Notes on Application of WKB Method

The notes here presented are of the modifications introduced in the application of WKB method. The problems of two- and three-dimensional harmonic oscillator potential are revisited by WKB and the new formulation of quantization rule respectively. It is found that the energy spectrum of the radial harmonic oscillator, which is reproduced exactly by the standard WKB method with the Langer modification, is also reproduced exactly without the Langer modification via the new quantization rule approach. An alternative way to obtain the non-integral Maslov index for three-dimensional harmonic oscillator is proposed.     

Energy spectrum of the trigonometric Rosen-Morse potential using an improved quantization rule

The improved quantization rule simplifies the calculation of the energy levels for the exactly solvable quantum system. In this Letter we calculate the energy levels of the Schrödinger equation with the symmetric and asymmetric trigonometric Rosen-Morse potentials by the improved quantization rule.     

Quantum coadjoint orbits in gl(n)*

We present a doubleU _h(gl(n, ℂ))-equivariant quantization on semisimple coadjoint orbits of the group GL(n, ℂ) as a quotient of the extended reflection equation algebra by relations which are given explicitly. Such a quantization is a two-parameter family including an explicit GL(n)-equivariant quantization of the Kirillov-Kostant-Souriau Poisson bracket. Presented at the 11th Colloquium “Quantum Groups and Integrable Systems”, Prague, 20–22 June 2002.     

Weyl-Wigner-Moyal formalism for Fermi classical systems

The Weyl-Wigner-Moyal formalism of fermionic classical systems with a finite number of degrees of freedom is considered. The Weyl correspondence is studied by computing the relevant Stratonovich-Weyl quantizer. The Moyal -product, Wigner functions and normal ordering are obtained for generic fermionic systems. Finally, this formalism is used to perform the deformation quantization of the Fermi oscillator and the supersymmetric quantum mechanics.     

Arbitrary l-state solutions of the rotating Morse potential through the exact quantization rule method

By using the exact quantization rule, for non-zero l   values we present analytical solutions of the radial Schrödinger equation for the rotating Morse potential in the frame of the Pekeris approximation. The energy levels of all the bound states are easily calculated from this quantization rule. Especially, the intractable normalized wavefunctions are also obtained. The numerical calculations for three typical diatomic molecules HCl, CO and LiH are compared with those obtained by other methods such as the supersymmetry, the Fourier gird Hamiltonian, the asymptotic iteration, the variational, the Nikiforov–Uvarov, the shifted 1/N$1/N$ expansion and the modified shifted 1/N$1/N$ expansion. It is found that the results obtained by the present method are in good agreement with those obtained by other approximate methods.     

Coherent states of a particle in a magnetic field and the Stieltjes moment problem

A solution to a version of the Stieltjes moment problem is presented. Using this solution, we construct a family of coherent states of a charged particle in a uniform magnetic field. We prove that these states form an overcomplete set that is normalized and resolves the unity. By the help of these coherent states we construct the Fock-Bergmann representation related to the particle quantization. This quantization procedure takes into account a circle topology of the classical motion.     

Star products and geometric algebra

The formalism of geometric algebra can be described as deformed super analysis. The deformation is done with a fermionic star product, that arises from deformation quantization of pseudoclassical mechanics. If one then extends the deformation to the bosonic coefficients of superanalysis one obtains quantum mechanics for systems with spin. This approach clarifies on the one hand the relation between Grassmann and Clifford structures in geometric algebra and on the other hand the relation between classical mechanics and quantum mechanics. Moreover it gives a formalism that allows to handle classical and quantum mechanics in a consistent manner.     

A quantum exactly solvable non-linear oscillator with quasi-harmonic behaviour

The quantum version of a non-linear oscillator, previously analyzed at the classical level, is studied. This is a problem of quantization of a system with position-dependent mass of the form m = (1 + λx²)⁻¹ and with a λ-dependent non-polynomial rational potential. This λ-dependent system can be considered as a deformation of the harmonic oscillator in the sense that for λ → 0 all the characteristics of the linear oscillator are recovered. First, the λ-dependent Schrödinger equation is exactly solved as a Sturm-Liouville problem, and the λ-dependent eigenenergies and eigenfunctions are obtained for both λ > 0 and λ < 0. The λ-dependent wave functions appear as related with a family of orthogonal polynomials that can be considered as λ-deformations of the standard Hermite polynomials. In the second part, the λ-dependent Schrödinger equation is solved by using the Schrödinger factorization method, the theory of intertwined Hamiltonians, and the property of shape invariance as an approach. Finally, the new family of orthogonal polynomials is studied. We prove the existence of a λ-dependent Rodrigues formula, a generating function and λ-dependent recursion relations between polynomials of different orders.     

Topological Properties of Spatial Coherence Function

The topological properties of the spatial coherence function are investigated rigorously. The phase singular structures （coherence vortices） of coherence function can be naturally deduced from the topological current, which is an abstract mathematical object studied previously. We find that coherence vortices are characterized by the Hopf index and Brouwer degree in topology. The coherence flux quantization and the linking of the closed coherence vortices are also studied from the topological properties of the spatial coherence function.     

f-oscillators deformation for Moyal algebras

Using general construction of star-product the q-deformed Wigner-Weyl-Moyal quantization procedure is elaborated. The q-deformed Groenewold kernel determining the product of quantum observables is given in explicit form for small nonlinearities corresponding to nonlinear vibrations of classical and quantum q-oscillators. The deformation of Groenewold kernel related to general kinds of nonlinear vibrations described by f-oscillators are considered.     

