http://cds.cern.ch/record/262851/files/9405029.pdf

Supersymmetry and quantum mechanics

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An elementary introduction is given to the subject of supersymmetry in quantum mechanics which can be understood and appreciated by any one who has taken a first course in quantum mechanics. We demonstrate with explicit examples that given a solvable problem in quantum mechanics with n bound states, one can construct n new exactly solvable Hamiltonians having n − 1, n − 2,…, 0 bound states. The relationship between the eigenvalues, eigenfunctions and scattering matrix of the supersymmetric partner potentials is derived and a class of reflectionless potentials are explicitly constructed. We extend the operator method of solving the one‐dimensional harmonic oscillator problem to a class of potentials called shape‐invariant potentials. It is worth emphasizing that this class includes almost all the solvable problems that are found in the standard text books on quantum mechanics. Further, we show that given any potential with at least one bound state, one can very easily construct one continuous parameter family of potentials having same eigenvalues and s‐matrix. The supersymmetry inspired WKB approximation (SWKB) is also discussed and it is shown that unlike the usual WKB, the lowest order SWKB approximation is exact for the shape‐invariant potentials and further, this approximation is not only exact for large quantum numbers but by construction, it is also exact for the ground state. Finally, we also construct new exactly solvable periodic potentials by using the machinery of supersymmetric quantum mechanics.

Supersymmetry in quantum mechanics

It is well known that the harmonic oscillator potential can be solved by using raising and lowering operators. This operator method can be generalized with the help of supersymmetry and the concept of ‘‘shape‐invariant’’ potentials. This generalization allows one to calculate the energy eigenvalues and eigenfunctions of essentially all known exactly solvable potentials in a simple and elegant manner.

Supersymmetry, Shape Invariance and Exactly Solvable Potentials

The authors give explicit point canonical transformations which map twelve types of shape invariant potentials (which are known to be exactly solvable) into two potential classes. The eigenfunctions in these two classes are given by hypergeometric and confluent hypergeometric functions respectively.

https://iopscience.iop.org/article/10.1088/0305-4470/25/13/013/pdf

Mapping of shape invariant potentials under point canonical transformations

Exactness of supersymmetric WKB spectra for shape-invariant potentials

We discuss in some detail the self-similar potentials of Shabat [Inverse Prob. 8, 303 (1992)] and Spiridonov [Phys. Rev. Lett. 69, 298 (1992)] which are reflectionless and have an infinite number of bound states. We demonstrate that these self-similar potentials are in fact shape-invariant potentials within the formalism of supersymmetric quantum mechanics. In particular, using a scaling Ansatz for the change of parameters, we obtain a large class of new, reflectionless, shape-invariant potentials of which the Shabat-Spiridonov ones are a special case. These new potentials can be viewed as q deformations of the single-soliton solution corresponding to the Rosen-Morse potential. Explicit expressions for the energy eigenvalues, eigenfunctions, and transmission coefficients for these potentials are obtained. We show that these potentials can also be obtained numerically. Included as an intriguing case is a shape-invariant double-well potential whose supersymmetric partner potential is only a single well. Our class of exactly solvable Hamiltonians is further enlarged by examining two new directions: (i) changes of parameters which are different from the previously studied cases of translation and scaling and (ii) extending the usual concept of shape invariance in one step to a multistep situation. These extensions can be viewed as q deformations of the harmonic oscillator or multisoliton solutions corresponding to the Rosen-Morse potential.

/pdf/new-exactly-solvable-hamiltonians-shape-invariance-and-self-59bh51cvkm.pdf

New exactly solvable Hamiltonians: Shape invariance and self-similarity.

Motivated by an idea of Dutra (1993), we obtain a new class of one-dimensional conditionally exactly solvable potentials for which the entire spectra can be obtained in an algebraic manner provided one of the potential parameters is assigned a fixed negative value. It is shown that using shape-invariant potentials as input, one may generate different classes of such potentials even in more than one dimension. We also illustrate that WKB and supersymmetry inspired WKB methods provide very good approximations for these potentials with the latter doing comparatively better.

Ranabir Dutt

Papers

Supersymmetry, Shape Invariance and Exactly Solvable Potentials

Mapping of shape invariant potentials under point canonical transformations

Exactness of supersymmetric WKB spectra for shape-invariant potentials

New exactly solvable Hamiltonians: Shape invariance and self-similarity.

New class of conditionally exactly solvable potentials in quantum mechanics