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Journal of Physics Conference Series

A Time-Dependent Anharmonic Oscillator

Alonso Contreras-Astorga

Published May 26, 2017• 15 citations
Publisher

Abstract

Supersymmetric Quantum Mechanics is commonly used to generate time independent Hamiltonians with a desired spectrum. This technique can be generalized to construct time dependent potentials. In this work, the harmonic oscillator and a coherent state are taken to perform a generalized SUSY transformation in order to obtain a time dependent anharmonic oscillator.

DOI
10.1088/1742-6596/839/1/012019

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Supersymmetric Quantum Mechanics is commonly used to generate time independent Hamiltonians with a desired spectrum. This technique can be generalized to construct time dependent potentials. In this work, the harmonic oscillator and a coherent state are taken to perform a generalized SUSY transformation in order to obtain a time dependent anharmonic oscillator.
The harmonic oscillator is a quantum mechanical system that represents one of the most basic potentials. In order to understand the behavior of a particle within this system, the time-independent Schrödinger equation was solved; in other words, its eigenfunctions and eigenvalues were found. The first goal of this study was to construct a family of single parameter potentials and corresponding eigenfunctions with a spectrum similar to that of the harmonic oscillator. This task was achieved by means of supersymmetric quantum mechanics, which utilizes an intertwining operator that relates a known Hamiltonian with another whose potential is to be built. Secondly, a generalization of the technique was used to work with the time-dependent Schrödinger equation to construct new potentials and corresponding solutions.
Starting with a time-independent Hamiltonian h and an appropriately chosen solution of the von Neumann equation $i\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}(t)=[h,\ensuremath{\rho}(t)]$ we construct its binary-Darboux partner h1(t){h}_{1}(t)h1​(t) and an exact scattering solution of $i{\stackrel{\ifmmode \dot{}\else \.{}\fi{}}{\ensuremath{\rho}}}_{1}(t)=[{h}_{1}(t),{\ensuremath{\rho}}_{1}(t)],$ where h1(t){h}_{1}(t)h1​(t) is time dependent and not isospectral to h. The method is analogous to supersymmetric quantum mechanics but is based on a different version of a Darboux transformation. We illustrate the technique by the example where h corresponds to a one-dimensional harmonic oscillator. The resulting h1(t){h}_{1}(t)h1​(t) represents a scattering of a solitonlike pulse on a three-level system.
A mixed supersymmetric-algebraic approach to construction of the minimum uncertainty coherent states of anharmonic oscillators is presented. It permits generating not only the well-known coherent states of the harmonic and Morse oscillators but also the so far unknown coherent states of the Wei Hua, Kratzer-Fues and generalized Morse and Kratzer-Fues oscillators. The method can be applied also to generate superpotentials indispensable for deriving the Schrödinger equation in the supersymmetric form amenable to direct solution in the SUSYQM scheme.
A mixed supersymmetric–algebraic approach to the construction of the minimum uncertainty coherent states of anharmonic oscillators is presented. It permits generating not only the well-known coherent states of the harmonic and Morse oscillators but also the so far unknown coherent states of the Wei Hua and generalized Morse and Kratzer–Fues oscillators. The method can be applied to generate superpotentials indispensable for deriving the Schrödinger equation in the supersymmetric form amenable to direct solution in the SUSYQM scheme.
System-specific coherent states are constructed based on the formulation of supersymmetric quantum mechanics for arbitrary quantum systems. By regarding the superpotential as a generalized displacement variable, we identity the ground state of a quantum system as the minimizer of the supersymmetric Heisenberg uncertainty product. A special case is the ground state of the standard harmonic oscillator. One constructs standard coherent states by applying a shift operator to a 'fiducial function', taken as the ground state Gaussian. By analogy, we use the ground state for any other system as a new fiducial function, generating from its shifts new dynamically-adapted, overcomplete coherent states. The discretized system-specific coherent states can serve as a dynamically-adapted basis for bound state calculations. Accurate computational results for the Morse potential, the double well potential and the two-dimensional anharmonic oscillator systems demonstrate that the system-specific coherent states can provide rapidly-converging approximations for excited state energies and wave functions.