# Supersymmetric Quantum Mechanics and Solvable Models

^{1}

^{2}

^{*}

**Classification:**PACS 03.65.-w, 47.10.-g, 11.30.Pb

## 1. Introduction

## 2. Supersymmetric Quantum Mechanics

_{±}are products of the operator and its adjoint , their eigenvalues are either zero or positive [13]. The ground state eigenvalue of one of these Hamiltonians must be zero in order to have unbroken supersymmetry. Without loss of generality, we choose that Hamiltonian to be . Thus, we have

#### 2.1. Example

## 3. Shape Invariance in Supersymmetric Quantum Mechanics

_{±}, the shape invariance condition becomes

#### 3.1. Determination of Eigenvalues

#### 3.2 Determination of Eigenfunctions

## 4. Shape Invariance and Potential Algebra

#### 4.1. Building the Algebra

#### 4.2. Obtaining the Energy Spectrum from Algebra Representations

**Figure 1.**Generic behaviors of . Case (

**a**) corresponds to a finite, and (

**b**) to an infinite representation of the potential algebra.

## 5. How Do We Find Additive Shape Invariant Superpotentials?

#### 5.1. Known -Independent Shape Invariant Superpotentials

Name | Superpotential |
---|---|

Harmonic Oscillator | |

Coulomb | |

-D oscillator | |

Morse | |

Rosen-Morse I | |

Rosen-Morse II | |

Eckart | |

Scarf I | |

Scarf II | |

Gen. Pöschl-Teller |

#### 5.2. New Proof of Completeness of the Conventional Shape-Invariant Superpotentials

#### 5.2.1. Case 1: X1 Is a Constant and

#### 5.2.2. Case 2: Is Constant

#### 5.2.3. Case 3: and Are not Constant, but and

#### 5.3. -Dependent Superpotentials

## 6. Summary and Conclusions

## Acknowledgements

## References and Notes

- Darboux, G. Leçons sur la Théorie Général des Surfaces, 2nd ed; Gauthier-Villars: Paris, France, 1912. [Google Scholar]
- Schrödinger, E. A method of determining quantum-mechanical eigenvalues and eigenfunctions. Proc. R. Ir. Acad.
**1940**, A46, 9–16. [Google Scholar] - Schrödinger, E. Further studies on solving eigenvalue problems by factorization. Proc. R. Ir. Acad.
**1941**, A46, 183–206. [Google Scholar] - Schrödinger, E. The factorization of the hypergeometric equation. Proc. R. Ir. Acad.
**1941**, A47, 53–54. [Google Scholar] - Infeld, L.; Hull, T.E. The factorization method. Rev. Mod. Phys.
**1951**, 23, 21–68. [Google Scholar] - Witten, E. Dynamical breaking of supersymmetry. Nucl. Phys.
**1981**, B185, 513–554. [Google Scholar] - Solomonson, P.; Van Holten, J.W. Fermionic coordinates and supersymmetry in quantum mechanics. Nucl. Phys.
**1982**, B196, 509–531. [Google Scholar] - Cooper, F.; Freedman, B. Aspects of supersymmetric quantum mechanics. Ann. Phys.
**1983**, 146, 262–288. [Google Scholar] - Note the constant has been added to the usual harmonic oscillator potential to insure that the groundstate energy of the system remains at zero. This constant allows us to factorize the Hamiltonian as a product of operators and .
- Cooper, F.; Khare, A.; Sukhatme, U. Supersymmetry in Quantum Mechanics; World Scientific: Singapore, 2001. [Google Scholar]
- Gangopadhyaya, A.; Mallow, J.; Rasinariu, C. Supersymmetric Quantum Mechanics: An Introduction; World Scientific: Singapore, 2010. [Google Scholar]
- Thus, is an eigenstate of with an eigenvalue .
- .
- Bender, C.M.; Boettcher, S. Real spectra in non-hermitian hamiltonians having PT symmetry. Phys. Rev. Lett.
