Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics
Two generalizations of Kempf's quadratic canonical commutation relation in one dimension are considered. The first one is the most general quadratic commutation relation. The corresponding nonzero minimal uncertainties in position and momentum are determined and the effect on the energy spectru...
Збережено в:
| Опубліковано в: : | Symmetry, Integrability and Geometry: Methods and Applications |
|---|---|
| Дата: | 2007 |
| Автори: | , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Інститут математики НАН України
2007
|
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/147794 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics / C. Quesne, V.M. Tkachuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 48 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1862739303221690368 |
|---|---|
| author | Quesne, C. Tkachuk, V.M. |
| author_facet | Quesne, C. Tkachuk, V.M. |
| citation_txt | Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics / C. Quesne, V.M. Tkachuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 48 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | Two generalizations of Kempf's quadratic canonical commutation relation in one dimension are considered. The first one is the most general quadratic commutation relation. The corresponding nonzero minimal uncertainties in position and momentum are determined and the effect on the energy spectrum and eigenfunctions of the harmonic oscillator in an electric field is studied. The second extension is a function-dependent generalization of the simplest quadratic commutation relation with only a nonzero minimal uncertainty in position. Such an uncertainty now becomes dependent on the average position. With each function-dependent commutation relation we associate a family of potentials whose spectrum can be exactly determined through supersymmetric quantum mechanical and shape invariance techniques. Some representations of the generalized Heisenberg algebras are proposed in terms of conventional position and momentum operators x, p. The resulting Hamiltonians contain a contribution proportional to p4 and their p-dependent terms may also be functions of x. The theory is illustrated by considering Pöschl-Teller and Morse potentials.
|
| first_indexed | 2025-12-07T20:08:51Z |
| format | Article |
| fulltext | |
| id | nasplib_isofts_kiev_ua-123456789-147794 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T20:08:51Z |
| publishDate | 2007 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Quesne, C. Tkachuk, V.M. 2019-02-16T08:15:13Z 2019-02-16T08:15:13Z 2007 Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics / C. Quesne, V.M. Tkachuk // Symmetry, Integrability and Geometry: Methods and Applications. — 2007. — Т. 3. — Бібліогр.: 48 назв. — англ. 1815-0659 2000 Mathematics Subject Classification: 37N20; 81R15 https://nasplib.isofts.kiev.ua/handle/123456789/147794 Two generalizations of Kempf's quadratic canonical commutation relation in one dimension are considered. The first one is the most general quadratic commutation relation. The corresponding nonzero minimal uncertainties in position and momentum are determined and the effect on the energy spectrum and eigenfunctions of the harmonic oscillator in an electric field is studied. The second extension is a function-dependent generalization of the simplest quadratic commutation relation with only a nonzero minimal uncertainty in position. Such an uncertainty now becomes dependent on the average position. With each function-dependent commutation relation we associate a family of potentials whose spectrum can be exactly determined through supersymmetric quantum mechanical and shape invariance techniques. Some representations of the generalized Heisenberg algebras are proposed in terms of conventional position and momentum operators x, p. The resulting Hamiltonians contain a contribution proportional to p4 and their p-dependent terms may also be functions of x. The theory is illustrated by considering Pöschl-Teller and Morse potentials. V.M.T. thanks the National Fund for Scientific Research (FNRS), Belgium, for financial support. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics Article published earlier |
| spellingShingle | Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics Quesne, C. Tkachuk, V.M. |
| title | Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics |
| title_full | Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics |
| title_fullStr | Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics |
| title_full_unstemmed | Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics |
| title_short | Generalized Deformed Commutation Relations with Nonzero Minimal Uncertainties in Position and/or Momentum and Applications to Quantum Mechanics |
| title_sort | generalized deformed commutation relations with nonzero minimal uncertainties in position and/or momentum and applications to quantum mechanics |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/147794 |
| work_keys_str_mv | AT quesnec generalizeddeformedcommutationrelationswithnonzerominimaluncertaintiesinpositionandormomentumandapplicationstoquantummechanics AT tkachukvm generalizeddeformedcommutationrelationswithnonzerominimaluncertaintiesinpositionandormomentumandapplicationstoquantummechanics |