Quantum integrals of motion for variable quadratic Hamiltonians

Cordero-Soto, Ricardo; Suazo, Erwin; Suslov, Sergeĭ K.

doi:10.1016/j.aop.2010.02.020

Cited by 53 publications

(122 citation statements)

References 133 publications

Supporting

Mentioning

115

Contrasting

Order By: Relevance

“…Quantum systems with quadratic Hamiltonians (see, for example, [2], [5], [6], [10], [14], [15], [18], [20], [26], [46], [64], [65], [66], [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.…”

Section: Discussionmentioning

confidence: 99%

“…[54] (see also [10], [15], [59] and the references therein for important special cases). An application to the electromagnetic-field quantization and a generalization of the coherent states are discussed in Refs.…”

Section: −(β(T)x+ε(t))mentioning

confidence: 99%

“…On the other hand, the "dynamic harmonic oscillator states" (1.2)-(1.9) are eigenfunctions, 10) of the time-dependent quadratic invariant,…”

Section: −(β(T)x+ε(t))mentioning

confidence: 99%

“…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].…”

Section: −(β(T)x+ε(t))mentioning

confidence: 99%

See 3 more Smart Citations

On a hidden symmetry of quantum harmonic oscillators

Lopez

Suslov

Vega-Guzmán

2013

Journal of Difference Equations and Applications

Self Cite

View full text Add to dashboard Cite

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...

show abstract

Section: Discussionmentioning

confidence: 99%

Section: −(β(T)x+ε(t))mentioning

confidence: 99%

“…On the other hand, the "dynamic harmonic oscillator states" (1.2)-(1.9) are eigenfunctions, 10) of the time-dependent quadratic invariant,…”

Section: −(β(T)x+ε(t))mentioning

confidence: 99%

Section: −(β(T)x+ε(t))mentioning

confidence: 99%

See 2 more Smart Citations

On a hidden symmetry of quantum harmonic oscillators

Lopez

Suslov

Vega-Guzmán

2013

Journal of Difference Equations and Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…A goal of this paper is to make a modest step in this direction (see also [32,53] and the references therein). We use explicit solutions from recent papers on variable quadratic Hamiltonians in nonrelativistic quantum mechanics [49,[54][55][56][57][58][59][60][61] to describe steady-state and transient solutions to linear cable equations derived for membrane compartments with a non-necessarily constant or monotonically changing radius and propose, en passage, a new hyperbolic representation for the neurite compartments.…”

Section: Introductionmentioning

confidence: 99%

A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter

2014

View full text Add to dashboard Cite

Tree-like structures are ubiquitous in nature. In particular, neuronal axons and dendrites have tree-like geometries that mediate electrical signaling within and between cells. Electrical activity in neuronal trees is typically modeled using coupled cable equations on multi-compartment representations, where each compartment represents a small segment of the neuronal membrane. The geometry of each compartment is usually defined as a cylinder or, at best, a surface of revolution based on a linear approximation of the radial change in the neurite. The resulting geometry of the model neuron is coarse, with non-smooth or even discontinuous jumps at the boundaries between compartments. We propose a hyperbolic approximation to model the geometry of neurite compartments, a branched, multi-compartment extension, and a simple graphical approach to calculate steady-state solutions of an associated system of coupled cable equations. A simple case of transient solutions is also briefly discussed.

show abstract

Time‐dependent photon statistics in variable media

Kryuchkov

Suazo

Suslov

2018

Math Methods in App Sciences

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Quantum integrals of motion for variable quadratic Hamiltonians

Cited by 53 publications

References 133 publications

On a hidden symmetry of quantum harmonic oscillators

On a hidden symmetry of quantum harmonic oscillators

A Graphical Approach to a Model of a Neuronal Tree with a Variable Diameter

Time‐dependent photon statistics in variable media

Contact Info

Product

Resources

About