Laguerre–Gaussian Wave Propagation in Parabolic Media

We consider canonically conjugated generalized space and linear momentum operatorsx q andp q in quantum mechanics, associated to a generalized translation operator which produces infinitesimal deformed displacements controlled by a deformation parameter q. A canonical transformation (x,p) → (x q ,p q ) leads the Hamiltonian of a position-dependent mass particle in usual space to another Hamiltonian of a particle with constant mass in a conservative force field of the deformed space. The equation of motion for the classical phase space (x, p) may be expressed in terms of the deformed (dual) q-derivative. We revisit the problem of a q-deformed oscillator in both classical and quantum formalisms. Particularly, this canonical transformation leads a particle with positiondependent mass in a harmonic potential to a particle with constant mass in a Morse potential. The trajectories in phase spaces (x, p) and (x q , p q ) are analyzed for different values of the deformation parameter. Lastly, we compare the results of the problem in classical and quantum formalisms through the principle of correspondence and the WKB approximation.

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“…The canonical transformation given by Eqs. ( 16) leads to the new Hamiltonian (see, for instance, [38])…”

Section: Deformed Classical Formalismmentioning

confidence: 99%

A position-dependent mass harmonic oscillator and deformed space

Costa

Borges

2018

Journal of Mathematical Physics

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“…with N n the normalization constant given in (4). Interestingly, the nonstationary eigenfunctions (16) have found applications in wave propagations optical models, where the wave-packets are described by self-focusing Laguerre-Gaussian modes [42].…”

Section: Nonstationary Quantum Invariantmentioning

confidence: 99%

“…It is clear that G(t) = σ −2 (t) uniquely solves (42). With G(t) and ( 40), the time-dependent Hamiltomian Ĥ2 (t) is then given as in (31).…”

Section: A Determining ĥ2 (T)mentioning

confidence: 99%

Nonstationary deformed singular oscillator: quantum invariants and the factorization method

Zelaya

2020

Preprint

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New families of time-dependent potentials related with the stationary singular oscillator are introduced. This is achieved after noticing that a non stationary quantum invariant can be constructed for the singular oscillator. Such invariant depends on coefficients that are related to solutions of an Ermakov equation, the latter becomes essential since it guarantees the regularity of the solutions at each time. In this form, after applying the factorization method to the quantum invariant, rather than the Hamiltonian, one manages to introduce the time parameter into the transformation, leading to factorized operators which are the constants of motion of the new time-dependent potentials. Under the appropriate limit, the initial quantum invariant reduces to the stationary singular oscillator Hamiltonian, in such case, one recovers the families of potentials obtained through the conventional factorization method and previously reported in the literature. In addition, some special limits are discussed such that the singular barrier of the potential vanishes, leading to non-singular time-dependent potentials.

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“…In particular, shape invariance relation in the framework of SUSYQM has been examined and a deformed version of it within PDM was set up [16,17]. Further, invariants were found embracing spectrum generating algebras [18] and consistency of indefinite effective mass was shown to exist [19].…”

Section: Introductionmentioning

confidence: 99%

Dirac Equation with Morse Potential under the Influence of Position-Dependent Mass and Local Fermi Velocity

Bagchi

Ghosh

2023

J. Phys.: Conf. Ser.

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We solve the one-dimensional Dirac equation by taking into account the possibility of position-dependence in the mass function. We also take the Fermi velocity to act as a local variable and examine the combined effects of the two on the solvability of the Dirac equation with respect to the Morse potential. Our results for the wave functions and the energy levels corresponding to such an extended scheme are furnished in closed forms.

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Laguerre–Gaussian Wave Propagation in Parabolic Media

Cited by 8 publications

References 34 publications

A position-dependent mass harmonic oscillator and deformed space

A position-dependent mass harmonic oscillator and deformed space

Nonstationary deformed singular oscillator: quantum invariants and the factorization method

Dirac Equation with Morse Potential under the Influence of Position-Dependent Mass and Local Fermi Velocity

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