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| Main Author: | |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2411.04338 |
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Table of Contents:
- In this comment, we obtain the complete energy levels for Dvornikov's paper, that is, the energy levels dependent on two quantum numbers, namely, the radial quantum number (given by $N$) and the angular quantum number (given by $J_z$). In particular, what motivated us to do this was the fact that the quantized energy levels for particles (fermions or bosons) in polar, cylindrical, or spherical coordinates depend on two quantum numbers: a radial quantum number and an angular quantum number. From this, the following question/doubt arose: Why do the energy levels in Dvornikov's paper only depend on one quantum number? That is, Where did the angular quantum number given by $J_z$ go? So, using Studenikin's paper as a starting point (as well as others in the literature), we write one of the equations from Dvornikov's paper in matrix form. Next, we use the four-component Dirac spinor and obtain a set/system of four coupled first-order differential equations. From the first two equations with $m\to 0$ (massless neutrino or ultrarelativistic regime), we obtain a (compact) second-order differential equation for the last two spinor components. So, solving this equation, we obtain the neutrino energy levels, which explicitly depend on both $N$ and $J_z$. Finally, we note that for $J_z>0$ (positive angular momentum) with $u=+1$ (component $ψ_3$), we obtain exactly the particular energy levels of Dvornikov's paper.