Time Evolution (2 min)

Consider a wave function $\psi(x)$ of the infinite potential well on the interval $[0,L]$. Consider the case when the wave function at time $t=0$ is one of the eigenstates of energy $E_n$, i.e. $\Psi(x,t=0)= \psi_n(x)$ and check that the time evolution of a wave function that is an energy eigenstate is just given by multiplication with the time-dependent phase factor $e^{-iE_nt/\hbar}$, that is

$$\Psi(x,t=0)= \psi_n(x) \leadsto \Psi(x,t)= \psi_n(x)e^{-iE_nt/\hbar}.$$ (10)