We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Sch... more We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Schr ödinger equation with a time-dependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier transform and a relation with representations of the Heisenberg-Weyl group N (3) in a certain special case first, and then is extended to the general case. A three parameter extension of the classical Fourier integral is discussed as a by-product. Motion of a particle with a spin in uniform perpendicular magnetic and electric fields is considered as an application; a transition amplitude between Landau levels is evaluated in terms of Charlier polynomials. In addition, we also solve an initial value problem to a similar diffusion-type equation.
Journal of Difference Equations and Applications, Apr 1, 2013
We consider a six-parameter family of the square integrable wave functions for the simple harmoni... more We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented.
We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a vie... more We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a viewpoint of the Ermakov-type system. A six parameter family of the square integrable oscillator wave functions, which seems cannot be obtained by the standard separation of variables, is presented as an example. The invariance group of the generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schrödinger group of the free particle.
We present quadratic dynamic invariants and evaluate the Berry phase for the time-dependent Schrö... more We present quadratic dynamic invariants and evaluate the Berry phase for the time-dependent Schrödinger equation with the most general variable quadratic Hamiltonian.
We discuss basic potentials of the nonrelativistic and relativistic quantum mechanics that can be... more We discuss basic potentials of the nonrelativistic and relativistic quantum mechanics that can be integrated in the Nikiforov and Uvarov paradigm with the aid of a computer algebra system. This approach may help the readers to study modern analytical methods of quantum physics. Building on ideas of de Broglie and Einstein, I tried to show that the ordinary differential equations of mechanics, which attempt to define the co-ordinates of a mechanical system as functions of the time, are no longer applicable for "small" systems; instead there must be introduced a certain partial differential equation, which defines a variable ψ ("wave function") as a function of the co-ordinates and the time.
A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz is dis... more A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz is discussed. Simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed. Models of q-harmonic oscillators are being developed in connection with quantum groups and their various applications ( see, for example, Refs. [1-5]). The q-analogs of boson operators were introduced explicitly in Refs. [1,3] and [5], where the corresponding wave functions were found in terms of the continuous q-Hermite polynomials of Rogers and in terms of the Stieltjes-Wigert polynomials , respectively. Here we introduce one more explicit realization of q-creation and q-annihilation operators with the aid of q-Charlier polynomials of .
One more model of a q-harmonic oscillator based on the qorthogonal polynomials of Al-Salam and Ca... more One more model of a q-harmonic oscillator based on the qorthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed.
Journal of Computational and Applied Mathematics, Apr 1, 1996
Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier tra... more Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier transform. Askey, Atakishiyev and Suslov used this approach to obtain a q-Fourier transform by using the continuous q-Hermite polynomials. Rahman and Suslov extended this result by taking the Askey-Wilson polynomials, considered to be the most general continuous classical orthogonal polynomials. The theory of q-Fourier transformation is further extended here by considering a nonsymmetric version of the Poisson kernel with Askey-Wilson polynomials. This approach enables us to obtain some new results, for example, the complex and real orthogonalities of these kernels.
We construct the Green functions (or Feynman's propagators) for the Schrödinger equations of the ... more We construct the Green functions (or Feynman's propagators) for the Schrödinger equations of the form iψ t + 1 4 ψ xx ± tx 2 ψ = 0 in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and momentum representations. Particular solutions of the corresponding nonlinear Schrödinger equations with variable coefficients are also found. A special case of the quantum parametric oscillator is studied in detail first. The Green function is explicitly given in terms of Airy functions and the corresponding transition amplitudes are found in terms of a hypergeometric function. The general case of quantum parametric oscillator is considered then in a similar fashion. A group theoretical meaning of the transition amplitudes and their relation with Bargmann's functions is established.
We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a vie... more We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a viewpoint of the Ermakov-type system. A six parameter family of the square integrable oscillator wave functions, which seems cannot be obtained by the standard separation of variables, is presented as an example. The invariance group of the generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schrödinger group of the free particle.
Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalizat... more Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalization of Hahn's approach. In [3], [4], and [16] a number of new systems of orthogonal polynomials (i.e., 4F3 and 4q~3 polynomials) were introduced. The main properties of these polynomials have been established on the basis of the theory of hypergeometric and basic hypergeometric series [9], [11]. As is clear now, the Askey-Wilson [3], [4] and Wilson [16] polynomials are the most general extensions of the Jacobi, Laguerre, and Hermite polynomials known at present. The families of polynomials, considered in [1], [3], [4], and [16], together with their special and limiting cases, form a mathematical entity-the classical orthogonal polynomials of a discrete variable on nonuniform lattices [1], [15]. The introduction of such an object made it possible to generalize all the fundamental properties characteristic of Hahn's approach [12], [10]. We shall disc/ass these properties for the most general system-the Askey-Wilson polynomials, in the framework of an approach developed in [15], [7]. 1. Hypergeometrie-Type Difference Equation Consider the hypergeometric-type difference equation in the self-adjoint form (for details, see [15]) A [cr(z)p(z)~l+2p(z)y(z)=O ' (1) Vxl(z) (2) A[cr(z)p(z)] = p(z)~(z)Vx~(z), on the lattice x(z) = cosh 2o9z = ~q" + q-Z), q = e2~,, where xl(z) = x(z + 89 and AT(z) = Vf(z + 1) = f(z + 1)-f(z).
We derive the recurrence relations for relativistic Coulomb integrals directly from the integral ... more We derive the recurrence relations for relativistic Coulomb integrals directly from the integral representations with the help of computer algebra methods. In order to manage the computational complexity of this problem, we employ holonomic closure properties in a sophisticated way.
We present quadratic dynamical invariant and evaluate Berry's phase for the timedependent Schrödi... more We present quadratic dynamical invariant and evaluate Berry's phase for the timedependent Schrödinger equation with the most general variable quadratic Hamiltonian.
We discuss a new completely integrable case of the time-dependent Schrödinger equation in R n wit... more We discuss a new completely integrable case of the time-dependent Schrödinger equation in R n with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator recently considered by Meiler, Cordero-Soto, and Suslov. A second pair of dual Hamiltonians is found in the momentum representation. Our examples show that in mathematical physics and quantum mechanics a change in the direction of time may require a total change of the system dynamics in order to return the system back to its original quantum state. Particular solutions of the corresponding nonlinear Schrödinger equations are obtained. A Hamiltonian structure of the classical integrable problem and its quantization are also discussed.
We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Sch... more We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Schroedinger equation with a time-dependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier transform and a relation with representations of the Heisenberg-Weyl group N(3) in a certain special case first, and then is extended to the general case. A three parameter extension of the classical Fourier integral is discussed as a by-product. Motion of a particle with a spin in uniform perpendicular magnetic and electric fields is considered as an application; a transition amplitude between Landau levels is evaluated in terms of Charlier polynomials. In addition, we also solve an initial value problem to a similar diffusion-type equation.
We introduce a double sum extension of a very well-poised series and extend to this the transform... more We introduce a double sum extension of a very well-poised series and extend to this the transformations of Bailey and Sears as well as the 6 û 5 summation formula of F. H. Jackson and the q-Dixon sum. We also give q-integral representations of the double sum. Generalizations of the Nassrallah-Rahman integral are also found.
We derive closed formulas for mean values of all powers of r in nonrelativistic and relativistic ... more We derive closed formulas for mean values of all powers of r in nonrelativistic and relativistic Coulomb problems in terms of the Hahn and Chebyshev polynomials of a discrete variable. A short review on special functions and solution of the Coulomb problems in quantum mechanics is given.
We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Sch... more We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Schr ödinger equation with a time-dependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier transform and a relation with representations of the Heisenberg-Weyl group N (3) in a certain special case first, and then is extended to the general case. A three parameter extension of the classical Fourier integral is discussed as a by-product. Motion of a particle with a spin in uniform perpendicular magnetic and electric fields is considered as an application; a transition amplitude between Landau levels is evaluated in terms of Charlier polynomials. In addition, we also solve an initial value problem to a similar diffusion-type equation.
