Quantization of the Bateman damping system with conformable derivative
- URL: http://arxiv.org/abs/2301.02769v1
- Date: Sat, 7 Jan 2023 02:37:30 GMT
- Title: Quantization of the Bateman damping system with conformable derivative
- Authors: Tariq AlBanwa, Ahmed Al-Jamel, Eqab.M.Rabei and Mohamed.Al-Masaeed
- Abstract summary: The conformable Bateman Lagrangian for the damped harmonic oscillator system is proposed.
The corresponding conformable-Lagrange equations of motion and fractional Hamiltonian are then obtained.
It is found that the energy eigenvalues are real and there are sort of gradual ordering in the behavior of the probability densities.
- Score: 0.0
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- Abstract: In this work, the conformable Bateman Lagrangian for the damped harmonic
oscillator system is proposed using the conformable derivative concept. In
other words, the integer derivatives are replaced by conformable derivatives of
order $\alpha$ with $0<\alpha\leq 1$. The corresponding conformable
Euler-Lagrange equations of motion and fractional Hamiltonian are then
obtained. The system is then canonically quantized and the conformable
Schrodinger equation is constructed. The fractional-order dependence of the
energy eigenvalues $E_n ^\alpha$ and eigenfunctions $\psi_n ^\alpha$ are
obtained using using suitable transformations and the extended fractional
Nikiforov-Uvarov method. The corresponding conformable continuity equation is
also derived and the probability density and probability current are thus
suitably defined. The probability density evolution as well as its dependence
on $\alpha$ is plotted and analyzed for various situations. It is found that
the energy eigenvalues are real and there are sort of gradual ordering in the
behavior of the probability densities.
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