**1998**, 80, 5243–5246. [Google Scholar] - Bender, C.M.; Brody, D.C.; Jones, H.F. Complex extension of quantum mechanics. Phys. Rev. Lett.
**2002**, 89, 270401–1. [Google Scholar] - Bender, C.M.; Berry, M.V.; Mandilara, A. Generalized PT symmetry and real spectra. J. Phys. A
**2002**, 35, L467–L471. [Google Scholar] - Znojil, M. SI potentials with PT symmetry. J. Phys. A
**2000**, 33, L61–L62. [Google Scholar] - Levai, Z. Exact analytic study of the PT-symmetry-breaking mechanism. Czech. J. Phys.
**2004**, 54, 77–84. [Google Scholar] - Znojil, M. Matching method and exact solvability of discrete Pt-symmetric square wells. J. Phys. A
**2006**, 39, 10247–10261. [Google Scholar] - Quesne, C.; Bagchi, B.; Mallik, S.; Bila, H.; Jakubsky, V. PT supersymmetric partner of a short-range square well. Czech. J. Phys.
**2005**, 55, 1161–1166. [Google Scholar] - Bagchi, B.; Quesne, C.; Roychoudhury, R. Isospectrality of conventional and new extended potentials, second-order supersymmetry and role of PT Symmetry. Pramana
**2009**, 73, 337–347. [Google Scholar] - Miller, W., Jr. Lie Theory and Special Functions (Mathematics in Science and Engineering); Academic Press: New York,NY,USA, 1968. [Google Scholar]
- Gendenshtein, L.E. Derivation of exact spectra of the schrodinger equation by means of supersymmetry. JETP Lett.
**1983**, 38, 356–359. [Google Scholar] - Gendenshtein, L.E.; Krive, I.V. Supersymmetry in quantum mechanics. Sov. Phys. Usp.
**1985**, 28, 645–666. [Google Scholar] - Barclay, D.; Dutt, R.; Gangopadhyaya, A.; Khare, A.; Pagnamenta, A.; Sukhatme, U. New exactly solvable hamiltonians: Shape invariance and self-similarity. Phys. Rev. A
**1993**, 48, 2786–2797. [Google Scholar] - Spiridonov, V.P. Exactly solvable potentials and quantum algebras. Phys. Rev. Lett.
**1992**, 69, 398–401. [Google Scholar] - Sukhatme, U.P.; Rasinariu, C.; Khare, A. Cyclic shape invariant potentials. Phys. Lett.
**1997**, A234, 401–409. [Google Scholar] - Gangopadhyaya, A.; Mallow, J.V.; Sukhatme, U.P. Supersymmetry and Integrable Models:. In Proceedings of Workshop on Supersymmetry and Integrable Models; Aratyn, H., Imbo, T.D., Keung, W.-Y., Sukhatme, U., Eds.; Springer-Verlag.
- Balantekin, A.B. Algebraic approach to shape invariance. Phys. Rev. A
**1998**, 57, 4188–4191. [Google Scholar] - Gangopadhyaya, A.; Mallow, J.V.; Sukhatme, U.P. Translational shape invariance and the inherent potential algebra. Phys. Rev. A
**1998**, 58, 4287–4292. [Google Scholar] - Chaturvedi, S.; Dutt, R.; Gangopadhyaya, A.; Panigrahi, P.; Rasinariu, C.; Sukhatme, U. Algebraic shape invariant models. Phys. Lett.
**1998**, A248, 109–113. [Google Scholar] - Balantekin, A.; Candido Ribeiro, M.; Aleixo, A. Algebraic nature of shape-invariant and self-similar potentials. J. Phys. A
**1999**, 32, 2785–2790. [Google Scholar] - We assume that as , the supersymmetry remains unbroken.
- Dutt, R.; Khare, A.; Sukhatme, U. Supersymmetry, shape invariance and exactly solvable potentials. Am. J. Phys.
**1998**, 56, 163. [Google Scholar] - Cooper, F.; Ginocchio, J.; Khare, A. Relationship between supersymmetry and solvable potentials. Phys. Rev. D
**1987**, 36, 2458. [Google Scholar] - In the last line we have used the fact that . This implies that for any analytical function , we have .
- Veselov, A.P.; Shabat, A.B. Dressing chains and spectral theory of the Schrödinger operator. Funct. Anal. Appl.
**1993**, 27, 81–96. [Google Scholar] - These constraints are: .
- We have used .
- Dutt, R.; Gangopadhyaya, A.; Rasinariu, C.; Sukhatme, U. Coordinate realizations of deformed Lie algebras with three generators. Phys. Rev. A
