Phase velocity and group velocity

Quantum mechanics is a special kind of statistical mechanics. We can use the relation between phase velocity and group velocity as an example. Suppose the velocity of a particle is 1 km/s. In quantum mechanics, group velocity is equal to the particle velocity, which is 1 km/s in this case. Phase velocity is half of group velocity. So phase velocity is 0.5 km/s. In quantum mechanics, energy calculation is based on phase velocity. For example, an electron can absorb or release energy from interaction with photon. In the process it will absorb or release a photon. The corresponding frequency of electron rotation and frequency of light photons are equal, if we use phase velocity of electron in calculation.

Why is phase velocity half of group velocity? The simplest answer is that it can be derived from the Schrodinger equation. But can we give an intuitive answer? 

Let's consider a free particle. In Newtonian mechanics, a free particle moves in a straight line. If the particle velocity is 1 km/s, it is 1 km/s. But in Schrodinger equation, a free particle does not move in a straight line. Try to figure out the typical path of a free particle under Schrodinger equation. It contains random movements. But what are the characteristics of the random movements?