Explicitly time-dependent Hamiltonians are ubiquitous in applications of quantum theory. It is therefore necessary to solve the time-dependent Schrödinger equation directly. The system’s dynamics is associated with a unitary time-evolution operator (a propagator), formally given as an infinite Dyson series. Time-dependent observables are invariant under unitary time-dependent transformations, where it is sometimes useful to transform the time-evolution from the states into the corresponding operators. This is carried out in part (in full) by transforming to the interaction (Heisenberg) picture. The corresponding equations of motion for the time-dependent operators are introduced. For quadratic potential energy functions, the time evolution of quantum expectation values coincides with the corresponding classical dynamics. This is demonstrated and analyzed in detail for Gaussian wave packets and a coherent state. Finally, we derive exact and approximate expressions for time-dependent transition probabilities and transition rates between quantum states. The validity of time-dependent perturbation theory is analyzed by comparison to exact dynamics.

The postulates of quantum mechanics associate the time-evolution of a system with its time-dependent Schrödinger equation. We start by examining different solutions to this equation for a particle, represented in terms of a Gaussian wave packet, in different scenarios: free, scattered from a potential energy barrier, or trapped in a potential energy well. In some cases, we encounter stationary solutions, in which the probability density does not change in time (a standing wave). These solutions are identified as eigenfunctions of a system Hamiltonian. The properties of the Hamiltonian as a Hermitian operator are introduced, and particularly, the fact that its proper eigenfunctions can compose an orthonormal set, and that the corresponding eigenvalues are real-valued. Learning that all operators that relate to measurables are Hermitian, and that their eigenvalues relate to the measured values, we conclude that the eigenvalues of the energy operator (Hamiltonian) are the energy levels of the quantum system.