**1999**, 60, 3482–3486. [Google Scholar] [CrossRef] - Rocek, M. Representation theory of the nonlinear SU (2) algebra. Phys. Lett. B
**1991**, 255, 554–557. [Google Scholar] - Adams, B.G.; Cizeka, J.; Paldus, J. Lie algebraic methods and their applications to simple quantum systems. Advances in Quantum Chemistry, 19th ed; Academic Press: New York,NY,USA, 1987. [Google Scholar]
- In some cases these are additive constants and subtracted away. If we provide a common floor to all potentials, demanding that their groundstate energies be zero, we will find that all known solvable potentials pick up a dependent term.
- Shabat, A. The infinite-dimensional dressing dynamical system. Inverse Probl.
**1992**, 8, 303–308. [Google Scholar] - Gangopadhyaya, A.; Mallow, J.V. Generating shape invariant potentials. Int. J. Mod. Phys. A
**2008**, 23, 4959–4978. [Google Scholar] - Bougie, J.; Gangopadhyaya, A.; Mallow, J.V. Generation of a complete set of additive shape-invariant potentials from an euler equation. Phys. Rev. Lett.
**2010**, 210402–1. [Google Scholar] - Bougie, J.; Gangopadhyaya, A.; Mallow, J.V. Method for generating additive shape invariant potentials from an euler equation. J. Phys. A
**2012**, 44, 275307–1. [Google Scholar] - Normalizability of the groundstate requires that be greater than zero. Since an increase in decreases , there can only be a finite number of increases.
- By substituting into Equation (50) we find that shape invariance requires that .
- Quesne, C. Exceptional orthogonal polynomials, exactly solvable potentials and supersymmetry. J. Phys. A
**2008**, 41, 392001–1. [Google Scholar] - Quesne, C. Solvable rational potentials and exceptional orthogonal polynomials in supersymmetric quantum mechanics. Sigma
**2009**, 5, 084–1. [Google Scholar] - Odake, S.; Sasaki, R. Infinitely many shape invariant discrete quantum mechanical systems and new exceptional orthogonal polynomials related to the wilson and Askey-Wilson polynomials. Phys. Lett. B
**2009**, 682, 130–136. [Google Scholar] - Odake, S.; Sasaki, R. Another set of infinitely many exceptional (Xl) laguerre polynomials. Phys. Lett. B
**2010**, 684, 173–176. [Google Scholar] - Tanaka, T. N-fold supersymmetry and quasi-solvability associated with X-2-laguerre polynomials. J. Math. Phys.
**2010**, 51, 032101–1. [Google Scholar] - Sree Ranjani, S.; Panigrahi, P.; Khare, A.; Kapoor, A.; Gangopadhyaya, A. Exceptional orthogonal polynomials, QHJ formalism and SWKB quantization condition. J. Phys. A
**2012**, 055210–1. [Google Scholar] - Shiv Chaitanya, K.; Sree Ranjani, S.; Panigrahi, P.; Radhakrishnan, R.; Srinivasan, V. Exceptional polynomials and SUSY quantum mechanics. Available online: http://arxiv.org/pdf/1110.3738.pdf (accessed on 2 August 2012).

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Bougie, J.; Gangopadhyaya, A.; Mallow, J.; Rasinariu, C.
Supersymmetric Quantum Mechanics and Solvable Models. *Symmetry* **2012**, *4*, 452-473.
https://doi.org/10.3390/sym4030452

**AMA Style**

Bougie J, Gangopadhyaya A, Mallow J, Rasinariu C.
Supersymmetric Quantum Mechanics and Solvable Models. *Symmetry*. 2012; 4(3):452-473.
https://doi.org/10.3390/sym4030452

**Chicago/Turabian Style**

Bougie, Jonathan, Asim Gangopadhyaya, Jeffry Mallow, and Constantin Rasinariu.
2012. "Supersymmetric Quantum Mechanics and Solvable Models" *Symmetry* 4, no. 3: 452-473.
https://doi.org/10.3390/sym4030452