Linearised coherent states for non-rational SUSY extensions of the harmonic oscillator

Abstract

In this work, we derive two equivalent non-rational extensions of the quantum harmonic oscillator using two different supersymmetric transformations. For these extensions, we built ladder operators as the product of the intertwining operators related with these equivalent supersymmetric transformations, which results in two-step ladder operators. We linearised these operators to obtain operators of the same nature that follow a linear commutation relation. After the linearisation, we derive coherent states as eigenstates of the annigilation operator and analyse some relevant mathematical and physical properties, such as the completeness relation, mean-energy values, temporal stability, time evolution of the probability densities, and Wigner distributions. From these properties, we conclude that these coherent states present both classical and quantum behaviour.

Keywords

• supersymmetric quantum mechanics
• non-rational extensions
• linearised ladder operators
• coherent states