Abstract
To get around the difficulty that the singular nature of the potential function V(x) = V0/|x|α introduces into physics, the regularization methods are used. However, they affect the singular nature of the problem, and so we discuss here how quantum tunneling behaves if the original singular nature of the Schrodinger equation remains unperturbed. For this purpose, I am starting from the precondition that the singular terms are mutually compensated in the current probability density and the current can be considered continuous. As a result, it is obtained that the mild-singular potential (with 0 < α < 1) has finite but unusual tunnel transparency, in particular, a non-zero value at zero energy of the incident particle. The transparency of a Coulomb potential well oscillates infinitely at zero energy, and only the strongly singular potential (with α > 1) repeats the property of the regularized prototypes: to be impenetrable.
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Funding
This work was financially supported by the Science Committee of the Ministry of Education and Science of the Republic of Armenia within the framework of the Laboratory for Research and Modeling of Quantum Phenomena at Yerevan State University.
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Translated by V. Musakhanyan
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Muradyan, A.Z. On Quantum Tunneling of a Singular Potential. J. Contemp. Phys. 56, 91–97 (2021). https://doi.org/10.3103/S1068337221020122
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DOI: https://doi.org/10.3103/S1068337221020122