Higher-order nonlocal approach to classical and quantum mechanics through non-standard Lagrangians: The case of quantum wells and the quantum states of neutron in earth’s gravitational field

It is well-known that any dynamical system governed by a differential equation containing time derivatives higher than second order unavoidably holds unbounded energy solutions, dubbed ghosts that appear in the Hamiltonian. They correspond to instabilities displayed at the classical level. In this study, we show first that it is possible to construct in classical mechanics, characterized by non-standard Lagrangians and nonlocal-in-time kinetic energy, higher-order derivative theories that avoid the Ostrogradsky ghost. Second, we show that in the realm of quantum mechanics, higher-order discretized energies emerge in the theory which may lead to extended quantum mechanics formalism. The problem of quantum wells has been treated where we showed that negative energy and negative action, complexified energies and complexified actions may emerge in our formalism. We have also discussed the quantum motion of a neutron in the Earth’s gravitational field. It was observed that within the realm of higher-order non-standard Lagrangians, the quantum energies of the neutrons are higher than the energy levels obtained in the basic formalism.