Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator

Fernández‐Guasti, M.; Moya‐Cessa, H.

doi:10.1103/physreva.67.063803

Cited by 26 publications

(23 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…I am grateful to Julio Guerrero for pointing me to references [33][34][35] introducing similar concepts as discussed here using abstract conservation laws.…”

Section: Note Added In Proofsmentioning

confidence: 95%

Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes

Steuernagel

2014

Eur. Phys. J. Plus

View full text Add to dashboard Cite

Abstract. In quantum physics the free particle and the harmonically trapped particle are arguably the most important systems a physicist needs to know about. It is little known that, mathematically, they are one and the same. This knowledge helps us to understand either from the viewpoint of the other.Here we show that all general time-dependent solutions of the free-particle Schrödinger equation can be mapped to solutions of the Schrödinger equation for harmonic potentials, both the trapping oscillator and the inverted "oscillator". This map is fully invertible and therefore induces an isomorphism between both types of system, they are equivalent. A composition of the map and its inverse allows us to map from one harmonic oscillator to another with a different spring constant and different center position. The map is independent of the state of the system, consisting only of a coordinate transformation and multiplication by a form factor, and can be chosen such that the state is identical in both systems at one point in time. This transition point in time can be chosen freely, the wave function of the particle evolving in time in one system before the transition point can therefore be linked up smoothly with the wave function for the other system and its future evolution after the transition point. Such a cut-and-paste procedure allows us to describe the instantaneous changes of the environment a particle finds itself in. Transitions from free to trapped systems, between harmonic traps of different spring constants or center positions, or, from harmonic binding to repulsive harmonic potentials are straightforwardly modelled. This includes some time-dependent harmonic potentials. The mappings introduced here are computationally more efficient than either state-projection or harmonic oscillator propagator techniques conventionally employed when describing instantaneous (non-adiabatic) changes of a quantum particle's environment.

show abstract

“…I am grateful to Julio Guerrero for pointing me to references [33][34][35] introducing similar concepts as discussed here using abstract conservation laws.…”

Section: Note Added In Proofsmentioning

confidence: 95%

Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes

Steuernagel

2014

Eur. Phys. J. Plus

View full text Add to dashboard Cite

show abstract

“…E n,l = 2ω 2n + l + 3 2 , n = 0, 1, 2, 3, ..., l = 0, 1, 2, 3, ..., m = −l to l. (15) From these solutions we construct a complete set of squeezed coherent states for the three dimensional system (12). Since the potential under consideration is a spherically symmetric one, we separate radial part from the angular part and investigate each one of them separately.…”

Section: Present Workmentioning

confidence: 99%

Ladder operators and squeezed coherent states of a three-dimensional generalized isotonic nonlinear oscillator

Ruby¹,

Karthiga²,

Senthilvelan³

2012

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Abstract. We explore squeezed coherent states of a 3-dimensional generalized isotonic oscillator whose radial part is the newly introduced generalized isotonic oscillator whose bound state solutions have been shown to admit the recently discovered X 1 -Laguerre polynomials. We construct a complete set of squeezed coherent states of this oscillator by exploring the squeezed coherent states of the radial part and combining the latter with the squeezed coherent states of the angular part. We also prove that the three mode squeezed coherent states resolve the identity operator. We evaluate Mandel's Q-parameter of the obtained states and demonstrate that these states exhibit sub-Possionian and super-Possionian photon statistics. Further, we illustrate the squeezing properties of these states, both in the radial and angular parts, by choosing appropriate observables in the respective parts. We also evaluate Wigner function of these three mode squeezed coherent states and demonstrate squeezing property explicitly.

show abstract

“…5 Another approach has been to build up an arbitrary prole through piecewise integration of simpler proles that can be analytically solved. 6 On the other hand, amplitude and phase methods have been successfully employed in dynamical systems and quantum mechanics 7,8 but are not so widespread in optics and electromagnetic propagation. Iterative methods can be readily implemented to solve the nonlinear amplitude equation.…”

Section: Introductionmentioning

confidence: 99%

Stratified media: nonlinear ODE is better

Fernández‐Guasti

Diamant

2011

SPIE Proceedings

View full text Add to dashboard Cite

Propagation of light in stratied media is described with Maxwell's partial dierential equations (PDE). Separation of variables allow to decouple the linear PDE's to obtain second order non autonomous linear ODEs for the electric and magnetic elds. In the last decades, the problem has been tackled with matrices whose elements are linearly independent solutions of the elds.In our approach, although counter-intuitive, the linear dierential equations are transformed into a non-linear ODE. To this end, the eld is written in terms of amplitude and phase variables. An Ermakov invariant then permits the decoupling of the amplitude and phase nonlinear equations. The amplitude or Milne nonlinear equation is then solved numerically. This method has important advantages: i) initial or nal conditions are easily imposed, ii) important physical quantities such as the reectivity are readily obtained, iii) no further approximations have to be made iv) complex proles can be modeled with arbitrary degree of precision. The abrupt and adiabatic limits are obtained but most importantly, intermediate more realistic cases can also be tackled, for example, adsorption between thin lm layers. Novel eects are addressed such as enhanced reectivity at derivative discontinuities where the refractive index is continuous.

show abstract

Amplitude and phase representation of quantum invariants for the time-dependent harmonic oscillator

Cited by 26 publications

References 19 publications

Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes

Equivalence between free quantum particles and those in harmonic potentials and its application to instantaneous changes

Ladder operators and squeezed coherent states of a three-dimensional generalized isotonic nonlinear oscillator

Stratified media: nonlinear ODE is better

Contact Info

Product

Resources

About