Quantization of the dynamics of a particle on a double cone by preserving Noether symmetries

Abstract

The classical quantization of the motion of a free particle and that of an harmonic oscillator on a double cone are achieved by a quantization scheme [M. C. Nucci, Theor. Math. Phys. 168 (2011) 994], that preserves the Noether point symmetries of the underlying Lagrangian in order to construct the Schrödinger equation. The result is different from that given in [K. Kowalski, J. Rembielński, Ann. Phys. 329 (2013) 146]. A comparison of the different outcomes is provided.Keywords: Lie and Noether symmetries; motion of a particle on a double cone; classical quantization.1 Such as normal-ordering [4,5] and Weyl quantization [6].

Algebraic entropy for systems of quad equations

“…During my Ph.D. I was also left free to have my own scientific collaborations, for instance my ongoing collaboration with Maria Clara Nucci, which resulted in a few more works [30][31][32], or my new collaboration with Davide Chiuchiù [12,27] on a completely different topic than the rest of my Ph.D. thesis.…”

The nonlinear pendulum always oscillates

“…Find the linearizing transformation which does not change the time, as prescribed in nonrelativistic quantum mechanics. 6) with Ψ = Ψ(y, u). We now check the classical consistency of the Schrödinger equation (4.6).…”

“…Then, we determine three Lagrangians that admit the highest number of Noether point symmetries with the help of the Jacobi last multiplier [8]. Then, we recall the quantization method that preserves the Noether point symmetries as described for the first time in [19,20], reformulated in [4] for problems that are linearizable by Lie point symmetries (as in the present case), and successfully applied to various classical problems: second-order Riccati equation [21], dynamics of a charged particle in a uniform magnetic field and a non-isochronous Calogero's goldfish system [20], an equation related to a Calogero's goldfish equation [22], two nonlinear equations somewhat related to the Riemann problem [23], a Liénard I nonlinear oscillator [4], a family of Liénard II nonlinear oscillators [5], N planar rotors and an isochronous Calogero's goldfish system [24], a particle on a double cone [6]. Consequently, as a mathematical divertissement, we quantize the second-order differential equation determining the phase-space trajectories of the nonlinear pendulum.…”