Harmonic states for the free particle

Guerrero, Julio Becerra; López-Ruiz, Francisco F.; Aldaya, V.; Cossío, Francisco González de

doi:10.1088/1751-8113/44/44/445307

Cited by 48 publications

(59 citation statements)

References 39 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…In the case Ω = 0, the expressions (38) coincide with the creation and annihilation operators reported by Nienhuis and Allen for the HG modes in free space [25]. Those operators have also appeared as the conserved creation and annihilation operators of the harmonic states for the free particle [37], and as generalized (invariant) ladder operators of the parametric harmonic oscillator [38].…”

Section: Ladder Operators Of Hermite-gaussian Modessupporting

confidence: 72%

Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium

Cruz

Gress

2017

Annals of Physics

View full text Add to dashboard Cite

h i g h l i g h t s• Ladder and shift operators for the Hermite-Gaussian modes are constructed.• These operators are used to obtain the generators of the dynamical algebras.• The Barut-Girardello and Perelomov coherent states are determined.• The uncertainty relations are considered in connection to the beam quality factor. a r t i c l e i n f o A group-theoretical approach to the paraxial propagation of Hermite-Gaussian modes based on the factorization method is presented. It is shown that the su(1, 1) and the su(2) algebras generate the spectrum of propagation constants at any fixed transversal plane. The complete set of HG modes is decomposed into hierarchies that are used to establish the representation spaces of SU(1, 1) and SU(2). The corresponding families of generalized coherent states are constructed and the variances of the quadratures and canonical variables are determined.

show abstract

Section: Ladder Operators Of Hermite-gaussian Modessupporting

confidence: 72%

Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium

Cruz

Gress

2017

Annals of Physics

View full text Add to dashboard Cite

show abstract

“…Describing these attempts, we have to cite Refs. [5,6,7,8] devoted to this problem. In our opinion, no single one of these studies completely solves the problem under consideration.…”

Section: Introductionmentioning

confidence: 99%

Coherent and semiclassical states of a free particle

Багров¹,

Гитман²,

Pereira³

2014

Phys.-Usp.

View full text Add to dashboard Cite

Coherent states (CS) were first introduced and studied in detail for bound motion and discrete-spectrum systems like harmonic oscillators and similar systems with a quadratic Hamiltonian. However, the problem of constructing CS has still not been investigated in detail for the simplest and physically important case of a free particle, for which, besides being physically important, the CS problem is of didactic value in teaching quantum mechanics, with the CS regarded as examples of wave packets representing semiclassical motion. In this paper, we essentially follow the Malkin-Dodonov-Man'ko method to construct the CS of a free nonrelativistic particle. We give a detailed discussion of the properties of the CS obtained, in particular, the completeness relations, the minimization of uncertainty relations, and the evolution of the corresponding probability density. We describe the physical conditions under which free-particle CS can be considered semiclassical states.

show abstract

“…One may imagine other possible applications, for example, in molecular spectroscopy [46], theory of crystals, quantum optics [24], [66], and cavity quantum electrodynamics [12], [13], [22], [35], [67]. We believe in a dynamic character of the nature [28].…”

Section: Discussionmentioning

confidence: 99%

“…as presented in the textbooks [21], [23], [24], [29], [38], [47]. The dependence on the quantum number n, which disappears from the Ehrenfest theorem [16], [27], is coming back at the level of the higher moments of the distribution.…”

Section: Examplementioning

confidence: 99%

On a hidden symmetry of quantum harmonic oscillators

Lopez

Suslov

Vega-Guzmán

2013

Journal of Difference Equations and Applications

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

Harmonic states for the free particle

Cited by 48 publications

References 39 publications

Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium

Group approach to the paraxial propagation of Hermite–Gaussian modes in a parabolic medium

Coherent and semiclassical states of a free particle

On a hidden symmetry of quantum harmonic oscillators

Contact Info

Product

Resources

About