Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential
The solution of a quantum periodic potential in solid state physics is relevant because one can understand how electrons behave in a corresponding crystal. In this paper, the analytical solution of the energy formulation for a one-dimensional periodic potential that meets certain boundary conditions...
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MDPI AG
2024-01-01
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| author | F. Mendoza-Villa Juan A. Ramos-Guivar R. M. Espinoza-Bernardo |
| author_facet | F. Mendoza-Villa Juan A. Ramos-Guivar R. M. Espinoza-Bernardo |
| author_sort | F. Mendoza-Villa |
| collection | DOAJ |
| description | The solution of a quantum periodic potential in solid state physics is relevant because one can understand how electrons behave in a corresponding crystal. In this paper, the analytical solution of the energy formulation for a one-dimensional periodic potential that meets certain boundary conditions is developed in a didactic and detailed way. In turn, the group speed and effective mass are also calculated from the transcendental energy equation of a potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>. From this, a comparison is made with periodic potentials with known analytical solutions, such as the Dirac delta, as well as rectangular and triangular potentials. Finally, some limits are proposed in which these periodic potentials of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> approach the periodic Dirac delta potential of positive intensity. Therefore, the calculations described in this paper can be used as the basis for more-complex one-dimensional potentials and related simulations. |
| first_indexed | 2024-04-24T17:54:27Z |
| format | Article |
| id | doaj.art-5d7437049d1b4cfdbc8b66005ad6a22b |
| institution | Directory Open Access Journal |
| issn | 2624-8174 |
| language | English |
| last_indexed | 2024-04-24T17:54:27Z |
| publishDate | 2024-01-01 |
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| record_format | Article |
| series | Physics |
| spelling | doaj.art-5d7437049d1b4cfdbc8b66005ad6a22b2024-03-27T14:00:23ZengMDPI AGPhysics2624-81742024-01-0161759310.3390/physics6010006Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta PotentialF. Mendoza-Villa0Juan A. Ramos-Guivar1R. M. Espinoza-Bernardo2Grupo de Investigación de Nanotecnología Aplicada para Biorremediación Ambiental, Energía, Biomedicina y Agricultura (NANOTECH), Facultad de Ciencias Físicas, Universidad Nacional Mayor de San Marcos, Av. Venezuela Cdra 34 S/N, Ciudad Universitaria, Lima 15081, PeruGrupo de Investigación de Nanotecnología Aplicada para Biorremediación Ambiental, Energía, Biomedicina y Agricultura (NANOTECH), Facultad de Ciencias Físicas, Universidad Nacional Mayor de San Marcos, Av. Venezuela Cdra 34 S/N, Ciudad Universitaria, Lima 15081, PeruGrupo de Métodos Computacionales Aplicado a Nanomateriales (GMCAN), Facultad de Ciencias Físicas, Universidad Nacional Mayor de San Marcos, P.O. Box 14-0149, Lima 15081, PeruThe solution of a quantum periodic potential in solid state physics is relevant because one can understand how electrons behave in a corresponding crystal. In this paper, the analytical solution of the energy formulation for a one-dimensional periodic potential that meets certain boundary conditions is developed in a didactic and detailed way. In turn, the group speed and effective mass are also calculated from the transcendental energy equation of a potential <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula>. From this, a comparison is made with periodic potentials with known analytical solutions, such as the Dirac delta, as well as rectangular and triangular potentials. Finally, some limits are proposed in which these periodic potentials of the form <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>V</mi><mo>=</mo><mi>V</mi><mo>(</mo><mi>x</mi><mo>)</mo></mrow></semantics></math></inline-formula> approach the periodic Dirac delta potential of positive intensity. Therefore, the calculations described in this paper can be used as the basis for more-complex one-dimensional potentials and related simulations.https://www.mdpi.com/2624-8174/6/1/6Dirac delta potentialone-dimensional periodic potentialcomputational physicsrectangular potentialtriangular potentialasymmetric potential |
| spellingShingle | F. Mendoza-Villa Juan A. Ramos-Guivar R. M. Espinoza-Bernardo Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential Physics Dirac delta potential one-dimensional periodic potential computational physics rectangular potential triangular potential asymmetric potential |
| title | Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential |
| title_full | Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential |
| title_fullStr | Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential |
| title_full_unstemmed | Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential |
| title_short | Generalized One-Dimensional Periodic Potential Wells Tending to the Dirac Delta Potential |
| title_sort | generalized one dimensional periodic potential wells tending to the dirac delta potential |
| topic | Dirac delta potential one-dimensional periodic potential computational physics rectangular potential triangular potential asymmetric potential |
| url | https://www.mdpi.com/2624-8174/6/1/6 |
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