Splitting a wave packet into parts moving in different directions
The lower part of this movie shows a Gaussian function which is at rest at the origin. This describes a particle with average position and average momentum zero.
The curve shows the absolute value of the wave function, the phase is visualized by a color. (Click here or on the color icon above to learn more about the colormap).
The initial state (at time t=0) is a real-valued (red) Gaussian function. Because of the uncertainty relation, the actual values of the position and the momentum have an uncertainty. The uncertainty of the initial position will increase in time due to the uncertainty of the momentum distribution. This explains the spreading of the wave packet. Due to the time-inversion invariance, the wave packet at time -t (in the past) is just the complex-conjugate of the wave packet at t.
The upper parts of the movie show the unique decomposition of the wave packet into parts which move in the right (resp. left) direction. The sum of these (orthogonal) parts gives the Gaussian wave packet below. This shows that even in the simplest case the state of the particle describes a superposition of two very distinct possibilities.
In the book "Visual Quantum Mechanics" you will learn how to perform this decomposition for an arbitrary wave function.