Reconstructing the Schrödinger groups

Lopez, Raquel M.; Suslov, Sergeĭ K.; Vega-Guzmán, José

doi:10.1088/0031-8949/87/03/038112

Cited by 18 publications

(46 citation statements)

References 45 publications

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“…At the same time, the probability density |ψ (x, t)| 2 of the solution (1.2) is obviously moving with time, somewhat contradicting to the standard textbooks [21], [23], [38], [47], [55], -an elementary Mathematica simulation reveals such space oscillations for the simplest "dynamic oscillator states" [44], [45] (see Appendix A for the Mathematica source code). The same is true for the probability distribution of the particle linear momentum due to the Heisenberg Uncertainty Principle [27].…”

Section: Discussionmentioning

confidence: 95%

“…These "missing" solutions can be derived analytically in a unified approach to generalized harmonic oscillators (see, for example, [9], [10], [39] and the references therein). They are also verified by a direct substitution with the aid of Mathematica computer algebra system [33], [45], [62]. (The simplest special case µ 0 = β 0 = 1 and α 0 = γ 0 = δ 0 = ε 0 = κ 0 = 0 reproduces the textbook solution obtained by the separation of variables [21], [23], [38], [47]; see also the original Schrödinger paper [55]; and the shape-preserving oscillator evolutions occur when α 0 = 0 and β 0 = 1.…”

Section: −(β(T)x+ε(t))mentioning

confidence: 99%

“…We also provide an interesting computer-animated feature of these solutions -the phase space oscillations of the electron density and the corresponding probability distribution of the particle linear momentum. As a result, a dynamic visualization of the fundamental Heisenberg Uncertainty Principle [27] is given [45], [62]. …”

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confidence: 99%

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On a hidden symmetry of quantum harmonic oscillators

Lopez

Suslov

Vega-Guzmán

2013

Journal of Difference Equations and Applications

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Abstract. 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.The purpose of this Letter is to elaborate on a "missing" class of solutions to the time-dependent Schrödinger equation for the simple harmonic oscillator in one dimension. We also provide an interesting computer-animated feature of these solutions -the phase space oscillations of the electron density and the corresponding probability distribution of the particle linear momentum. As a result, a dynamic visualization of the fundamental Heisenberg Uncertainty Principle [27] is given [45], [62]. Symmetry and Hidden SolutionsThe time-dependent Schrödinger equation for the simple harmonic oscillator,has the following six-parameter family of square integrable solutionswhere H n (x) are the Hermite polynomials [51] and On the other hand, the "dynamic harmonic oscillator states" (1.2)-(1.9) are eigenfunctions,of the time-dependent quadratic invariant,with the required operator identity [15], [54]:(1.12)Here, the time-dependent annihilation a (t) and creation a † (t) operators are explicitly given bywith p = i −1 ∂/∂x in terms of our solutions (1.4)-(1.9). These operators satisfy the canonical commutation relation, 14) and the oscillator-type spectrum (1.10) of the dynamic invariant E can be obtained in a standard way by using the Heisenberg-Weyl algebra of the rasing and lowering operators (a "second quantization" [1], [42], the Fock states): Here, ψ n (x, t) = e i(2n+1)γ(t) Ψ n (x, t) (1.16) is the relation to the wave functions (1.2) with ϕ n (t) = − (2n + 1) [68] and the references therein) have attracted substantial attention over the years because of their great importance in many advanced quantum problems. Examples are coherent and squeezed states, uncertainty relations, Berry's phase, quantization of mechanical systems and Hamiltonian cosmology. More applications include, but are not limited to charged particle traps and motion in uniform magnetic fields, molecular spectroscopy and polyatomic molecules in varying external fields, crystals through which an electron is passing and exciting the oscillator modes, and other mode interactions with external fields. Quadratic Hamiltonians have particular applications in quantum electrodynamics because the electromagnetic field can be represented as a set of generalized driven harmonic oscillators [13], [20].The maximal kinematical invariance group of the simple harmonic oscillator [50] provides the six-parameter family of solutions, namely (1.2) and (1.3)-(1.9), for an arbitrary choice of the initial data (of the corresp...

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Section: Discussionmentioning

confidence: 95%

Section: −(β(T)x+ε(t))mentioning

confidence: 99%

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On a hidden symmetry of quantum harmonic oscillators

Lopez

Suslov

Vega-Guzmán

2013

Journal of Difference Equations and Applications

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“…For the latter case, in [36], [37] and [38], multiparameter solutions in the spirit of Marhic in [30] have been presented. The parameters for the Riccati system arose originally in the process of proving convergence to the initial data for the Cauchy initial value problem Equation (1) with h(t) = 0 and in the process of finding a general solution of a Riccati system [38] and [39].…”

Section: Introductionmentioning

confidence: 99%

“…The parameters for the Riccati system arose originally in the process of proving convergence to the initial data for the Cauchy initial value problem Equation (1) with h(t) = 0 and in the process of finding a general solution of a Riccati system [38] and [39]. In addition, Ermakov systems with solutions containing parameters [36] have been used successfully to construct solutions for the generalized harmonic oscillator with a hidden symmetry [37], and they have also been used to present Galilei transformation, pseudoconformal transformation and others in a unified manner, see [37]. More recently, they have been used in [40] to show spiral and breathing solutions and solutions with bending for the paraxial wave equation.…”

Section: Introductionmentioning

confidence: 99%

On Solutions for Linear and Nonlinear Schrödinger Equations with Variable Coefficients: A Computational Approach

et al. 2016

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Abstract:In this work, after reviewing two different ways to solve Riccati systems, we are able to present an extensive list of families of integrable nonlinear Schrödinger (NLS) equations with variable coefficients. Using Riccati equations and similarity transformations, we are able to reduce them to the standard NLS models. Consequently, we can construct bright-, dark-and Peregrine-type soliton solutions for NLS with variable coefficients. As an important application of solutions for the Riccati equation with parameters, by means of computer algebra systems, it is shown that the parameters change the dynamics of the solutions. Finally, we test numerical approximations for the inhomogeneous paraxial wave equation by the Crank-Nicolson scheme with analytical solutions found using Riccati systems. These solutions include oscillating laser beams and Laguerre and Gaussian beams.

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Time‐dependent photon statistics in variable media

Kryuchkov

Suazo

Suslov

2018

Math Methods in App Sciences

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We find explicit solutions of the Heisenberg equations of motion for a quadratic Hamiltonian, which describes a generic model of variable media in the case of multiparameter squeezed input photon configuration. The corresponding probability amplitudes and photon statistics are also derived in the Schrödinger picture in an abstract operator setting of the quantum electrodynamics; a comparison discussion is made in Heisenberg's picture as well. The unitary transformation and an extension of the squeeze/evolution operator are introduced formally. The time‐dependent photon probability amplitudes with respect to the Fock basis are indeed derived in an operator form. Further, explicit expressions for the matrix elements of the displacement and squeeze operators are derived in terms of hypergeometric functions and solutions of a certain Ermakov‐type system. In the Supporting Information, we provide a computer algebra verification of the derivation of the Ermakov‐system and of the solutions of the Heisenberg equations.

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Reconstructing the Schrödinger groups

Cited by 18 publications

References 45 publications

On a hidden symmetry of quantum harmonic oscillators

On a hidden symmetry of quantum harmonic oscillators

On Solutions for Linear and Nonlinear Schrödinger Equations with Variable Coefficients: A Computational Approach

Time‐dependent photon statistics in variable media

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