The Dirac impenetrable barrier in the limit point of the Klein energy zone
We reanalyze the problem of a 1D Dirac single particle colliding with the electrostatic potential step of height V _0 with a positive incoming energy that tends to the limit point of the so-called Klein energy zone, i.e. E → V _0 − m c ^2 , for a given V _0 . In such a case, the particle is actually...
| Main Author: | |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
IOP Publishing
2023-01-01
|
| Series: | Journal of Physics Communications |
| Subjects: | |
| Online Access: | https://doi.org/10.1088/2399-6528/acb8ff |
| _version_ | 1797844573500735488 |
|---|---|
| author | Salvatore De Vincenzo |
| author_facet | Salvatore De Vincenzo |
| author_sort | Salvatore De Vincenzo |
| collection | DOAJ |
| description | We reanalyze the problem of a 1D Dirac single particle colliding with the electrostatic potential step of height V _0 with a positive incoming energy that tends to the limit point of the so-called Klein energy zone, i.e. E → V _0 − m c ^2 , for a given V _0 . In such a case, the particle is actually colliding with an impenetrable barrier. In fact, V _0 → E + m c ^2 , for a given relativistic energy E ( < V _0 ), is the maximum value that the height of the step can reach and that ensures the perfect impenetrability of the barrier. Nevertheless, we note that, unlike the nonrelativistic case, the entire eigensolution does not completely vanish, either at the barrier or in the region under the step, but its upper component does satisfy the Dirichlet boundary condition at the barrier. More importantly, by calculating the mean value of the force exerted by the impenetrable wall on the particle in this eigenstate and taking its nonrelativistic limit, we recover the required result. We use two different approaches to obtain the latter two results. In one of these approaches, the corresponding force on the particle is a type of boundary quantum force. Throughout the article, various issues related to the Klein energy zone, the transmitted solutions to this problem, and impenetrable barriers related to boundary conditions are also discussed. In particular, if the negative-energy transmitted solution is used, the lower component of the scattering solution satisfies the Dirichlet boundary condition at the barrier, but the mean value of the external force when V _0 → E + m c ^2 does not seem to be compatible with the existence of the impenetrable barrier. |
| first_indexed | 2024-04-09T17:24:30Z |
| format | Article |
| id | doaj.art-0c1996df2c734c69a6bc8439ca653e11 |
| institution | Directory Open Access Journal |
| issn | 2399-6528 |
| language | English |
| last_indexed | 2024-04-09T17:24:30Z |
| publishDate | 2023-01-01 |
| publisher | IOP Publishing |
| record_format | Article |
| series | Journal of Physics Communications |
| spelling | doaj.art-0c1996df2c734c69a6bc8439ca653e112023-04-18T14:10:41ZengIOP PublishingJournal of Physics Communications2399-65282023-01-017202500510.1088/2399-6528/acb8ffThe Dirac impenetrable barrier in the limit point of the Klein energy zoneSalvatore De Vincenzo0https://orcid.org/0000-0002-5009-053XThe Institute for Fundamental Study (IF), Naresuan University , Phitsanulok 65000, ThailandWe reanalyze the problem of a 1D Dirac single particle colliding with the electrostatic potential step of height V _0 with a positive incoming energy that tends to the limit point of the so-called Klein energy zone, i.e. E → V _0 − m c ^2 , for a given V _0 . In such a case, the particle is actually colliding with an impenetrable barrier. In fact, V _0 → E + m c ^2 , for a given relativistic energy E ( < V _0 ), is the maximum value that the height of the step can reach and that ensures the perfect impenetrability of the barrier. Nevertheless, we note that, unlike the nonrelativistic case, the entire eigensolution does not completely vanish, either at the barrier or in the region under the step, but its upper component does satisfy the Dirichlet boundary condition at the barrier. More importantly, by calculating the mean value of the force exerted by the impenetrable wall on the particle in this eigenstate and taking its nonrelativistic limit, we recover the required result. We use two different approaches to obtain the latter two results. In one of these approaches, the corresponding force on the particle is a type of boundary quantum force. Throughout the article, various issues related to the Klein energy zone, the transmitted solutions to this problem, and impenetrable barriers related to boundary conditions are also discussed. In particular, if the negative-energy transmitted solution is used, the lower component of the scattering solution satisfies the Dirichlet boundary condition at the barrier, but the mean value of the external force when V _0 → E + m c ^2 does not seem to be compatible with the existence of the impenetrable barrier.https://doi.org/10.1088/2399-6528/acb8ffrelativistic quantum mechanicsDirac equationimpenetrable barrierboundary conditionsforce operator |
| spellingShingle | Salvatore De Vincenzo The Dirac impenetrable barrier in the limit point of the Klein energy zone Journal of Physics Communications relativistic quantum mechanics Dirac equation impenetrable barrier boundary conditions force operator |
| title | The Dirac impenetrable barrier in the limit point of the Klein energy zone |
| title_full | The Dirac impenetrable barrier in the limit point of the Klein energy zone |
| title_fullStr | The Dirac impenetrable barrier in the limit point of the Klein energy zone |
| title_full_unstemmed | The Dirac impenetrable barrier in the limit point of the Klein energy zone |
| title_short | The Dirac impenetrable barrier in the limit point of the Klein energy zone |
| title_sort | dirac impenetrable barrier in the limit point of the klein energy zone |
| topic | relativistic quantum mechanics Dirac equation impenetrable barrier boundary conditions force operator |
| url | https://doi.org/10.1088/2399-6528/acb8ff |
| work_keys_str_mv | AT salvatoredevincenzo thediracimpenetrablebarrierinthelimitpointofthekleinenergyzone AT salvatoredevincenzo diracimpenetrablebarrierinthelimitpointofthekleinenergyzone |