# Property of the time evolution operator preserving the norm of the wavefunction

Since the time evolution unitary operator preserves norm, if applied to any system say electron whizzing around its orbitals, no matter what time we consider would it always have the same probability amplitude and distribution?

We can now consider how this probability distribution, associated to the squared modulus of the wavefunction, is evolving in time. As you correctly stated, the time evolution operator $$U(t) = e^{-i\frac{\mathcal{H}}{\hbar} t}$$ where $$i$$ is the imaginary unit, $$\mathcal{H}$$ is the hamiltonian of the system, $$\hbar$$ is the reduced Planck constant and $$t$$ the time, is a unitary operator. This means that it will preserve the norm of the wavefunction that is, actually, the squared modulus of the whole wavefunction. This has an important consequence: for any subsequent time instant, the integral of the squared modulus of the wavefunction over the whole space is one. From a physical point of view, this means that we are again sure to find the particle in the considered space also after the application of the time evolution operator, allowing to make use, again, of the same interpretation, giving a physical meaning to the quantity we obtain. It is important to notice that this does not mean that the wavefunction itself (and the associated distribution) has not changed during time. After a certain time interval $$t$$, the wavefunction associated can be different, the wavepacket associated to a given particle can have spread and its center can have moved from point $$A$$ to point $$B$$, but we know that integrating the squared modulus of the wavefunction over a certain region of space we can obtain again the probability of finding the particle in that region.