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Physica Scripta

Supersymmetric quantum mechanics and statistical physics: Schrödinger-like thermodynamic equation

J. Morales, José Juan Peña

Published Jun 22, 2006• 5 citations

Abstract

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.

DOI: 10.1088/0031-8949/74/1/010

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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.

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.

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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.

Abstract Chapter 6 introduces quantum-mechanical ensemble theory by proving the asymptotic equivalence of the quantum-mechanical, microcanonical ensemble average with the quantum grand canonical ensemble average for many-particle systems, based on the method of Darwin and Fowler. The procedures involved identify the grand partition function, entropy and other statistical thermodynamic variables, including the grand potential, Helmholtz free energy, thermodynamic potential, Gibbs free energy, Enthalpy and their relations in accordance with the fundamental laws of thermodynamics. Accompanying saddle-point integrations define temperature (inverse thermal energy) and chemical potential (Fermi energy). The concomitant emergence of quantum statistical mechanics and Bose–Einstein and Fermi–Dirac distribution functions are discussed in detail (including Bose condensation). The magnetic moment is derived from the Helmholtz free energy and is expressed in terms of a one-particle retarded Green’s function with an imaginary time argument related to inverse thermal energy. This is employed in a discussion of diamagnetism and the de Haas-van Alphen effect.

Wigner–Dunkl quantum mechanics can be viewed as deformed quantum mechanics, where the commutator between canonical momentum and position operators contains in addition a reflection operator. In the present work we carry over the general concepts of supersymmetric (SUSY) quantum mechanics to the Wigner–Dunkl quantum formulation. We investigate the SUSY ground states, the SUSY transformations and introduce a generalized shape invariance allowing for explicit solutions. In this way, exact solutions for a certain class of SUSY potentials can be provided. As example the harmonic oscillator is considered. We also take a look into a singular oscillator model, which does not belong to the shape-invariance class. Nevertheless, exact solutions can be provided, showing the peculiarities of that model in the SUSY Wigner–Dunkl quantum mechanics. A modification of that singular model is presented which restores shape invariance

We present here a supersymmetric (SUSY) approach for determining excitation energies within the context of a quantum Monte Carlo scheme. By using the fact that SUSY quantum mechanics gives rises to a series of isospectral Hamiltonians, we show that Monte Carlo ground-state calculations in the SUSY partners can be used to reconstruct accurately both the spectrum and states of an arbitrary Schrodinger equation. Since the ground state of each partner potential is nodeless, we avoid any "node" problem typically associated with the Monte Carlo technique. Although we provide an example of using this approach to determine the tunneling states in a double-well potential, the method is applicable to any 1D potential problem. We conclude by discussing the extension to higher dimensions.