Doubly uniform semiclassical quantization formula for resonances

The radial Schrödinger equation with an effective potential containing a single well and a single barrier is treated with an improved uniform semiclassical method. The improved quantization formula for complex energies (or resonances) contains a correction term that originates from a uniform treatment of the classically forbidden region near the origin in addition to the more familiar uniform treatment of the barrier region. In the present case the origin has a second-order pole, due to the centrifugal barrier potential term, and/or a Coulomb-type singularity, and these terms dominate the region inside the innermost classical turning point.Numerical results for first-order and third-order approximate complex resonance energies are compared with those of a standard (first- and third-order) barrier-uniform semiclassical method and also with those of ‘exact’ numerical computations.The improved quantization formula provides results in significantly better agreement with the exact results as the angular momentum quantum number l approaches zero.     

Semiclassical and quantum motions on the non-commutative plane

We study the canonical and the coherent state quantizations of a particle moving in a magnetic field on the non-commutative plane. Using a θ-modified action, we perform the canonical quantization and analyze the gauge dependence of the theory. We compare coherent states quantizations obtained through Malkin-Man'ko states and circular squeezed states. The relation between these states and the “classical” trajectories is investigated, and we present numerical explorations of some semiclassical quantities.     

Newton-Leibniz integration for ket-bra operators in quantum mechanics and derivation of entangled state representations

Newton-Leibniz integration rule only applies to commuting functions of continuum variables, while operators made of Dirac’s symbols (ket versus bra, e.g., |q〉〈q| of continuous parameter q) in quantum mechanics are usually not commutative. Therefore, integrations over the operators of type |〉〈| cannot be directly performed by Newton-Leibniz rule. We invented an innovative technique of integration within an ordered product (IWOP) of operators that made the integration of non-commutative operators possible. The IWOP technique thus bridges this mathematical gap between classical mechanics and quantum mechanics, and further reveals the beauty and elegance of Dirac’s symbolic method and transformation theory. Various applications of the IWOP technique, including constructing the entangled state representations and their applications, are presented.     

Use of harmonic inversion techniques in the periodic orbit quantization of integrable systems

Harmonic inversion has already been proven to be a powerful tool for the analysis of quantum spectra and the periodic orbit orbit quantization of chaotic systems. The harmonic inversion technique circumvents the convergence problems of the periodic orbit sum and the uncertainty principle of the usual Fourier analysis, thus yielding results of high resolution and high precision. Based on the close analogy between periodic orbit trace formulae for regular and chaotic systems the technique is generalized in this paper for the semiclassical quantization of integrable systems. Thus, harmonic inversion is shown to be a universal tool which can be applied to a wide range of physical systems. The method is further generalized in two directions: firstly, the periodic orbit quantization will be extended to include higher order corrections to the periodic orbit sum. Secondly, the use of cross-correlated periodic orbit sums allows us to significantly reduce the required number of orbits for semiclassical quantization, i.e., to improve the efficiency of the semiclassical method. As a representative of regular systems, we choose the circle billiard, whose periodic orbits and quantum eigenvalues can easily be obtained. Received 24 February 2000 and Received in final form 22 May 2000     

Quantization of motion with friction quadratic in velocity

The motion of a point object through a viscous field is considered. The friction is assumed to depend quadratically on velocity of the particle. The inverse problem of the variational calculus is solved and the Weyl quantization procedure is employed to write a Schrödinger equation. The solution of this equation shows that the quantum mechanical wave function is oscillatory for small values of the friction. Contrarily, for large values of the friction, the wave function resembles the solution of von Neumann shock problem.     

Correlations Existing in Three-Qubit Product States

Analytic results of the relationship between local noncommutativity and non-violations of Svetlichny inequalities for three-qubit separable states are obtained. It is shown that the converse trade-off relations presented by Seevinck and Uffinck [Phys. Rev. A 2007 76 042105] do not always hold for three-qubit states, and that there exists some correlation even though the state is the simple product state.     

Arrival time in quantum field theory

Via the proper-time eigenstates (event states) instead of the proper-mass eigenstates (particle states), free-motion time-of-arrival theory for massive spin-1/2 particles is developed at the level of quantum field theory. The approach is based on a position-momentum dual formalism. Within the framework of field quantization, the total time-of-arrival is the sum of the single event-of-arrival contributions, and contains zero-point quantum fluctuations because the clocks under consideration follow the laws of quantum mechanics.     

The WKB method for the Dirac equation with the vector and scalar potentials

The WKB approximation is developed for the Dirac equation with the spherically symmetrical vector and scalar potentials. The relativistic wavefunctions are constructed, new quantization rule containing the spin-orbital interaction is obtained. For spherically symmetrical model of the Stark effect the quasi-classical spectrum of relativistic hydrogen-like atom is calculated. Application of the WKB method to the mass spectrum of the hydrogen-like quark systems was done.     

Infinite quantum well: A coherent state approach

A new family of 2-component vector-valued coherent states for the quantum particle motion in an infinite square well potential is presented. They allow a consistent quantization of the classical phase space and observables for a particle in this potential. We then study the resulting position and (well-defined) momentum operators. We also consider their mean values in coherent states and their quantum dispersions.     

Modified Form of Wigner Functions for Non-Hamiltonian Systems

Quantization of non-Hamiltonian systems （such as damped systems） often gives rise to complex spectra and corresponding resonant states, therefore a standard form calculating Wigner functions cannot lead to static quasiprobability distribution functions. We show that a modified form of the Wigner functions satisfies a ＊-genvalue equation and can be derived from deformation quantization for such systems.