Journal of Difference Equations and Applications, Apr 1, 2013
We consider a six-parameter family of the square integrable wave functions for the simple harmoni... more We consider a six-parameter family of the square integrable wave functions for the simple harmonic oscillator, which cannot be obtained by the standard separation of variables. They are given by the action of the corresponding maximal kinematical invariance group on the standard solutions. In addition, the phase space oscillations of the electron position and linear momentum probability distributions are computer animated and some possible applications are briefly discussed. A visualization of the Heisenberg Uncertainty Principle is presented.
We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a vie... more We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a viewpoint of the Ermakov-type system. A six parameter family of the square integrable oscillator wave functions, which seems cannot be obtained by the standard separation of variables, is presented as an example. The invariance group of the generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schrödinger group of the free particle.
We present quadratic dynamic invariants and evaluate the Berry phase for the time-dependent Schrö... more We present quadratic dynamic invariants and evaluate the Berry phase for the time-dependent Schrödinger equation with the most general variable quadratic Hamiltonian.
We discuss basic potentials of the nonrelativistic and relativistic quantum mechanics that can be... more We discuss basic potentials of the nonrelativistic and relativistic quantum mechanics that can be integrated in the Nikiforov and Uvarov paradigm with the aid of a computer algebra system. This approach may help the readers to study modern analytical methods of quantum physics. Building on ideas of de Broglie and Einstein, I tried to show that the ordinary differential equations of mechanics, which attempt to define the co-ordinates of a mechanical system as functions of the time, are no longer applicable for "small" systems; instead there must be introduced a certain partial differential equation, which defines a variable ψ ("wave function") as a function of the co-ordinates and the time.
A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz is dis... more A model of a q-harmonic oscillator based on q-Charlier polynomials of Al-Salam and Carlitz is discussed. Simple explicit realization of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are found. A connection of the kernel of this transform with biorthogonal rational functions is observed. Models of q-harmonic oscillators are being developed in connection with quantum groups and their various applications ( see, for example, Refs. [1-5]). The q-analogs of boson operators were introduced explicitly in Refs. [1,3] and [5], where the corresponding wave functions were found in terms of the continuous q-Hermite polynomials of Rogers and in terms of the Stieltjes-Wigert polynomials , respectively. Here we introduce one more explicit realization of q-creation and q-annihilation operators with the aid of q-Charlier polynomials of .
One more model of a q-harmonic oscillator based on the qorthogonal polynomials of Al-Salam and Ca... more One more model of a q-harmonic oscillator based on the qorthogonal polynomials of Al-Salam and Carlitz is discussed. The explicit form of q-creation and q-annihilation operators, q-coherent states and an analog of the Fourier transformation are established. A connection of the kernel of this transform with a family of self-dual biorthogonal rational functions is observed.
Journal of Computational and Applied Mathematics, Apr 1, 1996
Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier tra... more Wiener used the Poisson kernel for the Hermite polynomials to deal with the classical Fourier transform. Askey, Atakishiyev and Suslov used this approach to obtain a q-Fourier transform by using the continuous q-Hermite polynomials. Rahman and Suslov extended this result by taking the Askey-Wilson polynomials, considered to be the most general continuous classical orthogonal polynomials. The theory of q-Fourier transformation is further extended here by considering a nonsymmetric version of the Poisson kernel with Askey-Wilson polynomials. This approach enables us to obtain some new results, for example, the complex and real orthogonalities of these kernels.
We construct the Green functions (or Feynman's propagators) for the Schrödinger equations of the ... more We construct the Green functions (or Feynman's propagators) for the Schrödinger equations of the form iψ t + 1 4 ψ xx ± tx 2 ψ = 0 in terms of Airy functions and solve the Cauchy initial value problem in the coordinate and momentum representations. Particular solutions of the corresponding nonlinear Schrödinger equations with variable coefficients are also found. A special case of the quantum parametric oscillator is studied in detail first. The Green function is explicitly given in terms of Airy functions and the corresponding transition amplitudes are found in terms of a hypergeometric function. The general case of quantum parametric oscillator is considered then in a similar fashion. A group theoretical meaning of the transition amplitudes and their relation with Bargmann's functions is established.
We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a vie... more We discuss the maximum kinematical invariance group of the quantum harmonic oscillator from a viewpoint of the Ermakov-type system. A six parameter family of the square integrable oscillator wave functions, which seems cannot be obtained by the standard separation of variables, is presented as an example. The invariance group of the generalized driven harmonic oscillator is shown to be isomorphic to the corresponding Schrödinger group of the free particle.
Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalizat... more Main properties of the Askey-Wilson polynomials are compactly given on the basis of a generalization of Hahn's approach. In [3], [4], and [16] a number of new systems of orthogonal polynomials (i.e., 4F3 and 4q~3 polynomials) were introduced. The main properties of these polynomials have been established on the basis of the theory of hypergeometric and basic hypergeometric series [9], [11]. As is clear now, the Askey-Wilson [3], [4] and Wilson [16] polynomials are the most general extensions of the Jacobi, Laguerre, and Hermite polynomials known at present. The families of polynomials, considered in [1], [3], [4], and [16], together with their special and limiting cases, form a mathematical entity-the classical orthogonal polynomials of a discrete variable on nonuniform lattices [1], [15]. The introduction of such an object made it possible to generalize all the fundamental properties characteristic of Hahn's approach [12], [10]. We shall disc/ass these properties for the most general system-the Askey-Wilson polynomials, in the framework of an approach developed in [15], [7]. 1. Hypergeometrie-Type Difference Equation Consider the hypergeometric-type difference equation in the self-adjoint form (for details, see [15]) A [cr(z)p(z)~l+2p(z)y(z)=O ' (1) Vxl(z) (2) A[cr(z)p(z)] = p(z)~(z)Vx~(z), on the lattice x(z) = cosh 2o9z = ~q" + q-Z), q = e2~,, where xl(z) = x(z + 89 and AT(z) = Vf(z + 1) = f(z + 1)-f(z).
We derive the recurrence relations for relativistic Coulomb integrals directly from the integral ... more We derive the recurrence relations for relativistic Coulomb integrals directly from the integral representations with the help of computer algebra methods. In order to manage the computational complexity of this problem, we employ holonomic closure properties in a sophisticated way.
We present quadratic dynamical invariant and evaluate Berry's phase for the timedependent Schrödi... more We present quadratic dynamical invariant and evaluate Berry's phase for the timedependent Schrödinger equation with the most general variable quadratic Hamiltonian.
We discuss a new completely integrable case of the time-dependent Schrödinger equation in R n wit... more We discuss a new completely integrable case of the time-dependent Schrödinger equation in R n with variable coefficients for a modified oscillator, which is dual with respect to the time inversion to a model of the quantum oscillator recently considered by Meiler, Cordero-Soto, and Suslov. A second pair of dual Hamiltonians is found in the momentum representation. Our examples show that in mathematical physics and quantum mechanics a change in the direction of time may require a total change of the system dynamics in order to return the system back to its original quantum state. Particular solutions of the corresponding nonlinear Schrödinger equations are obtained. A Hamiltonian structure of the classical integrable problem and its quantization are also discussed.
We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Sch... more We construct an explicit solution of the Cauchy initial value problem for the one-dimensional Schroedinger equation with a time-dependent Hamiltonian operator for the forced harmonic oscillator. The corresponding Green function (propagator) is derived with the help of the generalized Fourier transform and a relation with representations of the Heisenberg-Weyl group N(3) in a certain special case first, and then is extended to the general case. A three parameter extension of the classical Fourier integral is discussed as a by-product. Motion of a particle with a spin in uniform perpendicular magnetic and electric fields is considered as an application; a transition amplitude between Landau levels is evaluated in terms of Charlier polynomials. In addition, we also solve an initial value problem to a similar diffusion-type equation.
We introduce a double sum extension of a very well-poised series and extend to this the transform... more We introduce a double sum extension of a very well-poised series and extend to this the transformations of Bailey and Sears as well as the 6 û 5 summation formula of F. H. Jackson and the q-Dixon sum. We also give q-integral representations of the double sum. Generalizations of the Nassrallah-Rahman integral are also found.
We derive closed formulas for mean values of all powers of r in nonrelativistic and relativistic ... more We derive closed formulas for mean values of all powers of r in nonrelativistic and relativistic Coulomb problems in terms of the Hahn and Chebyshev polynomials of a discrete variable. A short review on special functions and solution of the Coulomb problems in quantum mechanics is given.
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Papers by Sergei Suslov