Animations of the wave function

Free Particle

${\Large\[i\hbar \frac{\partial}{\partial t}\psi(x,t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}\psi(x,t)\]}$

Standing still.

Moving to the right.

${\Large\[i\hbar \frac{\partial}{\partial t}\psi(x,t) = -\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2}\psi(x,t)\;\; \mathrm{BCs}=0\]}$

${\Large\[i\hbar \frac{\partial}{\partial t}\psi(x,t) = \left[-\frac{\hbar^2}{2m} \frac{\partial^2}{\partial x^2} + V(x)\right]\psi(x,t)\]}$

This video highlights the probability wave.

This one gives a clearer picture of the wave function itself. Note the “feedback” between the real and imaginary parts as it evolves.