Quantum dynamics of wave packets in a Morse potential: A dynamical system approach

We show how a dynamical systems approach can, somewhat unexpectedly, be relevant in the quantum dynamics featuring oscillations and escape in the Morse potential. We compare the dynamics resulting from the approach with the results obtained from a direct numerical integration of the relevant Schrödinger equation to support our claim. An interesting finding of the numerical investigation is the marked increase in the probability of obtaining a significant fraction (more than $50\%$) of the wave packet in the classically forbidden range beyond a critical energy of the packet. The fact that the dynamical systems approach shows an instability near that critical energy is a definite indication of the relevance of dynamical systems to the quantum dynamics. At lower energies, the calculated mean position $\langle x\rangle$ and variance $V$ from the dynamical system allow us to clearly establish the phenomenon of tunneling since the sum $\langle x\rangle +\sqrt{V}$ clearly exceeds, at various times, the classical bound on displacement for the corresponding energy.