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Supersymmetric Aspects of One Dimensional Quantum Mechanics

Edilson Ferreira Batista, Ronaldo Thibes

Published Jun 30, 2013• 0 citations
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Abstract

We review SUSY in non-relativistic quantum mechanics as a powerful tool for studying one dimensional potentials and discuss some applications. The Dirac operatorial factorization method for the harmonic oscillator in QM is generalized to a broader scope. A Schr¨odinger second order operator can be factorized into two first-order ones by solving a Riccati equation and determining the corresponding superpotential. We apply the technique to square well, P¨oschl-Teller and finite barrier potentials, analyzing the resulting energy spectra and the transmission and reflection coefficients. Both discrete and continuous energy spectrum cases are discussed.

DOI
10.22481/exon.v4i1.19473

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We review SUSY in non-relativistic quantum mechanics as a powerful tool for studying one dimensional potentials and discuss some applications. The Dirac operatorial factorization method for the harmonic oscillator in QM is generalized to a broader scope. A Schr¨odinger second order operator can be factorized into two first-order ones by solving a Riccati equation and determining the corresponding superpotential. We apply the technique to square well, P¨oschl-Teller and finite barrier potentials, analyzing the resulting energy spectra and the transmission and reflection coefficients. Both discrete and continuous energy spectrum cases are discussed.
Starting from the operator factorization in Quantum Mechanics and combined with the shape invariance of superpotential, the supersymmetric operator method can be used to exactly solve the energies and wavefunctions of bound states in a one dimensional potential well, Scarf(Ⅱ) potential. And the existence of its bound states was discussed. The similar one dimensional square well barrier was also investigated and conditions for existence of bound states were also found. Comparing other various square potential wells with continuous ones, the general discussion for existence condition of one dimensional bound states was given and it is found that this existence condition is closely related with the shapes of one dimensional potentials.
This undergraduate thesis contains one dimensional introductory treatment of Supersymmetric Quantum Mechanics (SUSY QM). The factorisation method is used to obtain Partner Hamiltonians, and the notions of partner potentials and superpotential are introduced. It demonstrated how energy spectra and eigenfunctions of partner potentials can be correlated for the cases of unbroken and broken supersymmetry. Generalisations of these cases attributed to Sets of Hamiltonians and shape invariant potentials are introduced. The devel oped formalisation is used to obtain energy eigenfunctions and spectra of partner potentials of some well-known quantum systems, namely the Infinite Square Well and the Free Particle. In these systems, the partner potentials are trigonometric and hyperbolic functions, respectively. A comparison of these two physically distinct but mathematically related partner potentials is provided. A new philosophical interpretation of SUSY QM is introduced, with an ontological discussion of supersymmetric notions and SUSY QM. The current observation problem of SUSY is discussed, and on that basis, two opposing arguments are presented. The method of Falsification by Karl Popper is briefly introduced, and an argument is produced regarding its application to SUSY.
We review supersymmetry (SUSY) in nonrelativistic quantum mechanics emphasizing algebraic aspects.We discuss the Hamiltonian subgroup implementing supersymmetry as well as the corresponding algebra of the SUSY generators.In the SUSYQM framework, the two distinct partner potentials are connected by a superpotential satisfying a Riccati differential equation.A full Hamiltonian operator in an extended Hilbert space is defined in order to render the supersymmetry manifest.As a result, the eigenfunctions of the original potentials are connected by generalized ladder operators.We provide an explicit realization of the abstract supersymmetry group for SUSYQM depending on one real and two Grassmann parameters.
In this work we present an introduction to Supersymmetry in the context of 1-dimensional Quantum Mechanics. For that purpose we develop the concept of hamiltonians factorization using the simple harmonic oscillator as an example, we introduce the supersymmetric oscilator and, next, we generalize these concepts to introduce the fundamentals of Supersymmetric Quantum Mechanics. We also discuss useful tools to solve problems in Quantum Mechanics which are intrinsecally related to Supersymmetry as hierarchy of hamiltonians and shape invariance. We present two approximation methods which will be specially useful: the well known Variational Method and the Logarithmic Perturbation Theory, the latter being closely related to the concepts of superpotentials and hierarchy of hamiltonians. Finally, we present problems related to superpotentials which are monomials in even powers of the x coordinate multiplied by the sign function epsilon(x), which seems to be a new class of problems in Supersymmetric Quantum Mechanics.
We review the higher‐order supersymmetric quantum mechanics (H‐SUSY QM), which involves differential intertwining operators of order greater than one. The iterations of first‐order SUSY transformations are used to derive in a simple way the higher‐order case. The second order technique is addressed directly, and through this approach unexpected possibilities for designing spectra are uncovered. The formalism is applied to the harmonic oscillator: the corresponding H‐SUSY partner Hamiltonians are ruled by polynomial Heisenberg algebras which allow a straight construction of the coherent states.
Supersymmetric quantum mechanics (SUSY QM) is used to determine the class of potentials that are solvable as well as to find their isospectral partners. To do that SUSY QM assumes the knowledge of the Witten superpotential as a particular solution of the Riccati equation involved in order to obtain the corresponding generalizations. Consequently, due to the fact that in statistical mechanics the internal energy is indeed known as a single temperature derivative of the Helmholtz free energy, a simple transformation has been proposed here in order to get, in an heuristic way, a Schrödinger-like thermodynamic (SLT) equation. In this proposal, the quantum Witten superpotential and the wavefunction correspond, in statistical physics, to the internal energy and to the canonical partition function respectively. With that, it is shown how all the results from SUSY QM can be incorporated in the statistical case, to find the generalized internal energy and partition functions as well as to generalize all the other related thermodynamical properties. As an example of the usefulness of the proposed SLT equation, the case of Boltzmann statistics has been considered explicitly and it was found that the corresponding generalized internal energy contain, as a particular case, the Bose–Einstein or the Fermi–Dirac probability distributions.