A wave packet is a certain set of waves having different frequencies that describe a formation possessing wave properties, generally limited in time and space. In quantum mechanics, a particle description in the form of wave packets (a set of solitons) is used.

In this section, wave propagation (the initial state of which is given by the Gaussian profile in the previous section) through the barrier is simulated. In other words, here we can analyze the temporal evolution of the wave packet in the barrier system.

In quantum mechanics, the expected value is the probabilistic expected value of the result (measurement) of an experiment. It can be considered as an average of all possible measurement results, weighted by their probability.
Consider an operator \( \hat{A} \). The expectation value is then \( \langle A \rangle = \langle \psi |\hat{A}| \psi \rangle \) in Dirac notation with \( | \psi \rangle \) a normalized state vector.