Constructing Dynamical Symmetries

Dynamical symmetries, operators that do not quite commute with the Hamiltonian, extend the role of ordinary symmetries. They also provide an interesting insight on constants of the motion. Motivated by progress in quantum technologies we discuss a practical algebraic approach to computing such time-dependent operators. Explicitly we expand them as a linear combination with time dependent coefficients of time-dependent Schrödinger operators. We propose possible applications in determining quantum mechanical distributions of maximal entropy and to the dynamics of systems of coherently coupled coherent two state systems. We suggest that this generates an Ising-like Hamiltonian where each ’spin’ is a state and therefore of relevance to quantum computing based on qubit architecture.