)=Lk^1(η^2)[(k 1)Lk^0(η^2)]^-1,their statistical properties are studied.     

Supersymmetric Pair Coherent States

Based on the permu table property between the two-mode boson-fermion annihilator for a supersymmetric harmonic oscillator, which is described by H = b^↑b + f^↑f, and the "charge" operators b^↑b - f^↑f we construct the supersymmetric pair coherent state. The super-lie algebra associated with the supersymmetric pair coherent state is analyzed.     

The soliton supermultiplet in 4 + 1 dimensions

We consider the SO(4) = SU(2) ? USp(2) Clifford algebra, obtained by the supersymmetry algebra for the N = 2 supersymmetric Yang-Mills theory in 4+1 dimensions, which, in the phase of unbroken gauge symmetry, has a topological charge as central charge. We find that, even if the Higgs mechanism is absent, the massive soliton supermultiplet contains the same number of states as the massless supermultiplet of elementary particles.     

非线性孪相干态的光子统计性质

利用f振子的湮没算符和产生算符的逆算符的性质，得到了SU(1，1)李代数的两个双模非厄密实现，在此基础上引入了两种非线性孪相干态，并讨论了它们的光子统计性质和相位特性。     

N ＝ 4 Superconformal Quantum Mechanics in One Dimension and Its Algebraic Structure

N = 4 superconformal quantum mechanics of nonrelativistic particles in the typical 1/x2-potentials, which holds not only supersymmetry but also dynamical conformal symmetry, is studied. The corresponding superconformal quantum mechanical algebra, which contains supersymmetric quantum mechanical algebra with four supercharges and conformal algebra as subalgebras, and its two canonical group chains are established.