El oscilador repulsivo

Abstract

The repulsive oscillator is characterized for having a potential (-1/2)$x^2$ that acts as a barrier; it is one of the few quantum systems whose solution is explicit. It shows several interesting aspects due to the potential barrier which separates its states into those that classically surmounts the barrier for having positive energy, from those with negative energy, which partially reflect and partially transmit through it. The energy spectrum is double, since the states can move to the right or to the left. Here we analyze this system in quantum phase space through the Wigner function, with the strategy of using $2\times2$ matrices to find the time evolution, factorizing it into a geometric and a dynamical transformation.