Yes, in quantum mechanics, the wave function is not only dependent on spatial coordinates but also evolves in time. The time evolution of the wave function is described by the Schrödinger equation, which is one of the fundamental equations of quantum mechanics.
The Schrödinger equation mathematically represents how the wave function of a system changes over time. It relates the time derivative of the wave function to its spatial derivatives and the total energy of the system. The equation incorporates the principle of wave-particle duality, expressing the behavior of particles as wave-like entities.
The time evolution of the wave function allows us to predict the probability distribution of different outcomes when making measurements at different points in time. The squared magnitude of the wave function, known as the probability density, gives the probability of finding a particle in a particular state or position at a specific time.
The time evolution of the wave function is typically deterministic in quantum mechanics, meaning that given an initial wave function and the system's Hamiltonian (the operator representing the total energy), we can calculate the wave function at any future time. However, when a measurement is made, the collapse of the wave function occurs, introducing an element of indeterminacy into the outcomes of measurements.
It is worth noting that there are also relativistic quantum mechanical theories, such as quantum field theory, that describe the behavior of quantum systems in a framework consistent with special relativity. In these theories, the wave function is replaced by a quantum field that evolves in both space and time.
In summary, the wave function in quantum mechanics does propagate in time, and its time evolution is governed by the Schrödinger equation. This temporal evolution allows us to make predictions about the behavior of quantum systems and the probabilities of measurement outcomes at different points in time.