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Journal of Physics G Nuclear and Particle Physics

Linearized collective Schrödinger equation and supersymmetry

Jürgen Schmiegel, Martin Greiner, G. Soff

Published Jan 1, 1999• 0 citations
Publisher

Abstract

The Hamiltonian obtained from the linearization of the free collective Schr?dinger equation can be understood as a supercharge. The inclusion of a scalar potential shows that the linearization procedure is equivalent to the factorization method applied in supersymmetric quantum mechanics. Finally a supersymmetric Hamiltonian is constructed which includes the collective Schr?dinger equation as well as the linearized collective Schr?dinger equation.

DOI
10.1088/0954-3899/25/5/103

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The Hamiltonian obtained from the linearization of the free collective Schr?dinger equation can be understood as a supercharge. The inclusion of a scalar potential shows that the linearization procedure is equivalent to the factorization method applied in supersymmetric quantum mechanics. Finally a supersymmetric Hamiltonian is constructed which includes the collective Schr?dinger equation as well as the linearized collective Schr?dinger equation.
Construction of a few Quantum mechanical Hamiltonians via Leˊ\acute{e}eˊvi-Leblond type Linearization:Spinor states and Supersymmetry
A number of new Leˊ\acute{e}eˊvi-Leblond type equations admitting four component spinor solutions have been proposed. The pair of linearized equations thus obtained in each case lead to Hamiltonians with characteristic features like L-S coupling and supersymmetry. The relevant momentum operators have often been understood in terms of Clifford algebraic bases producing Schro¨\ddot{o}o¨dinger Hamiltonians with L-S coupling. As for example Hamiltonians representing Rashba effect or three dimensional harmonic oscillator have been constructed. The supersymmetric nature of one dimensional oscillator has also been appreciated.
Usually, in supersymmetric theories, it is assumed that the time evolution of states is determined by the Hamiltonian, through the Schrödinger equation. Here we explore the superevolution of states in superspace, in which the supercharges are the principal operators. The superevolution equation is consistent with the Schrödinger equation, but it avoids the usual degeneracy between bosonic and fermionic states. We discuss superevolution in supersymmetric quantum mechanics and in a simple supersymmetric field theory.
The Dirac equation in one dimension with a Lorentz scalar potential is associated with a supersymmetric pair of Schr\"odinger Hamiltonians H1{\mathit{H}}_{1}H1​ and H2{\mathit{H}}_{2}H2​. The H1{\mathit{H}}_{1}H1​ and H2{\mathit{H}}_{2}H2​ share the same energy spectrum and scattering phases. The shared spectrum includes the lowest states unless the Dirac equation allows a zero mode (a zero-energy bound state). This situation is unlike the common examples of supersymmetric quantum mechanics. The Dirac equation admits a zero mode only if the scalar potential has certain ``topology.'' Various such features are illustrated through explicit examples. In particular, the phase equivalence between H1{\mathit{H}}_{1}H1​ and H2{\mathit{H}}_{2}H2​ is exploited to construct transparent potentials for the Dirac equation in one dimension.