Shape Invariant Potentials in Supersymmetric Quantum Cosmology
In this brief review, we comment on the concept of shape invariant potentials, which is an essential feature in many settings of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-06-01
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| Series: | Universe |
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| Online Access: | https://www.mdpi.com/2218-1997/8/6/316 |
| _version_ | 1797481671739572224 |
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| author | Shahram Jalalzadeh Seyed Meraj M. Rasouli Paulo Moniz |
| author_facet | Shahram Jalalzadeh Seyed Meraj M. Rasouli Paulo Moniz |
| author_sort | Shahram Jalalzadeh |
| collection | DOAJ |
| description | In this brief review, we comment on the concept of shape invariant potentials, which is an essential feature in many settings of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> supersymmetric quantum mechanics. To motivate its application within supersymmetric quantum cosmology, we present a case study to illustrate the value of this promising tool. Concretely, we take a spatially flat FRW model in the presence of a single scalar field, minimally coupled to gravity. Then, we extract the associated Schrödinger–Wheeler–DeWitt equation, allowing for a particular scope of factor ordering. Subsequently, we compute the corresponding supersymmetric partner Hamiltonians, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>2</mn></msub></semantics></math></inline-formula>. Moreover, we point out how the shape invariance property can be employed to bring a relation among several factor orderings choices for our Schrödinger–Wheeler–DeWitt equation. The ground state is retrieved, and the excited states easily written. Finally, the Hamiltonians, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>2</mn></msub></semantics></math></inline-formula>, are explicitly presented within a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> supersymmetric quantum mechanics framework. |
| first_indexed | 2024-03-09T22:17:52Z |
| format | Article |
| id | doaj.art-10074d4edbe342aaa188919a5a58abc3 |
| institution | Directory Open Access Journal |
| issn | 2218-1997 |
| language | English |
| last_indexed | 2024-03-09T22:17:52Z |
| publishDate | 2022-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Universe |
| spelling | doaj.art-10074d4edbe342aaa188919a5a58abc32023-11-23T19:19:10ZengMDPI AGUniverse2218-19972022-06-018631610.3390/universe8060316Shape Invariant Potentials in Supersymmetric Quantum CosmologyShahram Jalalzadeh0Seyed Meraj M. Rasouli1Paulo Moniz2Departamento de Física, Universidade Federal de Pernambuco, Recife 52171-900, PE, BrazilDepartamento de Física, Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, Rua Marquês d’Avila e Bolama, 6200-001 Covilhã, PortugalDepartamento de Física, Centro de Matemática e Aplicações (CMA-UBI), Universidade da Beira Interior, Rua Marquês d’Avila e Bolama, 6200-001 Covilhã, PortugalIn this brief review, we comment on the concept of shape invariant potentials, which is an essential feature in many settings of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> supersymmetric quantum mechanics. To motivate its application within supersymmetric quantum cosmology, we present a case study to illustrate the value of this promising tool. Concretely, we take a spatially flat FRW model in the presence of a single scalar field, minimally coupled to gravity. Then, we extract the associated Schrödinger–Wheeler–DeWitt equation, allowing for a particular scope of factor ordering. Subsequently, we compute the corresponding supersymmetric partner Hamiltonians, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>2</mn></msub></semantics></math></inline-formula>. Moreover, we point out how the shape invariance property can be employed to bring a relation among several factor orderings choices for our Schrödinger–Wheeler–DeWitt equation. The ground state is retrieved, and the excited states easily written. Finally, the Hamiltonians, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>1</mn></msub></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>H</mi><mn>2</mn></msub></semantics></math></inline-formula>, are explicitly presented within a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>=</mo><mn>2</mn></mrow></semantics></math></inline-formula> supersymmetric quantum mechanics framework.https://www.mdpi.com/2218-1997/8/6/316supersymmetric quantum mechanicsshape invariant potentialssupersymmetric quantum cosmology |
| spellingShingle | Shahram Jalalzadeh Seyed Meraj M. Rasouli Paulo Moniz Shape Invariant Potentials in Supersymmetric Quantum Cosmology Universe supersymmetric quantum mechanics shape invariant potentials supersymmetric quantum cosmology |
| title | Shape Invariant Potentials in Supersymmetric Quantum Cosmology |
| title_full | Shape Invariant Potentials in Supersymmetric Quantum Cosmology |
| title_fullStr | Shape Invariant Potentials in Supersymmetric Quantum Cosmology |
| title_full_unstemmed | Shape Invariant Potentials in Supersymmetric Quantum Cosmology |
| title_short | Shape Invariant Potentials in Supersymmetric Quantum Cosmology |
| title_sort | shape invariant potentials in supersymmetric quantum cosmology |
| topic | supersymmetric quantum mechanics shape invariant potentials supersymmetric quantum cosmology |
| url | https://www.mdpi.com/2218-1997/8/6/316 |
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